A324369 -id:A324369 - cp's OEIS frontend

A144845 Least number k such that all coefficients of k*B(n,x), the n-th Bernoulli polynomial, are integers.

1, 2, 6, 2, 30, 6, 42, 6, 30, 10, 66, 6, 2730, 210, 30, 6, 510, 30, 3990, 210, 2310, 330, 690, 30, 2730, 546, 42, 14, 870, 30, 14322, 462, 39270, 3570, 210, 6, 1919190, 51870, 2730, 210, 94710, 2310, 99330, 2310, 4830, 4830, 9870, 210, 46410, 6630, 14586, 858

Offset: 0

T. D. Noe, Sep 22 2008

nonn

The lcm of the terms in row n of A053383. It appears that the Bernoulli polynomial B(n,x) is irreducible for all even n.
This sequence appears in a paper of Bazsó & Mező, who use this sequence to give necessary and sufficient conditions for power sums to be integer polynomials. - Istvan Mezo, Mar 20 2016
In "The denominators of power sums of arithmetic progressions" Corollary 1, we give a simple way to compute a(n) without using Bernoulli polynomials. Namely, a(n) equals (product of the primes dividing n+1) times (product of the primes p <= (n+1)/(2+(n+1 mod 2)) not dividing n+1 such that the sum of the base-p digits of n+1 is at least p). - Bernd C. Kellner and Jonathan Sondow, May 15 2017

Bernd C. Kellner, Table of n, a(n) for n = 0..10000 (n = 0..1000 from T. D. Noe)
András Bazsó and István Mező, On the coefficients of power sums of arithmetic progressions, J. Number Th., 153 (2015), 117-123.
András Bazsó and István Mező, Some Notes on Alternating Power Sums of Arithmetic Progressions, J. Int. Seq., Vol. 21 (2018), Article 18.7.8.
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, The denominators of power sums of arithmetic progressions, Integers 18 (2018), #A95, 17 pp.; arXiv:1705.05331 [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.
Eric Weisstein's World of Mathematics, Bernoulli Polynomial.

Cf. A027642, A053383, A064538, A195441, A286515, A286516, A286517, A324369, A324370, A324371.

seq(denom(bernoulli(i,x)),i=0..51); # Peter Luschny, Jun 16 2012

(* From Bernd C. Kellner, Oct 18 2023: (Start) *)
(* Denominator formula *)
Table[Denominator[Together[BernoulliB[n, x]]], {n, 0, 51}]
(* Product formula *)
SD[n_, p_] := If[n < 1 || p < 2, 0, Plus@@IntegerDigits[n, p]]; rad[n_] := Times @@ Select[Divisors[n], PrimeQ]; (* A324370 *) DD2[n_] := Times @@ Select[Prime[Range[PrimePi[(n+1)/(2+Mod[n+1, 2])]]], !Divisible[n, #] && SD[n, #] >= # &];
DB[n_] := DD2[n+1] rad[n+1]; Table[DB[n], {n, 0, 51}]
(* (End) *)

a(n) = lcm(apply(x->denominator(x), Vec(bernpol(n)))); \\ Michel Marcus, Mar 03 2020

def A144845(n):
    return mul(prime_divisors(n+1) + [p for p in (2..(n+2)//(2+n%2))
    if is_prime(p) and not p.divides(n+1) and sum((n+1).digits(base=p)) >= p])
print([A144845(n) for n in (0..51)]) # Peter Luschny, Sep 12 2018

From Bernd C. Kellner, Oct 18 2023: (Start)
Let rad(n) = A007947(n) be the radical of n. Let (n)_m be the falling factorial. Let f^(m)(x) denote the m-th derivative of f(x).
a(n) = lcm(A195441(n-1), A027642(n)) = lcm(denom(B(n,x)-B_n), denom(B_n)) = denom(B(n,x)).
a(n) = lcm(A195441(n), rad(n+1)).
a(n) = lcm(a(n+1), rad(n+1)), if n >= 2 is even.
a(2n)/a(2n+1) = A286517(n), if n >= 1.
a(n) = A324369(n+1) * A324370(n+1) * A324371(n+1).
a(n) = A324370(n+1) * rad(n+1).
a(n) = rad(A064538(n)).
If n >= m >= 1, then denom(B^(m)(n,x)) = a(n-m)/gcd(a(n-m), (n)A324370(n-m+1)/gcd(A324370(n-m+1),%20(n)">m) = A324370(n-m+1)/gcd(A324370(n-m+1), (n){m-1}).
(See papers of Kellner and Kellner & Sondow.) (End)

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.

