A351578 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(f(i)) = A007814(f(j)) and A278222(f(i)) = A278222(f(j)), for all i, j >= 1, where f(k) = A109812(k).

1, 2, 3, 4, 5, 6, 7, 8, 9, 6, 7, 10, 6, 11, 12, 13, 14, 6, 15, 16, 15, 16, 17, 18, 7, 16, 19, 20, 21, 22, 18, 7, 16, 22, 6, 22, 18, 17, 23, 24, 25, 15, 16, 26, 16, 27, 28, 23, 29, 30, 23, 22, 6, 31, 16, 7, 32, 33, 15, 33, 18, 22, 18, 27, 33, 16, 22, 34, 23, 17, 25, 27, 16, 35, 36, 37, 38, 32, 28, 32, 39, 18, 40, 16

Offset

1,2

Comments

Restricted growth sequence transform of the ordered pair [A007814(A109812(n)), A046523(A005940(1+A109812(n)))].

The sequence allots a new distinct number for each newly encountered combination of the 2-adic valuation of A109812 (A351964), and the multiset of the lengths of 1-runs in the odd part of A109812 (A351965). See the examples.

For all i, j: a(i) = a(j) => A352889(i) = A352889(j).

Links

Antti Karttunen, Table of n, a(n) for n = 1..100000

Index entries for sequences related to binary expansion of n

Example

   n   A109812(n)  [base-2]   A351964(n)           Lengths of       a(n)
                              (# of trailing 0's)  1-runs       (allotted #)
-----+----------------------------------------------------------------------
   1 :          1       [1],  0                    [1]               1
   2 :          2      [10],  1                    [1]               2
   3 :          4     [100],  2                    [1]               3
   4 :          3      [11],  0                    [2]               4
   5 :          8    [1000],  3                    [1]               5
   6 :          5     [101],  0                    [1,1]             6
   7 :         10    [1010],  1                    [1,1]             7
   8 :         16   [10000],  4                    [1]               8
   9 :          6     [110],  1                    [2]               9
  10 :          9    [1001],  0                    [1,1]             6
  11 :         18   [10010],  1                    [1,1]             7
Because the combinations of the multiset of 1-runs in the binary expansion of A109812(n) and the number of trailing zeros in it (A351964) are unique for n = 1 .. 9, a unique increasing number (starting from 1) is allotted for each, and a(n) = n for n <= 9. On the other hand, at n=10, the binary expansion is [1001], for which these two measures are equal to that of binary expansion [101] found first time at n=6, therefore the rgs-transform allots for 10 the same number as for 6, and a(10) = a(6) = 6. At n=11, the binary expansion is [10010], where these two measures coincide with that of [1010] found first time at n=7, therefore a(10) = a(7) = 7.

Prog

(PARI)
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };
v109812 = readvec("b109812_to10e5.txt"); \\ Prepared from b-file data with gawk ' { print $2 } '
up_to = #v109812;
A109812(n) = v109812[n];
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A007814(n) = valuation(n, 2);
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
v351578 = rgs_transform(vector(up_to, n, [A007814(A109812(n)), A046523(A005940(1+A109812(n)))]));
A351578(n) = v351578[n];

Crossrefs

Cf. A007814, A109812, A278222 (A286622), A351963, A351964, A351965, A352888, A352889.

Sequence in context: A216196 A028901 A081597 * A328469 A373229 A347105

Adjacent sequences: A351575 A351576 A351577 * A351579 A351580 A351581

Keyword

nonn,base

Author

Antti Karttunen, Apr 07 2022

Status

approved