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

1,6

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.

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.

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)).

Example

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.

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}]

Prog

(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

Crossrefs

Cf. A007947, A144845, A195441, A324315, A324316, A324317, A324318, A324319, A324320, A324370, A324371, A324404, A324405.

Sequence in context: A206824 A293810 A356553 * A276781 A303759 A330754

Adjacent sequences: A324366 A324367 A324368 * A324370 A324371 A324372

Keyword

nonn,base,look

Author

Bernd C. Kellner and Jonathan Sondow, Feb 24 2019

Status

approved