Original entry on oeis.org

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

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 24 2019

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

			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.

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.

Cf. A002997, A094960, A144845, A195441, A324315, A324316, A324317, A324318, A324319, A324320, A324369, A324371, A324404, A324405, A366168, A366169, A366186, A366187, A366188.

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

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 *)

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

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)

A195441 a(n) = denominator(Bernoulli_{n+1}(x) - Bernoulli_{n+1}).

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 6, 3, 10, 2, 6, 2, 210, 30, 6, 3, 30, 10, 210, 42, 330, 30, 30, 30, 546, 42, 14, 2, 30, 2, 462, 231, 3570, 210, 6, 2, 51870, 2730, 210, 42, 2310, 330, 2310, 210, 4830, 210, 210, 210, 6630, 1326, 858, 66, 330, 110, 798, 114, 870, 30, 30, 6

Offset: 0

Peter Luschny, Sep 18 2011

nonn

If s(n) is the smallest number such that s(n)*(1^n + 2^n + ... + x^n) is a polynomial in x with integer coefficients then a(n)=s(n)/(n+1) (see A064538).
a(n) is squarefree, by the von Staudt-Clausen theorem on the denominators of Bernoulli numbers. - Kieren MacMillan and Jonathan Sondow, Nov 20 2015
Kellner and Sondow give a detailed analysis of this sequence and provide a simple way to compute the terms without using Bernoulli polynomials and numbers. They prove that a(n) is the product of the primes less than or equal to (n+2)/(2+(n mod 2)) such that the sum of digits of n+1 in base p is at least p. - Peter Luschny, May 14 2017
The equation a(n-1) = denominator(Bernoulli_n(x) - Bernoulli_n) = rad(n+1) has only finitely many solutions, where rad(n) = A007947(n) is the radical of n. It is conjectured that S = {3, 5, 8, 9, 11, 27, 29, 35, 59} is the full set of all such solutions. Note that (S\{8})+1 joined with {1,2} equals A094960. More precisely, the set S implies the finite sequence of A094960. See Kellner 2023. - Bernd C. Kellner, Oct 18 2023
As was observed in the example section of A318256: denominator(B_n(x)) = rad(n+1) if n is in {0, 1, 3, 5, 9, 11, 27, 29, 35, 59} = {A094960(n) - 1: 1 <= n <= 10}. - Peter Luschny, Oct 18 2023

Peter Luschny, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
Olivier Bordellès, Florian Luca, Pieter Moree, and Igor E. Shparlinski, Denominators of Bernoulli polynomials, Mathematika 64 (2018), 519-541.
Harald Hofstätter, Denominators of coefficients of the Baker-Campbell-Hausdorff series, arXiv:2010.03440 [math.NT], 2020. Mentions this sequence.
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, The denominators of power sums of arithmetic progressions, Integers 18 (2018), #A95, 17 pp.; arXiv:1705.05331 [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.

Cf. A002997, A064538, A094960, A144845, A286516, A286762, A286763, A318256, A324369, A324370, A324371.

using Nemo, Primes
function A195441(n::Int)
    n < 4 && return ZZ([1,1,2,1][n+1])
    P = primes(2, div(n+2, 2+n%2))
    prod([ZZ(p) for p in P if p <= sum(digits(n+1, base=p))])
end
println([A195441(n) for n in 0:59]) # Peter Luschny, May 14 2017

