A027760 -id:A027760 - cp's OEIS frontend

A067513 Number of divisors d of n such that d+1 is prime.

1, 2, 1, 3, 1, 3, 1, 3, 1, 3, 1, 5, 1, 2, 1, 4, 1, 4, 1, 4, 1, 3, 1, 5, 1, 2, 1, 4, 1, 5, 1, 4, 1, 2, 1, 7, 1, 2, 1, 5, 1, 4, 1, 4, 1, 3, 1, 6, 1, 3, 1, 4, 1, 4, 1, 4, 1, 3, 1, 8, 1, 2, 1, 4, 1, 5, 1, 3, 1, 4, 1, 8, 1, 2, 1, 3, 1, 4, 1, 6, 1, 3, 1, 7, 1, 2, 1, 5, 1, 6, 1, 4, 1, 2, 1, 7, 1, 2, 1, 5, 1, 4, 1

Offset: 1

Amarnath Murthy, Feb 12 2002

1, 2 and 4 are the only numbers such that for every divisor d, d+1 is a prime.
These and only these primes appear as prime divisors of any term of InvPhi(n) set if n is not empty, i.e., if n is from A002202. - Labos Elemer, Jun 24 2002
a(n) is the number of integers k such that n = k - k/p where p is one of the prime divisors of k. (See, e.g., A064097 and A333123, which are related to the mapping k -> k - k/p.) - Robert G. Wilson v, Jun 12 2022

			a(12) = 5 as the divisors of 12 are 1, 2, 3, 4, 6 and 12 and the corresponding primes are 2,3,5,7 and 13. Only 3+1 = 4 is not a prime.

T. D. Noe, Table of n, a(n) for n = 1..10000
Yuchen Ding, On a conjecture of R. M. Murty and V. K. Murty, arXiv:2208.06704 [math.NT], 2022.
Yuchen Ding, On a conjecture of R. M. Murty and V. K. Murty II, arXiv:2209.01087 [math.NT], 2022-2023.
Steve Fan and Carl Pomerance, Shifted-prime divisors, arXiv:2401.10427 [math.NT], 2024.
Mikhail R. Gabdullin, Moments of the shifted prime divisor function, arXiv preprint (2025). arXiv:2505.24050 [math.NT]
M. Ram Murty and V. Kumar Murty, A variant of the Hardy-Ramanujan theorem, Hardy-Ramanujan Journal, Vol. 44 (2021), pp. 32-40.
Karl Prachar, Über die Anzahl der Teiler einer natürlichen Zahl, welche die Form p-1 haben, Monatshefte für Mathematik, Vol. 59 (1955), pp. 91-97.

Even-indexed terms give A046886.
Cf. A000005, A001221, A001222, A002202, A027750, A064097, A141197, A185633, A202727, A202728, A322312, A322976, A333123, A346467, A355452.
Cf. A005408 (positions of 1's), A051222 (of 2's).

a067513 = sum . map (a010051 . (+ 1)) . a027750_row
-- Reinhard Zumkeller, Jul 31 2012

A067513 := proc(n)
    local a,d;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if isprime(d+1) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A067513(n),n=1..100) ; # R. J. Mathar, Aug 07 2022

a[n_] := Length[Select[Divisors[n]+1, PrimeQ]]
Table[Count[Divisors[n],?(PrimeQ[#+1]&)],{n,110}] (* _Harvey P. Dale, Feb 29 2012 *)
a[n_] := DivisorSum[n, 1 &, PrimeQ[# + 1] &]; Array[a, 100] (* Amiram Eldar, Jan 11 2025 *)

a(n)=sumdiv(n,d,isprime(d+1)) \\ Charles R Greathouse IV, Dec 23 2011

from sympy import divisors, isprime
def a(n): return sum(1 for d in divisors(n, generator=True) if isprime(d+1))
print([a(n) for n in range(1, 104)]) # Michael S. Branicky, Jul 12 2022

