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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.

%I A324370 #44 May 12 2024 15:04:15
%S A324370 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,
%T A324370 14,1,30,1,462,231,1190,105,6,1,51870,1365,70,21,2310,55,2310,105,322,
%U A324370 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
%N 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.
%C A324370 The product is finite, as the sum of the base-p digits of n is n if p > n.
%C A324370 a(198) = 2465 is the only term below 10^6 that is a Carmichael number (A002997).
%C A324370 It appears that a(n)=1 if and only if n is in A094960. - _Robert Israel_, Mar 30 2020
%C A324370 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
%H A324370 Robert Israel, <a href="/A324370/b324370.txt">Table of n, a(n) for n = 1..5000</a>
%H A324370 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 A324370 Bernd C. Kellner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Kellner/kell2.html">On the finiteness of Bernoulli polynomials whose derivative has only integral coefficients</a>, J. Integer Seq. 27 (2024), Article 24.2.8, 11 pp.; arXiv:<a href="https://arxiv.org/abs/2310.01325">2310.01325</a> [math.NT], 2023.
%H A324370 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 A324370 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 A324370 a(n) * A324369(n) = A195441(n-1) = denominator(Bernoulli_n(x) - Bernoulli_n).
%F A324370 a(n) * A324369(n) * A324371(n) = A144845(n-1) = denominator(Bernoulli_{n-1}(x)).
%F A324370 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
%F A324370 From _Bernd C. Kellner_, Oct 12 2023: (Start)
%F A324370 a(n) = denominator(Bernoulli_n(x)').
%F A324370 k-th derivative: let (n)_m be the falling factorial.
%F A324370 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)
%e A324370 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.
%p A324370 N:= 100: # for a(1)..a(N)
%p A324370 V:= Vector(N,1):
%p A324370 p:= 1:
%p A324370 for iter from 1 do
%p A324370    p:= nextprime(p);
%p A324370    if p >= N then break fi;
%p A324370    for n from p+1 to N do
%p A324370      if n mod p <> 0 and convert(convert(n,base,p),`+`)>= p then
%p A324370        V[n]:= V[n]*p
%p A324370      fi
%p A324370 od od:
%p A324370 convert(V,list); # _Robert Israel_, Mar 30 2020
%p A324370 # Alternatively, note that this formula is suggesting offset 0 and a(0) = 1:
%p A324370 seq(denom(diff(bernoulli(n, x), x)), n = 1..51); # _Peter Luschny_, Oct 13 2023
%t A324370 SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
%t A324370 DD2[n_] := Times @@ Select[Prime[Range[PrimePi[(n+1)/(2+Mod[n+1, 2])]]], !Divisible[n, #] && SD[n, #] >= # &];
%t A324370 Table[DD2[n], {n, 1, 100}]
%t A324370 (* From _Bernd C. Kellner_, Oct 12 2023 (Start) *)
%t A324370 (* Denominator of first derivative of BP *)
%t A324370 k = 1; Table[Denominator[Together[D[BernoulliB[n, x], {x, k}]]], {n, 1, 100}]
%t A324370 (* End *)
%o A324370 (Python)
%o A324370 from math import prod
%o A324370 from sympy.ntheory import digits
%o A324370 from sympy import primefactors, primerange
%o A324370 def a(n):
%o A324370     nonpf = set(primerange(1, n+1)) - set(primefactors(n))
%o A324370     return prod(p for p in nonpf if sum(digits(n, p)[1:]) >= p)
%o A324370 print([a(n) for n in range(1, 75)]) # _Michael S. Branicky_, Jul 03 2022
%Y A324370 Cf. A002997, A094960, A144845, A195441, A324315, A324316, A324317, A324318, A324319, A324320, A324369, A324371, A324404, A324405, A366168, A366169, A366186, A366187, A366188.
%K A324370 nonn,base
%O A324370 1,3
%A A324370 _Bernd C. Kellner_ and _Jonathan Sondow_, Feb 24 2019