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A372299 Primitive infinitary abundant numbers (definition 2): infinitary abundant numbers (A129656) having no proper infinitary divisors that are infinitary abundant numbers. +0
0
24, 30, 40, 42, 54, 56, 66, 70, 72, 78, 88, 96, 102, 104, 114, 138, 150, 174, 186, 222, 246, 258, 282, 294, 318, 354, 366, 402, 420, 426, 438, 474, 486, 498, 534, 540, 582, 606, 618, 642, 654, 660, 678, 726, 756, 762, 780, 786, 822, 834, 894, 906, 924, 942, 945, 960, 978, 990 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
EXAMPLE
24 is a term since it is an infinitary abundant number and none of its proper infinitary divisors, {1, 2, 3, 4, 6, 8, 12}, are infinitary abundant numbers.
The least infinitary abundant number that is not primitive is 120. It has 3 infinitary divisors, 24, 30, and 40, that are also infinitary abundant numbers.
MATHEMATICA
f[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]];
isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; iabQ[n_] := isigma[n] > 2*n; idivs[1] = {1};
idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, e_Integer} :> p^Select[Range[0, e], BitOr[e, #] == e &])];
q[n_] := Module[{d = idivs[n]}, Total[d] > 2*n && AllTrue[Most[d], !iabQ[#] &]]; Select[Range[1000], q]
PROG
(PARI) isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
idivs(n) = {my(f = factor(n), d = divisors(f), idiv = []); for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
isigma(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1)))} ;
is(n) = isigma(n) > 2*n && select(x -> x < n && isigma(x) > 2*x, idivs(n)) == [];
CROSSREFS
Subsequence of A129656.
A372298 is a subsequence.
Cf. A077609, A129657.
Similar sequences: A091191, A302574, A339940.
KEYWORD
nonn,easy,new
AUTHOR
Amiram Eldar, Apr 25 2024
STATUS
approved
A372298 Primitive infinitary abundant numbers (definition 1): infinitary abundant numbers (A129656) whose all proper infinitary divisors are infinitary deficient numbers. +0
0
40, 56, 70, 72, 88, 104, 756, 924, 945, 1092, 1188, 1344, 1386, 1428, 1430, 1596, 1638, 1760, 1870, 2002, 2016, 2080, 2090, 2142, 2176, 2210, 2394, 2432, 2470, 2530, 2584, 2720, 2750, 2944, 2990, 3040, 3128, 3190, 3200, 3230, 3250, 3400, 3410, 3496, 3712, 3770 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
EXAMPLE
40 is a term since it is an infinitary abundant number and all its proper infinitary divisors, {1, 2, 4, 5, 8, 10, 20}, are infinitary deficient numbers.
24 and 30, which are infinitary abundant numbers, are not primitive, because their are divisible by 6 which is an infinitary perfect number.
MATHEMATICA
f[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]];
isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; idefQ[n_] := isigma[n] < 2*n; idivs[1] = {1};
idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, e_Integer} :> p^Select[Range[0, e], BitOr[e, #] == e &])];
q[n_] := Module[{d = idivs[n]}, Total[d] > 2*n && AllTrue[Most[d], idefQ]]; Select[Range[4000], q]
PROG
(PARI) isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
idivs(n) = {my(f = factor(n), d = divisors(f), idiv = []); for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
isigma(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1)))} ;
is(n) = isigma(n) > 2*n && select(x -> x < n && isigma(x) >= 2*x, idivs(n)) == [];
CROSSREFS
Subsequence of A129656 and A372299.
A372300 is a subsequence.
Cf. A007357, A077609, A129657.
Similar sequences: A071395, A298973, A302573, A307112, A307114, A307115.
KEYWORD
nonn,easy,new
AUTHOR
Amiram Eldar, Apr 25 2024
STATUS
approved
A372300 Numbers k such that k and k+1 are both primitive infinitary abundant numbers (definition 1, A372298). +0
0
812889, 3181815, 20787584, 181480695, 183872535, 307510664, 337206344, 350158808, 523403264, 744074624, 868421504, 1063361144, 1955365125, 2076191864, 2578966215, 3672231255, 4185590408, 5032685384, 7158001304, 8348108535, 10784978295, 16264812135, 20917209495, 24514454055 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The corresponding sequence with definition 2 (A372299) coincides with this sequence for the first 24 terms.
LINKS
Table of n, a(n) for n=1..24.
PROG
(PARI) isidiv(d, f) = {my(bne, bde); if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
idivs(n) = {my(f = factor(n), d = divisors(f), idiv = []); for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
isigma(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1)))} ;
isab(n) = isigma(n) > 2*n;
isprim(n) = select(x -> x<n && isigma(x) >= 2*x, idivs(n)) == [];
lista(kmax) = {my(is1 = 0, is2); for(k = 2, kmax, is2 = isab(k); if(is1 && is2, if(isprim(k-1) && isprim(k), print1(k-1, ", "))); is1 = is2); }
CROSSREFS
Subsequence of A129656, A327635 and A372298.
