A089965 -id:A089965 - cp's OEIS frontend

A130190 Denominators of z-sequence for the Sheffer matrix (triangle) A094816 (coefficients of Poisson-Charlier polynomials).

1, 2, 6, 4, 15, 12, 42, 24, 90, 10, 33, 8, 910, 105, 90, 48, 255, 180, 3990, 420, 6930, 330, 345, 720, 13650, 273, 378, 28, 145, 20, 14322, 2464, 117810, 3570, 7, 24, 1919190, 1729, 2730, 840, 9471, 13860, 99330, 1540, 217350, 4830, 4935, 10080, 324870

Offset: 0

Wolfdieter Lang, Jun 01 2007

The numerators are given in A130189.
See A130189 for the W. Lang link on z-sequences for Sheffer matrices.
The prime factors of each a(n) are such that n!/a(n) has the prime, p = n+1, as the denominator of its reduced fraction, and if n+1 is not prime then n!/a(n) is an integer, except at n = 3, which has denominator = 2. Also see alternate formula for a(n) below. - Richard R. Forberg, Dec 28 2014
As implied above, at n = p-1 the largest prime factor of a(n) is p. For a(m), where m is an integer within the set given by A089965, the two largest prime factors of a(m) are m+1 and (m+1)/2. Furthermore, it appears, when n+1 is not a prime no prime factor of a(n) is greater than k/2, where k is the next higher value of n where n+1 is prime. Two examples at this upper limit of k/2 are n = 104 and 105, where the highest prime factor of a(n) is 53; it is then at n = k = 106 where n+1 is prime. - Richard R. Forberg, Jan 01 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A089965, A094816, A248716, A130189.

seq(denom(add(Stirling2(n,k)/(k+1),k=0..n)),n=0..20); # Peter Luschny, Apr 28 2009

Denominator[Table[(1/Exp[1])* Sum[Sum[j^n/k!, {j, 0, k}], {k, 0, Infinity}], {n, 0, 100}]] (* Richard R. Forberg, Dec 28 2014 *)
Table[Denominator[Sum[StirlingS2[n, k]/(k + 1), {k, 0, n}]], {n, 0, 50}] (* G. C. Greubel, Jul 10 2018 *)

a(n) = denominator(sum(k=0, n, stirling(n, k, 2)/(k+1))); \\ Michel Marcus, Jan 15 2015, after Maple

a(n) = denominator(z(n)),n>=0, with the e.g.f. for z(n) given in A130189.
Denominator of Sum_{k=0..n} A048993(n,k)/(k+1).  - Peter Luschny, Apr 28 2009
Alternate: a(n) = denominator((1/e)*Sum_{k>=0}*(Sum_{j=0..k} j^n/k!)). NOTE: Numerators are different from A130189, and given by A248716. - Richard R. Forberg, Dec 28 2014
This more generalized expression ((1/e)*Sum_{k>=0} (Sum_{j=0..k} (j+m)^n/k!)), gives the same denominators for any integer m. - Richard R. Forberg, Jan 14 2015

A278500 a(n) = largest k such that n+1 = a prime, n+2 = 2 * a prime, ..., n+k is k times a prime, a(n) = 0 if n+1 is not a prime.

Original entry on oeis.org

1, 2, 0, 2, 0, 1, 0, 0, 0, 1, 0, 3, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Antti Karttunen, Nov 30 2016

nonn

First 4 occurs at n=12720, first 5 occurs at n=19440. See A074200.

			a(12) = 3 as 13 = 1*prime, 14 = 2*prime, 15 = 3*prime.

Antti Karttunen, Table of n, a(n) for n = 1..21000

Cf. A072668 (positions of zeros), A006093 (nonzeros), A089965 (positions of terms >= 2), A278583 (of terms >= 3), A278585 (of terms >= 4).

Cf. A074200 (position of the first term >= n).

Table[If[CompositeQ[n + 1], 0, k = 1; While[Times @@ Boole@ Map[PrimeQ, MapIndexed[#1/First@ #2 &, (n + Range@ k)]] == 1, k++]; k - 1], {n, 120}] (* Michael De Vlieger, Dec 01 2016 *)

A278500(n) = { my(k=1); while((!((n+k)%k) && isprime((n+k)/k)), k = k+1); (k-1); }
for(n=1, 2^20, write("b278500.txt", n, " ", A278500(n)));

(define (A278500 n) (let loop ((k 1)) (let ((h (/ (+ n k) k))) (if (or (not (integer? h)) (zero? (A010051 h))) (- k 1) (loop (+ 1 k))))))

A278585 Numbers k such that k+1 is a prime, k+2 is twice a prime, k+3 is three times a prime, and k+4 is four times a prime.

Original entry on oeis.org

12720, 16920, 19440, 24480, 49680, 61560, 104160, 229320, 255360, 259680, 266400, 291720, 298200, 311040, 331920, 419400, 423480, 436800, 446880, 471240, 525240, 532800, 539400, 581520, 600600, 663600, 704160, 709920, 783720, 867000, 904800, 908040, 918360

Offset: 1

N. J. A. Sloane, Nov 30 2016

nonn

a(n) == 0 mod 120 (see comment in A163573). - Chai Wah Wu, Nov 30 2016

Chai Wah Wu, Table of n, a(n) for n = 1..10001

Equals A163573(n) - 1.
Cf. A036570, A074200.
Positions of terms >= 4 in A278500, thus a subsequence of A278583, A089965 and A006093.

Select[Range[920000],AllTrue[{#+1,(#+2)/2,(#+3)/3,(#+4)/4},PrimeQ]&] (* Harvey P. Dale, Aug 08 2021 *)

is(k)=k%120==0 && isprime(k+1) && isprime(k/2+1) && isprime(k/3+1) && isprime(k/4+1) \\ Charles R Greathouse IV, Dec 03 2016

from sympy import prime, isprime
A278585_list = [4*q-4 for q in (prime(i) for i in range(1,10000)) if isprime(4*q-3) and isprime(2*q-1) and (not (4*q-1) % 3) and isprime((4*q-1)//3)] # Chai Wah Wu, Nov 30 2016

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A130190 Denominators of z-sequence for the Sheffer matrix (triangle) A094816 (coefficients of Poisson-Charlier polynomials).

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A278500 a(n) = largest k such that n+1 = a prime, n+2 = 2 * a prime, ..., n+k is k times a prime, a(n) = 0 if n+1 is not a prime.

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A278585 Numbers k such that k+1 is a prime, k+2 is twice a prime, k+3 is three times a prime, and k+4 is four times a prime.

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