A195441 := n -> denom(bernoulli(n+1, x)-bernoulli(n+1)):
seq(A195441(i),i=0..59);
# Formula of Kellner and Sondow:
a := proc(n) local s; s := (p,n) -> add(i,i=convert(n,base,p));
select(isprime,[$2..(n+2)/(2+irem(n,2))]); mul(i,i=select(p->s(p,n+1)>=p,%)) end: seq(a(n), n=0..59); # Peter Luschny, May 14 2017

a[n_] := Denominator[Together[(BernoulliB[n + 1, x] - BernoulliB[n + 1])]]; Table[a[n], {n, 0, 59}] (* Jonathan Sondow, Nov 20 2015 *)
SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]]; DD[n_] := Times @@ Select[Prime[Range[PrimePi[(n+2)/(2+Mod[n, 2])]]], SD[n+1, #] >= # &]; Table[DD[n], {n, 0, 59}] (* Bernd C. Kellner, Oct 18 2023 *)

a(n) = {my(vp = Vec(bernpol(n+1, x)-bernfrac(n+1))); lcm(vector(#vp, k, denominator(vp[k])));} \\ Michel Marcus, Feb 08 2016

from math import prod
from sympy.ntheory.factor_ import primerange, digits
def A195441(n): return prod(p for p in primerange((n+2)//(2|n&1)+1) if sum(digits(n+1,p)[1:])>=p) # Chai Wah Wu, Oct 04 2023

A195441 = lambda n: mul([p for p in (2..(n+2)//(2+n%2)) if is_prime(p) and sum((n+1).digits(base=p))>=p])
print([A195441(n) for n in (0..59)]) # Peter Luschny, May 14 2017

a(n) = A064538(n)/(n+1). - Jonathan Sondow, Nov 12 2015
A001221(a(n)) = A001222(a(n)). - Kieren MacMillan and Jonathan Sondow, Nov 20 2015
a(2*n)/a(2*n+1) = A286516(n+1). - Bernd C. Kellner and Jonathan Sondow, May 24 2017
a(n) = A007947(A338025(n+1)). - Harald Hofstätter, Oct 10 2020
From Bernd C. Kellner, Oct 18 2023: (Start)
Note that the formulas here are shifted in index by 1 due to the definition of a(n) using index n+1!
a(n) = A324369(n+1) * A324370(n+1).
a(n) = A144845(n) / A324371(n+1).
a(n-1) = lcm(a(n), rad(n+1)), if n >= 3 is odd.
If n+1 is composite, then rad(n+1) divides a(n-1).
If m is a Carmichael number (A002997), then m divides both a(m-1) and a(m-2).
See papers of Kellner and Kellner & Sondow. (End)

Definition simplified by Jonathan Sondow, Nov 20 2015

A324316 Primary Carmichael numbers.

Original entry on oeis.org

1729, 2821, 29341, 46657, 252601, 294409, 399001, 488881, 512461, 1152271, 1193221, 1857241, 3828001, 4335241, 5968873, 6189121, 6733693, 6868261, 7519441, 10024561, 10267951, 10606681, 14469841, 14676481, 15247621, 15829633, 17098369, 17236801, 17316001, 19384289, 23382529, 29111881, 31405501, 34657141, 35703361, 37964809

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 21 2019

Squarefree integers m > 1 such that if prime p divides m, then the sum of the base-p digits of m equals p. It follows that m is then a Carmichael number (A002997).
Dickson's conjecture implies that the sequence is infinite, see Kellner 2019.
If m is a term and p is a prime factor of m, then p <= a*sqrt(m) with a = sqrt(66337/132673) = 0.7071..., where the bound is sharp.
The distribution of primary Carmichael numbers is A324317.
See Kellner and Sondow 2019 and Kellner 2019.
Primary Carmichael numbers are special polygonal numbers A324973. The rank of the n-th primary Carmichael number is A324976(n). See Kellner and Sondow 2019. - Jonathan Sondow, Mar 26 2019
The first term is the Hardy-Ramanujan number. - Omar E. Pol, Jan 09 2020

			1729 = 7 * 13 * 19 is squarefree, and 1729 in base 7 is 5020_7 = 5 * 7^3 + 0 * 7^2 + 2 * 7 + 0 with 5+0+2+0 = 7, and 1729 in base 13 is a30_13 with a+3+0 = 10+3+0 = 13, and 1729 in base 19 is 4f0_19 with 4+f+0 = 4+15+0 = 19, so 1729 is a member.

