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 A324370 Product of all primes p not dividing n such that the sum of the base-p digits of n is at least p, or 1 if no such prime exists. 27
 1, 1, 2, 1, 6, 1, 6, 3, 10, 1, 6, 1, 210, 15, 2, 3, 30, 5, 210, 21, 110, 15, 30, 5, 546, 21, 14, 1, 30, 1, 462, 231, 1190, 105, 6, 1, 51870, 1365, 70, 21, 2310, 55, 2310, 105, 322, 105, 210, 35, 6630, 663, 286, 33, 330, 55, 798, 57, 290, 15, 30, 1, 930930, 15015, 1430, 2145, 1122, 85, 82110, 2415, 70, 3, 330, 55, 21111090, 285285 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The product is finite, as the sum of the base-p digits of n is n if p > n. a(198) = 2465 is the only term below 10^6 that is a Carmichael number (A002997). It appears that a(n)=1 if and only if n is in A094960. - Robert Israel, Mar 30 2020 It turns out that a(n) equals the denominator of the first derivative of the Bernoulli polynomial B(n,x). So a(n)=1 if and only if n is in A094960, also impyling that n+1 is prime. A324370 is also involved in such formulas regarding higher derivatives. See Kellner 2023. - Bernd C. Kellner, Oct 12 2023 LINKS Robert Israel, Table of n, a(n) for n = 1..5000 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, On the finiteness of Bernoulli polynomials whose derivative has only integral coefficients, J. Integer Seq. 27 (2024), Article 24.2.8, 11 pp.; arXiv:2310.01325 [math.NT], 2023. 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) * A324369(n) = A195441(n-1) = denominator(Bernoulli_n(x) - Bernoulli_n). a(n) * A324369(n) * A324371(n) = A144845(n-1) = denominator(Bernoulli_{n-1}(x)). a(n+1) = A195441(n)/A324369(n+1) = A144845(n)/A007947(n+1) = A318256(n). Essentially the same as A318256. - Peter Luschny, Mar 05 2019 From Bernd C. Kellner, Oct 12 2023: (Start) a(n) = denominator(Bernoulli_n(x)'). k-th derivative: let (n)_m be the falling factorial. For n > k, a(n-k+1)/gcd(a(n-k+1), (n)_{k-1}) = denominator(Bernoulli_n(x)^(k)). Otherwise, the denominator equals 1. (End) EXAMPLE For p = 2, 3, and 5, the sum of the base p digits of 7 is 1+1+1 = 3 >= 2, 2+1 = 3 >= 3, and 1+2 = 3 < 5, respectively, so a(7) = 2*3 = 6. MAPLE N:= 100: # for a(1)..a(N) V:= Vector(N, 1): p:= 1: for iter from 1 do p:= nextprime(p); if p >= N then break fi; for n from p+1 to N do if n mod p <> 0 and convert(convert(n, base, p), `+`)>= p then V[n]:= V[n]*p fi od od: convert(V, list); # Robert Israel, Mar 30 2020 # Alternatively, note that this formula is suggesting offset 0 and a(0) = 1: seq(denom(diff(bernoulli(n, x), x)), n = 1..51); # Peter Luschny, Oct 13 2023 MATHEMATICA SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]]; DD2[n_] := Times @@ Select[Prime[Range[PrimePi[(n+1)/(2+Mod[n+1, 2])]]], !Divisible[n, #] && SD[n, #] >= # &]; Table[DD2[n], {n, 1, 100}] (* From Bernd C. Kellner, Oct 12 2023 (Start) *) (* Denominator of first derivative of BP *) k = 1; Table[Denominator[Together[D[BernoulliB[n, x], {x, k}]]], {n, 1, 100}] (* End *) PROG (Python) from math import prod from sympy.ntheory import digits from sympy import primefactors, primerange def a(n): nonpf = set(primerange(1, n+1)) - set(primefactors(n)) return prod(p for p in nonpf if sum(digits(n, p)[1:]) >= p) print([a(n) for n in range(1, 75)]) # Michael S. Branicky, Jul 03 2022 CROSSREFS Cf. A002997, A094960, A144845, A195441, A324315, A324316, A324317, A324318, A324319, A324320, A324369, A324371, A324404, A324405, A366168, A366169, A366186, A366187, A366188. Sequence in context: A286515 A166120 A318256 * A324193 A364829 A264859 Adjacent sequences: A324367 A324368 A324369 * A324371 A324372 A324373 KEYWORD nonn,base,changed AUTHOR Bernd C. Kellner and Jonathan Sondow, Feb 24 2019 STATUS approved

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