a(n) = 2 iff Bernoulli number B_{n} has denominator 6 (cf. A051222). - Vladeta Jovovic, Feb 13 2002
a(n) <= A141197(n). - Reinhard Zumkeller, Oct 06 2008
a(n) = A001221(A027760(n)). - Enrique Pérez Herrero, Dec 23 2011
a(n) = Sum_{k = 1..A000005(n)} A010051(A027750(n,k)+1). - Reinhard Zumkeller, Jul 31 2012
a(n) = A001221(A185633(n)) = A001222(A322312(n)). - Antti Karttunen, Jul 12 2022
Sum_{k=1..n} a(k) = n * (log(log(n)) + B) + O(n/log(n)), where B is a constant (Prachar, 1955). - Amiram Eldar, Jan 11 2025

Edited by Dean Hickerson, Feb 12 2002

A141459 a(n) = Product_{p-1 divides n} p, where p is an odd prime.

Original entry on oeis.org

1, 1, 3, 1, 15, 1, 21, 1, 15, 1, 33, 1, 1365, 1, 3, 1, 255, 1, 399, 1, 165, 1, 69, 1, 1365, 1, 3, 1, 435, 1, 7161, 1, 255, 1, 3, 1, 959595, 1, 3, 1, 6765, 1, 903, 1, 345, 1, 141, 1, 23205, 1, 33, 1, 795, 1, 399, 1, 435, 1, 177, 1, 28393365, 1, 3, 1, 255, 1, 32361, 1, 15, 1, 2343, 1, 70050435

Offset: 0

Paul Curtz, Aug 08 2008

nonn

Previous name was: A027760(n)/2 for n>=1, a(0) = 1.
Conjecture: a(n) = denominator of integral_{0..1}(log(1-1/x)^n) dx. - Jean-François Alcover, Feb 01 2013
Define the generalized Bernoulli function as B(s,z) = -s*z^s*HurwitzZeta(1-s,1/z) for Re(1/z) > 0 and B(0,z) = 1 for all z; further the generalized Bernoulli polynomials as Bp(m,n,z) = Sum_{j=0..n} B(j,m)*C(n,j)*(z-1)^(n-j) then the a(n) are denominators of Bp(2,n,1), i. e. of the generalized Bernoulli numbers in the case m=2. The numerators of these numbers are A157779(n). - Peter Luschny, May 17 2015
From Peter Luschny, Nov 22 2015: (Start)
a(n) are the denominators of the centralized Bernoulli polynomials 2^n*Bernoulli(n, x/2+1/2) evaluated at x=1. The numerators are A239275(n).
a(n) is the odd part of A141056(n).
a(n) is squarefree, by the von Staudt-Clausen theorem. (End)
Apparently a(n) = denominator(Sum_{k=0..n-1}(-1)^k*E2(n-1, k+1)/binomial(2*n-1, k+1)) where E2(n, k) denotes the second-order Eulerian numbers A340556. - Peter Luschny, Feb 17 2021

			The denominators of 1, 0, -1/3, 0, 7/15, 0, -31/21, 0, 127/15, 0, -2555/33, 0, 1414477/1365, ...

Robert Israel, Table of n, a(n) for n = 0..10000
Peter Luschny, Generalized Bernoulli Numbers and Polynomials

Cf. A027760, A141056, A141459, A157779, A157818, A160014, A226157, A239275, A340556.

Bfun := (s,z) -> `if`(s=0,1,-s*z^s*Zeta(0,1-s,1/z): # generalized Bernoulli function
Bpoly := (m,n,z) -> add(Bfun(j,m)*binomial(n,j)*(z-1)^(n-j),j=0..n): # generalized Bernoulli polynomials
seq(Bpoly(2,n,1),n=0..50): denom([%]);
# which simplifies to:
a := n -> 0^n+add(Zeta(1-j)*(2^j-2)*j*binomial(n,j), j=1..n):
seq(denom(a(n)), n=0..50); # Peter Luschny, May 17 2015
# Alternatively:
with(numtheory):
ClausenOdd := proc(n) local S, m;
S := map(i -> i + 1, divisors(n));
S := select(isprime, S) minus {2};
mul(m, m = S) end: seq(ClausenOdd(n), n=0..72); # Peter Luschny, Nov 22 2015
# Alternatively:
N:= 1000: # to get a(0) to a(N)
V:= Array(0..N, 1):
for p in select(isprime, [seq(i,i=3..N+1,2)]) do
  R:=[seq(j,j=p-1..N, p-1)]:
  V[R]:= V[R] * p;
od:
convert(V,list); # Robert Israel, Nov 22 2015