Bernd C. Kellner, Table of n, a(n) for n = 1..10000 (computed by using Pinch's database, see link below)
Bernd C. Kellner, On primary Carmichael numbers, #A38 Integers 22 (2022), 39 p.; arXiv:1902.11283 [math.NT], 2019.
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, #A52 Integers 21 (2021), 21 p.; arXiv:1902.10672 [math.NT], 2019.
R. G. E. Pinch, The Carmichael numbers up to 10^18, 2008.
Index entries for sequences related to Carmichael numbers.

Subsequence of A002997, A324315.
Least primary Carmichael number with n prime factors is A306657.
Cf. also A005117, A195441, A324317, A324318, A324319, A324320, A324369, A324370, A324371, A324404, A324405, A324973, A324976, A001235.

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
TestCP[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] == # &];
Select[Range[1, 10^7, 2], TestCP[#] &]

use ntheory ":all"; my $m; forsquarefree { $m=$; say if @ > 2 && is_carmichael($m) && vecall { $ == vecsum(todigits($m,$)) } @; } 1e7; # _Dana Jacobsen, Mar 28 2019

from sympy import factorint
from sympy.ntheory import digits
def ok(n):
    pf = factorint(n)
    if n < 2 or max(pf.values()) > 1: return False
    return all(sum(digits(n, p)[1:]) == p for p in pf)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jul 03 2022

a_1 + a_2 + ... + a_k = p if p is prime and m = a_1 * p + a_2 * p^2 + ... + a_k * p^k with 0 <= a_i <= p-1 for i = 1, 2, ..., k (note that a_0 = 0).

A064538 a(n) is the smallest positive integer such that a(n)*(1^n + 2^n + ... + x^n) is a polynomial in x with integer coefficients.

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 42, 24, 90, 20, 66, 24, 2730, 420, 90, 48, 510, 180, 3990, 840, 6930, 660, 690, 720, 13650, 1092, 378, 56, 870, 60, 14322, 7392, 117810, 7140, 210, 72, 1919190, 103740, 8190, 1680, 94710, 13860, 99330, 9240, 217350, 9660, 9870, 10080, 324870

Offset: 0

Floor van Lamoen, Oct 08 2001

a(n) is a multiple of n+1. - Vladimir Shevelev, Dec 20 2011
Let P_n(x) = 1^n + 2^n + ... + x^n = Sum_{i=1..n+1}c_i*x^i. Let P^*n(x) = Sum{i=1..n+1}(c_i/(i+1))*(x^(i+1)-x). Then b(n) = (n+1)*a(n+1)is the smallest positive integer such that b(n)*P^*n(x) is a polynomial with integer coefficients. Proof follows from the recursion P(n+1)(x) = x + (n+1)*P^*n(x). As a corollary, note that, if p is the maximal prime divisor of a(n), then p<=n+1. - _Vladimir Shevelev, Dec 21 2011
The recursion P_(n+1)(x) = x + (n+1)*P^*n(x) is due to Abramovich (1973); see also Shevelev (2007). - _Jonathan Sondow, Nov 16 2015
The sum S_m(n) = Sum_{k=0..n} k^m can be written as S_m(n) = n(n+1)(2n+1)P_m(n)/a(m) for even m>1, or S_m(n) = n^2*(n+1)^2*P_m(n)/a(m) for odd m>1, where a(m) is the LCM of the denominators of the coefficients of the polynomial P_m/a(m), i.e., the smallest integer such that P_m defined in this way has integer coefficients. (Cf. Michon link.) - M. F. Hasler, Mar 10 2013
a(n)/(n+1) is squarefree, by Faulhaber's formula and the von Staudt-Clausen theorem on the denominators of Bernoulli numbers. - Kieren MacMillan and Jonathan Sondow, Nov 20 2015
a(n) equals n+1 times the product of the primes p <= (n+2)/(2+(n mod 2)) such that the sum of the base-p digits of n+1 is at least p. - Bernd C. Kellner and Jonathan Sondow, May 24 2017

			1^3 + 2^3 + ... + x^3 = (x(x+1))^2/4 so a(3)=4.
1^4 + 2^4 + ... + x^4 = x(x+1)(2x+1)(3x^2+3x-1)/30, therefore a(4)=30.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprints), p. 804, Eq. 23.1.4.

