A286762 - OEIS

A286762

Indices k such that A195441(k) = A195441(k+1).

0, 21, 22, 45, 46, 57, 70, 94, 105, 118, 142, 147, 165, 171, 177, 187, 190, 214, 221, 222, 225, 237, 238, 261, 267, 281, 286, 291, 313, 315, 318, 334, 345, 350, 357, 358, 381, 382, 387, 403, 430, 437, 441, 448, 465, 477, 478, 501, 507, 538, 555, 558, 561, 565

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OFFSET

1,2

COMMENTS

k is in this sequence if and only if the primes p less than or equal to (k+2)/(2+(k mod 2)) such that the sum of digits of k+1 in base p is at least p are also the primes less than or equal to (k+3)/(2+((k+1) mod 2)) such that the sum of digits of k+2 in base p is at least p.

For the comment above and the fact that the sequence is infinite, see Thm. 2 in "Power-Sum Denominators" and Cor. 3 in "The denominators of power sums of arithmetic progressions". - Bernd C. Kellner and Jonathan Sondow, May 24 2017

LINKS

Table of n, a(n) for n=1..54.

Bernd C. Kellner, On a product of certain primes, J. Number Theory 179 (2017), 126-141; arXiv:1705.04303 [math.NT], 2017.

Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017.

Bernd C. Kellner and Jonathan Sondow, The denominators of power sums of arithmetic progressions, Integers 18 (2018), Article #A95, 17 pp.; arXiv:1705.05331 [math.NT], 2017.

EXAMPLE

21 and 22 are in this sequence because {2, 3, 5} is the set of primes which meet the given constraints. Let sd(n, p) denote the sum of digits of n in base p, then we have:

2 <= sd(22, 2) = 3; 3 <= sd(22, 3) = 4; 5 <= sd(22, 5) = 6;

2 <= sd(23, 2) = 4; 3 <= sd(23, 3) = 5; 5 <= sd(23, 5) = 7;

2 <= sd(24, 2) = 2; 3 <= sd(24, 3) = 4; 5 <= sd(24, 5) = 8.

All other candidates do not satisfy the requirements: sd(22,7) = 4; sd(22,11) = 2; sd(23,7) = 5; sd(24,7) = 6; sd(24,11) = 4; sd(24,13) = 12.

MATHEMATICA

-1 + SequencePosition[Table[Denominator[Together[(BernoulliB[n + 1, x] - BernoulliB[n + 1])]], {n, 0, 600}], w_ /; And[SameQ @@ w, Length@ w == 2]][[All, 1]] (* Michael De Vlieger, Sep 22 2017, after Jonathan Sondow at A195441 *)

PROG

(Julia)

function A286762_list(bound::Int)

L = fmpz[]; a = fmpz(0)

for n in 0:bound

u = A195441(n)

a == u && push!(L, n-1)

a = u

end

L end

println(A286762_list(566))

CROSSREFS