Cf. A372299.
Similar sequences: A283418, A330872, A361935.
KEYWORD
nonn,new
AUTHOR
Amiram Eldar, Apr 25 2024
STATUS
approved
A372311 Triangle read by rows: T(n, k) = n^k * Sum_{j=0..n} binomial(n - j, n - k) * Eulerian1(n, j). +0
0
1, 1, 1, 1, 6, 8, 1, 21, 108, 162, 1, 60, 800, 3840, 6144, 1, 155, 4500, 48750, 225000, 375000, 1, 378, 21672, 453600, 4354560, 19595520, 33592320, 1, 889, 94668, 3500658, 60505200, 536479440, 2371803840, 4150656720 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
Table of n, a(n) for n=0..35.
EXAMPLE
Triangle begins:
[0] 1;
[1] 1, 1;
[2] 1, 6, 8;
[3] 1, 21, 108, 162;
[4] 1, 60, 800, 3840, 6144;
[5] 1, 155, 4500, 48750, 225000, 375000;
[6] 1, 378, 21672, 453600, 4354560, 19595520, 33592320;
[7] 1, 889, 94668, 3500658, 60505200, 536479440, 2371803840, 4150656720;
MAPLE
S := (n, k) -> local j; add(eulerian1(n, j)*binomial(n-j, n-k), j = 0..n):
row := n -> local k; seq(S(n, k) * n^k, k = 0..n):
seq(row(n), n = 0..8);
PROG
(SageMath)
def A372311_row(n) :
x = polygen(ZZ, 'x')
A = []
for m in range(0, n + 1, 1) :
A.append((-x)^m)
for j in range(m, 0, -1):
A[j - 1] = j * (A[j - 1] - A[j])
return [n^k*c for k, c in enumerate(A[0])]
for n in (0..7) : print(A372311_row(n))
CROSSREFS
Cf. A061711 (main diagonal), A066524 (column 1), A372312 (row sums).
Cf. A163626, A173018 (eulerian1).
KEYWORD
nonn,tabl,new
AUTHOR
Peter Luschny, Apr 26 2024
STATUS
approved
A372312 Row sums of A372311. +0
0
1, 2, 15, 292, 10845, 653406, 58018051, 7123041416, 1155276253305, 239189245299010, 61550396579410431, 19268616527909790636, 7210621330821550184725, 3178541959877575827583334, 1630110354806890680999093435, 962286069560027427207269245456 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..15.
MAPLE
S := (n, k) -> local j; n^k*add(eulerian1(n, j)*binomial(n-j, n-k), j = 0..n):
seq(add(S(n, k), k = 0..n), n = 0..15);
CROSSREFS
Cf. A372311, A369435.
KEYWORD
nonn,new
AUTHOR
Peter Luschny, Apr 26 2024
STATUS
approved
A372310 Number of permutations of length n avoiding the pattern 1324 and with 1 appearing before n. +0
0
1, 3, 11, 45, 198, 919, 4446, 22239, 114347, 601722, 3229614, 17632437, 97707195, 548538588, 3115293151, 17875151109, 103511938302, 604392787819, 3555410248782, 21057224371290, 125484804821226, 752020468811244, 4530163818778839, 27419805899781843, 166694596163875206 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,2
COMMENTS
This sequence counts the number of permutations of size n written in one-line notation that avoid the pattern 1324 and have the 1 appearing before the n.
LINKS
Table of n, a(n) for n=2..26.
Juan B. Gil, Oscar A. Lopez, Michael D. Weiner, A positional statistic for 1324-avoiding permutations, arXiv:2311.18227 [math.CO], 2024.
FORMULA
G.f.: A(x) = (x*(B(x)-2))/(3-B(x)), where B(x) is the g.f. for A000139. (See arxiv paper by Gil, Lopez, Weiner).
G.f. satisfies 0 = x^4*(8*x-1)+x^2*(9*x-1)*(4*x-1)*A(x)+x*(6*x-1)*(9*x-2)*A(x)^2+(27*x^2-9*x+1)*A(x)^3.
EXAMPLE
For n=4, a(4)=11 is counting the permutations (in one-line notation): 1234, 1243, 1342, 1423, 1432, 2134, 2143, 2314, 3124, 3142, 3214.
MAPLE
f:= proc(n) f(n):= 2*(3*n)!/((2*n+1)!*(n+1)!) end:
a:= proc(n) option remember; `if`(n=1, 1,
add(a(n-i)*f(i), i=1..n))
end:
seq(a(n), n=2..26); # Alois P. Heinz, Apr 26 2024
CROSSREFS
Invert of shifted A000139.