a[n_] := If[OddQ[n], 1, Denominator[-2*(2^(n - 1) - 1)*BernoulliB[n]]]; Table[a[n], {n, 0, 72}] (* Jean-François Alcover, Jan 30 2013 *)
Table[Times @@ Select[Divisors@ n + 1, PrimeQ@ # && OddQ@ # &] + Boole[n == 0], {n, 0, 72}] (* Michael De Vlieger, Apr 30 2017 *)

A141056(n) =
{
    p = 1;
    if (n > 0,
        fordiv(n, d,
            r = d + 1;
            if (isprime(r) & r>2, p = p*r)
        )
    );
    return(p)
}
for(n=0, 72, print1(A141056(n), ", ")); \\ Peter Luschny, Nov 22 2015

def A141459_list(size):
    f = x / sum(x^(n*2+1)/factorial(n*2+1) for n in (0..2*size))
    t = taylor(f, x, 0, size)
    return [(factorial(n)*s).denominator() for n,s in enumerate (t.list())]
print(A141459_list(72)) # Peter Luschny, Jul 05 2016

a(2*n+1) = 1. a(2*n)= A001897(n).
a(n) = denominator(0^n + Sum_{j=1..n} zeta(1-j)*(2^j-2)*j*C(n,j)). - Peter Luschny, May 17 2015
Let P(x)= Sum_{n>=0} x^(2*n+1)/(2*n+1)! then a(n) = denominator( n! [x^n] x/P(x) ). - Peter Luschny, Jul 05 2016
a(n) = A157818(n)/4^n. See a comment under A157817, also for other Bernoulli numbers B[4,1] and B[4,3] with this denominator. - Wolfdieter Lang, Apr 28 2017

1 prepended and offset set to 0 by Peter Luschny, May 17 2015

New name from Peter Luschny, Nov 22 2015

A141417 (-1)^(n+1)A091137(n)a(0,n), where a(i,j) = Integral_{x=i..i+1} x(x-1)(x-2)...(x-j+1)/j! dx.

Original entry on oeis.org

-1, 1, 1, 1, 19, 27, 863, 1375, 33953, 57281, 3250433, 5675265, 13695779093, 24466579093, 132282840127, 240208245823, 111956703448001, 205804074290625, 151711881512390095, 281550972898020815, 86560056264289860203, 161867055619224199787, 20953816286242674495191, 39427936010479474495191

Offset: 0

Paul Curtz, Aug 05 2008

sign

This is row i=0 of an array defined as T(i,j) = (-1)^(i+j+1)*A091137(j)*a(i,j), columns j >= 0, which starts
  -1,   1,   1,    1,    19,    27,   863, ...
   1,  -3,   5,    1,    11,    11,   271, ...
  -1,   5, -23,    9,    19,    11,   191, ...
   1,  -7,  53,  -55,   251,    27,   271, ...
  -1,   9, -95,  161, -1901,   475,   863, ...
   1, -11, 149, -351,  6731, -4277, 19087, ...
  ...
The first two rows are related via T(0,j) = A027760(j)*T(0,j-1) - T(1,j).

P. Curtz, Integration .., note 12, C.C.S.A., Arcueil, 1969.

Cf. A141047, A140811, A140825.