Chai Wah Wu, Table of n, a(n) for n = 0..10000 (n = 0..1000 from T. D. Noe)
V. S. Abramovich, Power sums of natural numbers, Kvant 5 (1973), 22-25. (in Russian)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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, The denominators of power sums of arithmetic progressions, Integers 18 (2018), #A95, 17 pp.; arXiv:1705.05331 [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.
Meng Fai Lim, On the p-divisibility of even K-groups of the ring of integers of a cyclotomic field, arXiv:2308.04099 [math.NT], 2023.
Dr. Math, Summing n^k.
R. Mestrovic, A congruence modulo n^3 involving two consecutive sums of powers and its applications, arXiv:1211.4570 [math.NT], 2012.
R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
G. Michon, Faulhaber's Formula on NUMERICANA.com.
E. S. Rowland, Sums of Consecutive Powers
Vladimir Shevelev, A Short Proof of a Known Relation for Consecutive Power Sums, arXiv:0711.3692 [math.CA], 2007.
Eric Weisstein's World of Mathematics, Power Sum
Wikipedia, Faulhaber's Formula.

Cf. A144845, A195441, A256581, A286516, A286762, A286763, A324369, A324370, A324371.

A064538 := n -> denom((bernoulli(n+1,x)-bernoulli(n+1))/(n+1)): # Peter Luschny, Aug 19 2011
# Formula of Kellner and Sondow (2017):
a := proc(n) local s; s := (p,n) -> add(i,i=convert(n,base,p));
select(isprime,[$2..(n+2)/(2+irem(n,2))]);
(n+1)*mul(i,i=select(p->s(p,n+1)>=p,%)) end: seq(a(n), n=0..48); # Peter Luschny, May 14 2017

A064538[n_] := Denominator[ Together[ (BernoulliB[n+1, x] - BernoulliB[n+1])/(n+1)]];
Table[A064538[n], {n, 0, 44}] (* Jean-François Alcover, Feb 21 2012, after Maple *)

a(n) = {my(vp = Vec(bernpol(n+1, x)-bernfrac(n+1))/(n+1)); lcm(vector(#vp, k, denominator(vp[k])));} \\ Michel Marcus, Feb 07 2016

from _future_ import division
from sympy.ntheory.factor_ import digits, nextprime
def A064538(n):
    p, m = 2, n+1
    while p <= (n+2)//(2+ (n% 2)):
        if sum(d for d in digits(n+1,p)[1:]) >= p:
            m *= p
        p = nextprime(p)
    return m # Chai Wah Wu, Mar 07 2018

A064538 = lambda n: (n+1)*mul([p for p in (2..(n+2)//(2+n%2)) if is_prime(p) and sum((n+1).digits(base=p)) >= p])
print([A064538(n) for n in (0..48)]) # Peter Luschny, May 14 2017

a(n) = (n+1)*A195441(n). - Jonathan Sondow, Nov 12 2015
A001221(a(n)/(n+1)) = A001222(a(n)/(n+1)). - Kieren MacMillan and Jonathan Sondow, Nov 20 2015
rad(a(n)) = A007947(a(n)) = A144845(n) = A324369(n+1) * A324370(n+1) * A324371(n+1). - Bernd C. Kellner, Oct 12 2023

A324315 Squarefree integers m > 1 such that if prime p divides m, then the sum of the base p digits of m is at least p.