KEYWORD
nonn,new
AUTHOR
Michael D. Weiner, Apr 26 2024
STATUS
approved
A372195 Number of labeled simple graphs covering n vertices with a unique undirected cycle of length > 2. +0
0
0, 0, 0, 1, 15, 232, 3945 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
An undirected cycle in a graph is a sequence of distinct vertices, up to rotation and reversal, such that there are edges between all consecutive elements, including the last and the first.
LINKS
Table of n, a(n) for n=0..6.
EXAMPLE
The a(4) = 15 graphs:
12,13,14,23
12,13,14,24
12,13,14,34
12,13,23,24
12,13,23,34
12,13,24,34
12,14,23,24
12,14,23,34
12,14,24,34
12,23,24,34
13,14,23,24
13,14,23,34
13,14,24,34
13,23,24,34
14,23,24,34
MATHEMATICA
cyc[y_]:=Select[Join@@Table[Select[Join@@Permutations/@Subsets[Union@@y, {k}], And@@Table[MemberQ[Sort/@y, Sort[{#[[i]], #[[If[i==k, 1, i+1]]]}]], {i, k}]&], {k, 3, Length[y]}], Min@@#==First[#]&];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Union@@#==Range[n]&&Length[cyc[#]]==2&]], {n, 0, 5}]
CROSSREFS
For no cycles we have A105784 (for triangles A372168, non-covering A213434), unlabeled A144958 (for triangles A372169).
Counting triangles instead of cycles gives A372171 (non-covering A372172), unlabeled A372174 (non-covering A372194).
The unlabeled version is A372191, non-covering A372192.
The non-covering version is A372193, column k = 1 of A372176.
A000088 counts unlabeled graphs, labeled A006125.
A001858 counts acyclic graphs, unlabeled A005195.
A002807 counts cycles in a complete graph.
A006129 counts labeled graphs, unlabeled A002494.
A322661 counts covering loop-graphs, unlabeled A322700.
A372167 counts covering graphs by triangles (non-covering A372170), unlabeled A372173 (non-covering A263340).
Cf. A000169, A000272, A053530, A054548, A137916, A137917, A367863, A368597.
KEYWORD
nonn,more,new
AUTHOR
Gus Wiseman, Apr 25 2024
STATUS
approved
A372193 Number of labeled simple graphs on n vertices with a unique cycle of length > 2. +0
0
0, 0, 0, 1, 19, 317 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
An undirected cycle in a graph is a sequence of distinct vertices, up to rotation and reversal, such that there are edges between all consecutive elements, including the last and the first.
LINKS
Table of n, a(n) for n=0..5.
EXAMPLE
The a(4) = 19 graphs:
12,13,23
12,14,24
13,14,34
23,24,34
12,13,14,23
12,13,14,24
12,13,14,34
12,13,23,24
12,13,23,34
12,13,24,34
12,14,23,24
12,14,23,34
12,14,24,34
12,23,24,34
13,14,23,24
13,14,23,34
13,14,24,34
13,23,24,34
14,23,24,34
MATHEMATICA
cyc[y_]:=Select[Join@@Table[Select[Join@@Permutations /@ Subsets[Union@@y, {k}], And @@ Table[MemberQ[Sort/@y, Sort[{#[[i]], #[[If[i==k, 1, i+1]]]}]], {i, k}]&], {k, 3, Length[y]}], Min@@#==First[#]&];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Length[cyc[#]]==2&]], {n, 0, 5}]
CROSSREFS
For no cycles we have A001858 (covering A105784), unlabeled A005195 (covering A144958).
Counting triangles instead of cycles gives A372172 (non-covering A372171), unlabeled A372194 (non-covering A372174).
The unlabeled version is A372192, non-covering A372191.
The covering case is A372195, column k = 1 of A372175.
A000088 counts unlabeled graphs, labeled A006125.
A002807 counts cycles in a complete graph.
A006129 counts labeled graphs, unlabeled A002494.
A372167 counts graphs by triangles, non-covering A372170.
A372173 counts unlabeled graphs by triangles, non-covering A263340.
Cf. A000272, A054548, A121251, A137916, A213434, A322661, A372169, A372176.
KEYWORD
nonn,more,new
AUTHOR
Gus Wiseman, Apr 25 2024
STATUS
approved
A372176 Irregular triangle read by rows where T(n,k) is the number of labeled simple graphs on n vertices with exactly 2k directed cycles of length > 2. +0
0
1, 1, 2, 7, 1, 38, 19, 0, 6, 0, 0, 0, 1, 291, 317, 15, 220, 0, 0, 70, 55, 0, 0, 0, 0, 30, 15, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
A directed cycle in a simple (undirected) graph is a sequence of distinct vertices, up to rotation, such that there are edges between all consecutive elements, including the last and the first.