A091137 := proc(n) local a, i, p ; a := 1 ; for i from 1 do p := ithprime(i) ; if p > n+1 then break; fi; a := a*p^floor(n/(p-1)) ; od: a ; end proc:
A048994 := proc(n, k) combinat[stirling1](n, k) ; end proc:
a := proc(i,j) add(A048994(j,k)*x^k,k=0..j) ; int(%,x=i..i+1) ; %/j! ; end proc:
A141417 := proc(n) (-1)^(n+1)*A091137(n)*a(0,n) ; end proc:
seq(A141417(n),n=0..40) ; # R. J. Mathar, Nov 17 2010

(* a7 = A091137 *) a7[n_] := a7[n] = Times @@ Select[ Divisors[n]+1, PrimeQ]*a7[n-1]; a7[0]=1; a[n_] := (-1)^(n+1) * a7[n] * Integrate[ (-1)^n*Pochhammer[-x, n], {x, 0, 1}]/n!; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Aug 10 2012 *)

a(n):=if n=0 then -1 else num(n*(n+1)*sum(((-1)^(n-k)*stirling2(n+k,k)*binomial(2*n-1,n-k))/((n+k)*(n+k-1)),k,1,n)); /* Vladimir Kruchinin, Dec 12 2016 */

a(i,j) = a(i-1,j) + a(i-1,j-1), see reference page 33.
(q+1-j)*Sum_{j=0..q} a(i,j)*(-1)^(q-j) = binomial(i,q), see reference page 35.
a(n) = numerator(n*(n+1)*Sum_{k=1..n} ((-1)^(n-k)*Stirling2(n+k,k)*binomial(2*n-1,n-k))/((n+k)*(n+k-1))), n>0, a(0)=-1. - Vladimir Kruchinin, Dec 12 2016

Erroneous formula linking A091137 and A002196 removed, and more terms and program added by R. J. Mathar, Nov 17 2010

A176591 Bernoulli denominators A141056(n), with the exception a(1)=1.

Original entry on oeis.org

1, 1, 6, 2, 30, 2, 42, 2, 30, 2, 66, 2, 2730, 2, 6, 2, 510, 2, 798, 2, 330, 2, 138, 2, 2730, 2, 6, 2, 870, 2, 14322, 2, 510, 2, 6, 2, 1919190, 2, 6, 2, 13530, 2, 1806, 2, 690, 2, 282, 2, 46410, 2, 66, 2, 1590, 2, 798, 2, 870, 2, 354, 2, 56786730, 2, 6, 2, 510, 2, 64722, 2, 30, 2, 4686, 2, 140100870, 2, 6, 2, 30, 2

Offset: 0

Paul Curtz, Apr 21 2010

These are also the denominators of a sequence generated by inverse binomial transform of a modified Bernoulli sequence described in (with numerators in) A176328.

Antti Karttunen, Table of n, a(n) for n = 0..4096

Cf. A141056, A160014.

read("transforms") ; evb := [1, 0, seq(bernoulli(n), n=2..50)] ; BINOMIALi(evb) ; apply(denom, %) ; # R. J. Mathar, Dec 01 2010
seq(denom((bernoulli(i,1)+bernoulli(i,2))/2),i=0..50); # Peter Luschny, Jun 17 2012

a[n_] := If[OddQ[n], 2, BernoulliB[n] // Denominator]; a[1] = 1; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 29 2012 *)
Join[{1,1},BernoulliB[Range[2,80]]/.(0->1/2)//Denominator] (* Harvey P. Dale, Dec 31 2018 *)

A176591(n) = { my(p=1); if(n>1, fordiv(n, d, my(r=d+1); if(isprime(r), p = p*r))); return(p); }; \\ Antti Karttunen, Dec 20 2018, after code in A141056

a(n) = A141056(n), n <> 1.
a(n) = A027760(n), n>1.
a(2n) = A002445(n), a(2n+1)= A040000(n).

More terms from Antti Karttunen, Dec 20 2018

A027762 Denominator of Sum_{p prime, p-1 divides 2*n} 1/p.

Original entry on oeis.org

6, 30, 42, 30, 66, 2730, 6, 510, 798, 330, 138, 2730, 6, 870, 14322, 510, 6, 1919190, 6, 13530, 1806, 690, 282, 46410, 66, 1590, 798, 870, 354, 56786730, 6, 510, 64722, 30, 4686, 140100870, 6, 30, 3318, 230010, 498, 3404310, 6, 61410, 272118, 1410, 6, 4501770

Offset: 1

N. J. A. Sloane

From the von Staudt-Clausen theorem, denominator(B_2n) = product of primes p such that (p-1)|2n.

