A324371 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.

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, 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, 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

Offset: 1

Author

Bernd C. Kellner and Jonathan Sondow, Feb 25 2019

Keywords

• nonn
• base
• look

Comments

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

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

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• 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, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.

Crossrefs

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

Programs

Maple

f:= n -> convert(select(p -> convert(convert(n,base,p),`+`)Robert Israel, Apr 26 2020

Mathematica

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
DD3[n_] := Times @@ Select[LP[n], SD[n, #] < # &];
Table[DD3[n], {n, 1, 100}]

Python

from math import prod
from sympy.ntheory import digits
from sympy import primefactors as pf
def a(n): return prod(p for p in pf(n) if sum(digits(n, p)[1:]) < p)
print([a(n) for n in range(1, 85)]) # Michael S. Branicky, Jul 03 2022

Formula

a(n) * A324369(n) = A007947(n) = radical(n).

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