LINKS
Table of n, a(n) for n=0..50.
EXAMPLE
Triangle begins (zeros shown as dots):
1
1
2
7 1
38 19 . 6 ... 1
291 317 15 220 .. 70 55 .... 30 15 ........ 10 ............... 1
The T(4,3) = 6 graphs:
12,13,14,23,24
12,13,14,23,34
12,13,14,24,34
12,13,23,24,34
12,14,23,24,34
13,14,23,24,34
MATHEMATICA
cyc[y_]:=Select[Join@@Table[Select[Join@@Permutations/@Subsets[Union@@y, {k}], And@@Table[MemberQ[Sort/@y, Sort[{#[[i]], #[[If[i==k, 1, i+1]]]}]], {i, k}]&], {k, 3, Length[y]}], Min@@#==First[#]&];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Length[cyc[#]]==2k&]], {n, 0, 4}, {k, 0, Length[cyc[Subsets[Range[n], {2}]]]/2}]
CROSSREFS
Column k = 0 is A001858 (unlabeled A005195), covering A105784.
Row lengths are A002807 + 1.
Row sums are A006125, unlabeled A000088.
Counting edges instead of cycles gives A084546 (covering A054548), unlabeled A008406 (covering A370167).
Counting triangles instead of cycles gives A372170 (covering A372167), unlabeled A263340 (covering A372173).
The covering case is A372175.
Column k = 1 is A372193 (covering A372195), unlabeled A372192.
A006129 counts graphs, unlabeled A002494.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A000272, A053530, A137916, A144958, A213434, A367863, A372168, A372169, A372171, A372172, A372191.
KEYWORD
nonn,tabf,more,new
AUTHOR
Gus Wiseman, Apr 25 2024
STATUS
approved
A372175 Irregular triangle read by rows where T(n,k) is the number of labeled simple graphs covering n vertices with exactly 2k directed cycles of length > 2. +0
0
1, 0, 1, 3, 1, 19, 15, 0, 6, 0, 0, 0, 1, 155, 232, 15, 190, 0, 0, 70, 50, 0, 0, 0, 0, 30, 15, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
A directed cycle in a simple (undirected) graph is a sequence of distinct vertices, up to rotation, such that there are edges between all consecutive elements, including the last and the first.
LINKS
Table of n, a(n) for n=0..50.
EXAMPLE
Triangle begins (zeros shown as dots):
1
.
1
3 1
19 15 . 6 ... 1
155 232 15 190 .. 70 50 .... 30 15 .......... 10 .............. 1
Row n = 4 counts the following graphs:
12,34 12,13,14,23 . 12,13,14,23,24 . . . 12,13,14,23,24,34
13,24 12,13,14,24 12,13,14,23,34
14,23 12,13,14,34 12,13,14,24,34
12,13,14 12,13,23,24 12,13,23,24,34
12,13,24 12,13,23,34 12,14,23,24,34
12,13,34 12,13,24,34 13,14,23,24,34
12,14,23 12,14,23,24
12,14,34 12,14,23,34
12,23,24 12,14,24,34
12,23,34 12,23,24,34
12,24,34 13,14,23,24
13,14,23 13,14,23,34
13,14,24 13,14,24,34
13,23,24 13,23,24,34
13,23,34 14,23,24,34
13,24,34
14,23,24
14,23,34
14,24,34
MATHEMATICA
cycles[g_]:=Join@@Table[Select[Join@@Permutations /@ Subsets[Union@@g, {k}], Min@@#==First[#]&&And@@Table[MemberQ[Sort/@g, Sort[{#[[i]], #[[If[i==k, 1, i+1]]]}]], {i, k}]&], {k, 3, Length[g]}];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Union@@#==Range[n]&&Length[cycles[#]]==2k&]], {n, 0, 5}, {k, 0, Length[cycles[Subsets[Range[n], {2}]]]/2}]
CROSSREFS
Row lengths are A002807 + 1.
Row sums are A006129, unlabeled A002494.
Column k = 0 is A105784 (for triangles A372168, non-covering A213434), unlabeled A144958 (for triangles A372169).
Counting triangles instead of cycles gives A372167 (non-covering A372170), unlabeled A372173 (non-covering A263340).
The non-covering version is A372176.
Column k = 1 is A372195 (non-covering A372193, for triangles A372171), unlabeled A372191 (non-covering A372192, for triangles A372174).
A000088 counts unlabeled graphs, labeled A006125.
A001858 counts acyclic graphs, unlabeled A005195.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A000272, A053530, A121251, A137916, A137917, A372172, A372194.
KEYWORD
nonn,more,tabf,new
AUTHOR
Gus Wiseman, Apr 24 2024
STATUS
approved
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