%I #28 Jul 04 2022 01:32:16
%S 1,1,1,1,1,2,1,1,1,2,1,2,1,2,3,1,1,2,1,2,3,2,1,6,1,2,1,2,1,2,1,1,3,2,
%T 1,2,1,2,3,2,1,6,1,2,15,2,1,6,1,2,3,2,1,2,1,2,3,2,1,6,1,2,3,1,5,6,1,2,
%U 3,10,1,6,1,2,3,2,1,6,1,2,1,2,1,2,5,2,3,2,1,10,7,2,3,2,5,6,1
%N Product of all primes p dividing n such that the sum of the base p digits of n is at least p, or 1 if no such prime.
%C a(n) = n iff n divides denominator(Bernoulli_n(x) - Bernoulli_n) (see A195441).
%C a(n) = n iff n = 1 or n is in A324315.
%C a(n) = n if n is a Carmichael number (A002997).
%C See the section on Bernoulli polynomials in Kellner and Sondow 2019.
%H Amiram Eldar, <a href="/A324369/b324369.txt">Table of n, a(n) for n = 1..10000</a>
%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="http://math.colgate.edu/~integers/v52/v52.pdf">On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits</a>, Integers 21 (2021), #A52, 21 pp.; arXiv:<a href="https://arxiv.org/abs/1902.10672">1902.10672</a> [math.NT], 2019.
%F a(n) * A324371(n) = A007947(n) = radical(n).
%F a(n) * A324370(n) = A195441(n-1) = denominator(Bernoulli_n(x) - Bernoulli_n).
%F a(n) * A324370(n) * A324371(n) = A144845(n-1) = denominator(Bernoulli_{n-1}(x)).
%e 6 = 2 * 3, and 6 = 110_2 in base 2 with 1+1+0 >= 2, but 6 = 20_3 in base 3 with 2+0 = 2 < 3, so a(6) = 2.
%p g:= proc(n,p) convert(convert(n,base,p),`+`) >= p end proc:
%p f:= proc(n) local p;
%p convert(select(p -> g(n,p), numtheory:-factorset(n)),`*`)
%p end proc:
%p map(f, [$1..100]); # _Robert Israel_, Feb 28 2019
%t SD[n_, p_] := If[n < 2, 0, Plus @@ IntegerDigits[n, p]];
%t LP[n_] := Transpose[FactorInteger[n]][[1]];
%t DD1[n_] := Times @@ Select[LP[n], SD[n, #] >= # &];
%t Table[DD1[n], {n, 1, 100}]
%o (Python)
%o from math import prod
%o from sympy.ntheory import digits
%o from sympy import primefactors as pf
%o def a(n): return prod(p for p in pf(n) if sum(digits(n, p)[1:]) >= p)
%o print([a(n) for n in range(1, 98)]) # _Michael S. Branicky_, Jul 03 2022
%Y Cf. A007947, A144845, A195441, A324315, A324316, A324317, A324318, A324319, A324320, A324370, A324371, A324404, A324405.
%K nonn,base,look
%O 1,6
%A _Bernd C. Kellner_ and _Jonathan Sondow_, Feb 24 2019