A324371 - OEIS

%I #23 Jul 04 2022 01:32:00

%S 1,2,3,2,5,3,7,2,3,5,11,3,13,7,5,2,17,3,19,5,7,11,23,1,5,13,3,7,29,15,

%T 31,2,11,17,35,3,37,19,13,5,41,7,43,11,1,23,47,1,7,5,17,13,53,3,55,7,

%U 19,29,59,5,61,31,7,2,13,11,67,17,23,7,71,1,73,37,5,19,77,13,79,5,3,41,83,21

%N Product of all primes p dividing n such that the sum of the base p digits of n is less than p, or 1 if no such prime.

%C Does not contain any elements of A324315, and thus none of the Carmichael numbers A002997.

%C See the section on Bernoulli polynomials in Kellner and Sondow 2019.

%H Robert Israel, <a href="/A324371/b324371.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) * A324369(n) = A007947(n) = radical(n).

%F a(n) * A195441(n) = a(n) * A324369(n) * A324370(n) = A144845(n-1) = denominator(Bernoulli_{n-1}(x)).

%e For p = 2 and 3, the sum of the base p digits of 6 is 1+1+0 = 2 >= 2 and 2+0 = 2 < 3, respectively, so a(6) = 3.

%p f:= n -> convert(select(p -> convert(convert(n,base,p),`+`)<p,

%p numtheory:-factorset(n)),`*`):map(f, [$1..100]); # _Robert Israel_, Apr 26 2020

%t SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];

%t LP[n_] := Transpose[FactorInteger[n]][[1]];

%t DD3[n_] := Times @@ Select[LP[n], SD[n, #] < # &];

%t Table[DD3[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, 85)]) # _Michael S. Branicky_, Jul 03 2022

%Y Cf. A002997, A007947, A144845, A195441, A324315, A324316, A324317, A324318, A324319, A324320, A324369, A324370, A324404, A324405.

%K nonn,base,look

%O 1,2

%A _Bernd C. Kellner_ and _Jonathan Sondow_, Feb 25 2019