Original entry on oeis.org

231, 561, 1001, 1045, 1105, 1122, 1155, 1729, 2002, 2093, 2145, 2465, 2821, 3003, 3315, 3458, 3553, 3570, 3655, 3927, 4186, 4199, 4522, 4774, 4845, 4862, 5005, 5187, 5565, 5642, 5681, 6006, 6118, 6270, 6279, 6545, 6601, 6670, 6734, 7337, 7395, 7735, 8177, 8211, 8265, 8294, 8323, 8463, 8645, 8789, 8855, 8911, 9282, 9361, 9435, 9690, 9867

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 21 2019

The sequence is infinite, because it contains all Carmichael numbers (A002997).
If m is a term and p is a prime factor of m, then p <= a*sqrt(m) with a = sqrt(11/21) = 0.7237..., where the bound is sharp.
A term m must have at least 3 prime factors if m is odd, and must have at least 4 prime factors if m is even.
m is a term if and only if m > 1 divides denominator(Bernoulli_m(x) - Bernoulli_m) = A195441(m-1).
A term m is a Carmichael number iff s_p(m) == 1 (mod p-1) whenever prime p divides m, where s_p(m) is the sum of the base p digits of m.
See Kellner and Sondow 2019.

			231 = 3 * 7 * 11 is squarefree, and 231 in base 3 is 22120_3 = 2 * 3^4 + 2 * 3^3 + 1 * 3^2 + 2 * 3 + 0 with 2+2+1+2+0 = 7 >= 3, and 231 = 450_7 with 4+5+0 = 9 >= 7, and 231 = 1a0_11 with 1+a+0 = 1+10+0 = 11 >= 11, so 231 is a member.

Amiram Eldar, Table of n, a(n) for n = 1..10000
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, #A52 Integers 21 (2021), 21 pp.; arXiv:1902.10672 [math.NT], 2019.

Cf. A002997, A005117, A195441, A324316, A324317, A324318, A324319, A324320, A324369, A324370, A324371, A324404, A324405.

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
TestS[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # &];
Select[Range[10^4], TestS[#] &]

from sympy import factorint
from sympy.ntheory import digits
def ok(n):
    pf = factorint(n)
    if n < 2 or max(pf.values()) > 1: return False
    return all(sum(digits(n, p)[1:]) >= p for p in pf)
print([k for k in range(10**4) if ok(k)]) # Michael S. Branicky, Jul 03 2022

a_1 + a_2 + ... + a_k >= p for m = a_1 * p + a_2 * p^2 + ... + a_k * p^k with 0 <= a_i <= p-1 for i = 1, 2, ..., k (note that a_0 = 0).

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.

Original entry on oeis.org

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

Bernd C. Kellner and Jonathan Sondow, Feb 25 2019

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.

			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.

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.

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

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

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}]

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

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)).

A324319 Terms of A324315 (squarefree integers m > 1 such that if prime p divides m, then the sum of the base p digits of m is at least p) that are also hexagonal numbers (A000384) with index equal to their largest prime factor.

Original entry on oeis.org

231, 561, 3655, 5565, 8911, 10585, 13695, 23653, 32131, 45451, 59685, 74305, 108345, 115921, 157641, 243253, 248865, 302253, 314821, 334153, 371091, 392055, 417241, 458403, 505515, 546535, 688551, 702705, 795691, 821121, 915981, 932295, 1004653, 1145341, 1181953

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 23 2019

561, 8911, and 10585 are also Carmichael numbers (A002997).
The smallest primary Carmichael number (A324316) in the sequence is 8801128801 = 181 * 733 * 66337 = A000384(66337).
See the section on polygonal numbers in Kellner and Sondow 2019.
Subsequence of the special polygonal numbers A324973. - Jonathan Sondow, Mar 27 2019

			A324315(1) = 231 = 3 * 7 * 11 = 11 * (2 * 11 - 1) = A000384(11), so 231 is a member.

Amiram Eldar, Table of n, a(n) for n = 1..10000
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, #A52 Integers 21 (2021), 21 pp.; arXiv:1902.10672 [math.NT], 2019.

Cf. A000384, A002997, A195441, A324315, A324316, A324317, A324318, A324320, A324369, A324370, A324371, A324404, A324405, A324973.

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
HN[n_] := n(2n - 1);
TestS[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # &];
Select[HN@ Prime[Range[100]], TestS[#] &]

More terms from Amiram Eldar, Dec 05 2020

A324320 Terms of A324315 (squarefree integers m > 1 such that if prime p divides m, then the sum of the base p digits of m is at least p) that are also octagonal numbers (A000567) with index equal to their largest prime factor.

