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A372180
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Square array read by antidiagonals upwards in which T(n,m) is the n-th number whose symmetric representation of sigma consists of m copies of unimodal pattern 121 (separated by 0's if m > 1).
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2
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6, 12, 78, 20, 102, 1014, 24, 114, 1734, 12246, 28, 138, 2166, 12714, 171366, 40, 174, 3174, 13026, 501126, 1922622, 48, 186, 5046, 13182, 781926, 2057406, 28960854, 56, 222, 5766, 13494, 1679046, 2067546, 144825414, 300014754, 80, 246, 8214, 13962, 4243686, 2072382, 282275286, 300137214, 4174476774
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OFFSET
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1,1
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COMMENTS
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Every number in this sequence is even since the symmetric representation of sigma for an odd number q starts 101. Each number in column m of T(n,m) has 2*m odd divisors.
Since u(m) = 2 * 3 * 13^(m-1), m>=1, has 2m odd divisors and 1 < 3 < 4 < 4*3 < 13 < 3*13 < 4*13 < 3*4*13 < 13^2 < ..., the symmetric representation of sigma for u(m) consists of m copies of unimodal pattern 121. Therefore, every column in the table T(n,m), m>=1, contains infinitely many entries. Number u(m) is the smallest entry in the m-th column when m is prime.
In general: If m>1 then T(n,m) = 2^k * q, k>=1, q odd, has at least 4 odd divisors which satisfy
d_(2i+2) < 2^(k+1) * d_(2i+1) < 2^(k+1) * d_(2i+2) < d_(2i+3), i>=0,
with the odd divisors d_j of n in increasing order.
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LINKS
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FORMULA
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T(n,1) = 2^k * p with odd prime p satisfying p < 2^(k+1), see A370205.
T(n,2) = 2^k * p * q, k > 0, p and q prime, 2 < p < 2^(k+1) < 2^(k+1) * p < q, see A370206.
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EXAMPLE
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a(1) = T(1,1) = 6, its symmetric representation of sigma, SRS(6), has unimodal pattern 121 and a single unit of width 2 at the diagonal.
a(3) = T(1,2) = 78, SRS(78) has unimodal pattern 1210121;
a(10) = T(1,4) = 12246, SRS(12246) has unimodal pattern 121012101210121;
both symmetric representations of sigma have width 0 at the diagonal where two parts meets.
Each number in the m-th column has 2m odd divisors. T(1,9) = 4174476774.
-------------------------------------------------------------------------
n\m 1 2 3 4 5 6 7 8
-------------------------------------------------------------------------
1| 6 78 1014 12246 171366 1922622 28960854 300014754 ...
2| 12 102 1734 12714 501126 2057406 144825414 300137214 ...
3| 20 114 2166 13026 781926 2067546 282275286 300235182 ...
4| 24 138 3174 13182 1679046 2072382 888215334 300357642 ...
5| 28 174 5046 13494 4243686 2081742 3568939926 300431118 ...
6| 40 186 5766 13962 5541126 2091882 ... 300602562 ...
7| 48 222 8214 14118 8487372 2097966 300651546 ...
8| 56 246 10086 14898 11082252 2110134 300896466 ...
9| 80 258 10092 15054 11244966 2112162 301165878 ...
10| 88 282 11094 15366 16954566 2116218 301386306 ...
...
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MATHEMATICA
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divQ[k_, {d1_, d2_, d3_}] := d2<2^(k+1)d1&&2^(k+1)d2<d3
seqQ[t_, m_] := Module[{oP, k, dL}, oP=NestWhile[#/2&, t, EvenQ[#]&]; k=Log[2, t/oP]; dL=Divisors[oP]; Length[dL]==2m&&If[Length[dL]==2, Last[dL]<2^(k+1), AllTrue[Partition[dL, 3, 2], divQ[k, #]&]]]
a372180[{r_, s_}, m_] := Select[Range[r, s], seqQ[#, m]&] (* range r...s of numbers in column m *)
a372180[{1, 15366}, 4] (* computes column 4 in the table *)
a372180[{1, 2116218}, 6] (* computes column 6 in the table *)
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CROSSREFS
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Cf. A235791, A237048, A237270, A237271, A237591, A237593, A249223, A262045, A341969, A341971, A342592, A342594, A342595, A342596, A367377, A370209.
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KEYWORD
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AUTHOR
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STATUS
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approved
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