A324369 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.

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, 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, 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

Offset: 1

Author

Bernd C. Kellner and Jonathan Sondow, Feb 24 2019

Keywords

• nonn
• base
• look

Comments

a(n) = n iff n divides denominator(Bernoulli_n(x) - Bernoulli_n) (see A195441).

a(n) = n iff n = 1 or n is in A324315.

a(n) = n if n is a Carmichael number (A002997).

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

Examples

			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.
		

Links

• Amiram Eldar, 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. A007947, A144845, A195441, A324315, A324316, A324317, A324318, A324319, A324320, A324370, A324371, A324404, A324405.

Programs

Maple

g:= proc(n,p) convert(convert(n,base,p),`+`) >= p end proc:
f:= proc(n) local p;
      convert(select(p -> g(n,p), numtheory:-factorset(n)),`*`)
end proc:
map(f, [$1..100]); # Robert Israel, Feb 28 2019

Mathematica

SD[n_, p_] := If[n < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
DD1[n_] := Times @@ Select[LP[n], SD[n, #] >= # &];
Table[DD1[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, 98)]) # Michael S. Branicky, Jul 03 2022

Formula

a(n) * A324371(n) = A007947(n) = radical(n).

a(n) * A324370(n) = A195441(n-1) = denominator(Bernoulli_n(x) - Bernoulli_n).

a(n) * A324370(n) * A324371(n) = A144845(n-1) = denominator(Bernoulli_{n-1}(x)).