Original entry on oeis.org

1045, 2465, 2821, 15841, 20501, 34133, 51221, 68101, 89441, 116033, 118405, 162401, 170885, 216545, 300833, 364705, 439301, 472033, 530881, 642181, 687365, 746005, 970145, 976981, 997633, 1104133, 1148245, 1193221, 1231361, 1239061, 1398101, 1654661, 1971541

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 23 2019

2465 is also a Carmichael number (A002997).
2821 is also a primary Carmichael number (A324316).
See the section on polygonal numbers in Kellner and Sondow 2019.
Subsequence of the special polygonal numbers A324973. - Jonathan Sondow, Mar 27 2019

			A324315(4) = 1045 = 5 * 11 * 19 = 19 * (3 * 19 - 2) = A000567(19), so 1045 is a member.

Amiram Eldar, Table of n, a(n) for n = 1..10000
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, #A52 Integers 21 (2021), 21 pp.; arXiv:1902.10672 [math.NT], 2019.

Cf. A000567, A002997, A324315, A324316, A324317, A324318, A324319, A324369, A324370, A324371, A324404, A324405, A324973.

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
ON[n_] := n(3n - 2);
TestS[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # &];
Select[ON@ Prime[Range[100]], TestS[#] &]

More terms from Amiram Eldar, Dec 05 2020

A324404 Squarefree integers m > 1 such that if prime p divides m, then s_p(m) >= p and s_p(m) == 2 (mod p-1), where s_p(m) is the sum of the base p digits of m.

Original entry on oeis.org

1122, 3458, 5642, 6734, 11102, 13202, 17390, 17822, 21170, 22610, 27962, 31682, 46002, 58682, 61778, 79730, 82082, 93314, 105266, 106262, 125490, 127946, 136202, 150722, 153254, 177122, 182002, 202202, 203870, 214370, 231842, 252434, 274298, 278462, 305102, 315282

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 26 2019

For d >= 1 define S_d = (terms m in A324315 such that s_p(m) == d (mod p-1) if prime p divides m). Then S_1 is precisely the Carmichael numbers (A002997), S_2 is A324404, S_3 is A324405, and the union of all S_d for d >= 1 is A324315.
Subsequence of the 2-Knödel numbers (A050990). Generally, for d > 1 the terms of S_d that are greater than d form a subsequence of the d-Knödel numbers.
See Kellner and Sondow 2019.

			1122 = 2*3*11*17 is squarefree and equals 10001100010_2, 1112120_3, 930_11, and 3f0_17 in base p = 2, 3, 11, and 17. Then s_2(1122) = 1+1+1+1 = 4 >= 2, s_3(1122) = 1+1+1+2+1+2 = 8 >= 3, s_11(1122) = 9+3 = 12 >= 11, and s_17(1122) = 3+f = 3+15 = 18 >= 17. Also, s_2(1122) = 4 == 2 (mod 1), s_3(1122) = 8 == 2 (mod 2), s_11(1122) = 12 == 2 (mod 10), and s_17(1122) = 18 == 2 (mod 16), so 1122 is a member.

Amiram Eldar, Table of n, a(n) for n = 1..2200
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), Article #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019-2021.

Cf. A002997, A050990, A324315, A324316, A324317, A324318, A324319, A324320, A324369, A324370, A324371, A324405.

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
TestSd[n_, d_] := (n > 1) && (d > 0) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # && Mod[SD[n, #] - d, # - 1] == 0 &];
Select[Range[200000], TestSd[#, 2] &]

More terms from Amiram Eldar, Dec 05 2020

cp's OEIS Frontend

A144845 Least number k such that all coefficients of k*B(n,x), the n-th Bernoulli polynomial, are integers.

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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.

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A195441 a(n) = denominator(Bernoulli_{n+1}(x) - Bernoulli_{n+1}).

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A324316 Primary Carmichael numbers.

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A064538 a(n) is the smallest positive integer such that a(n)*(1^n + 2^n + ... + x^n) is a polynomial in x with integer coefficients.

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A324315 Squarefree integers m > 1 such that if prime p divides m, then the sum of the base p digits of m is at least p.

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