A286762 - OEIS

%I #41 Sep 10 2024 08:16:01

%S 0,21,22,45,46,57,70,94,105,118,142,147,165,171,177,187,190,214,221,

%T 222,225,237,238,261,267,281,286,291,313,315,318,334,345,350,357,358,

%U 381,382,387,403,430,437,441,448,465,477,478,501,507,538,555,558,561,565

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

%C 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.

%C 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

%H Bernd C. Kellner, <a href="https://doi.org/10.1016/j.jnt.2017.03.020">On a product of certain primes</a>, J. Number Theory 179 (2017), 126-141; arXiv:<a href="https://arxiv.org/abs/1705.04303">1705.04303</a> [math.NT], 2017.

%H Bernd C. Kellner and Jonathan Sondow, <a href="https://doi.org/10.4169/amer.math.monthly.124.8.695">Power-Sum Denominators</a>, Amer. Math. Monthly 124 (2017), 695-709; arXiv:<a href="https://arxiv.org/abs/1705.03857">1705.03857</a> [math.NT], 2017.

%H Bernd C. Kellner and Jonathan Sondow, <a href="https://doi.org/10.5281/zenodo.10682734">The denominators of power sums of arithmetic progressions</a>, Integers 18 (2018), Article #A95, 17 pp.; arXiv:<a href="https://arxiv.org/abs/1705.05331">1705.05331</a> [math.NT], 2017.

%e 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:

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

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

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

%e 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.

%t -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 *)

%o (Julia)

%o function A286762_list(bound::Int)

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

%o     for n in 0:bound

%o         u = A195441(n)

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

%o         a = u

%o     end

%o L end

%o println(A286762_list(566))

%Y Cf. A195441, A286516, A286763.

%K nonn

%O 1,2

%A _Peter Luschny_, May 14 2017