Same as A002445.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Th. 118.
H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.

R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
Index entries for sequences related to Bernoulli numbers.

Cf. A027761, A006954.

Cf. A027760.

a(n)=
{
    my(s=0);
    forprime (p=2, 2*n+1, if( (2*n)%(p-1)==0, s+=1/p ) );
    return( denominator(s) );
}
/* Joerg Arndt, May 06 2012 */

a(n) = A002445(n). [Joerg Arndt, May 06 2012]

a(n) = A027760(2*n). - Ridouane Oudra, Feb 22 2022

A185633 For odd n, a(n) = 2; for even n, a(n) = denominator of Bernoulli(n)/n; The number 2 alternating with the elements of A006953.

Original entry on oeis.org

2, 12, 2, 120, 2, 252, 2, 240, 2, 132, 2, 32760, 2, 12, 2, 8160, 2, 14364, 2, 6600, 2, 276, 2, 65520, 2, 12, 2, 3480, 2, 85932, 2, 16320, 2, 12, 2, 69090840, 2, 12, 2, 541200, 2, 75852, 2, 2760, 2, 564, 2, 2227680, 2, 132, 2, 6360

Offset: 1

Paul Curtz, Dec 18 2012

nonn

There is an integer sequence b(n) = A053657(n)/2^(n-1) = 1, 1, 6, 6, 360, 360, 45360, 45360, 5443200, 5443200,... which consists of the duplicated entries of A202367.

The ratios of this sequence are b(n+1)/b(n) = 1, 6, 1, 60, 1, 126 .... = a(n)/2, which is a variant of A036283.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A006953, A007395 (bisections).

Cf. A006863, A027760, A067513, A322312, A322315 (rgs-transform).

A185633 := proc(n)
    A053657(n+1)/A053657(n) ;
end proc: # R. J. Mathar, Dec 19 2012

max = 52; s = Expand[Normal[Series[(-Log[1-x]/x)^z, {x, 0, max}]]]; a[n_, k_] := Denominator[Coefficient[s, x^n*z^k]]; A053657 = Prepend[LCM @@@ Table[a[n, k], {n, max}, {k, n}], 1]; a[n_] := A053657[[n+1]]/A053657[[n]]; Table[a[n], {n, 1, max}] (* Jean-François Alcover, Dec 20 2012 *)

A185633(n) = if(n%2,2,denominator(bernfrac(n)/(n))); \\ Antti Karttunen, Dec 03 2018

A185633(n) = { my(m=1); fordiv(n, d, if(isprime(1+d), m *= (1+d)^(1+valuation(n,1+d)))); (m); }; \\ Antti Karttunen, Dec 03 2018

a(n) = A053657(n+1)/A053657(n).
a(2*n) = 2*A036283(n).
From Antti Karttunen, Dec 03 2018: (Start)
a(n) = Product_{d|n} [(1+d)^(1+A286561(n,1+d))]^A010051(1+d) - after Peter J. Cameron's Mar 25 2002 comment in A006863.
A007947(a(n)) = A027760(n)
A001221(a(n)) = A067513(n).
A181819(a(n)) = A322312(n).
(End)

Name edited by Antti Karttunen, Dec 03 2018

A193267 The number 1 alternating with the numbers A006953/A002445 (which are integers).

Original entry on oeis.org

1, 2, 1, 4, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 16, 1, 18, 1, 20, 1, 2, 1, 24, 1, 2, 1, 4, 1, 6, 1, 32, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 4, 1, 2, 1, 48, 1, 2, 1, 4, 1, 54, 1, 8, 1, 2, 1, 60, 1, 2, 1, 64, 1, 6, 1, 4, 1, 2, 1, 72, 1, 2, 1, 4, 1, 6, 1, 80, 1, 2, 1, 84, 1, 2, 1, 8, 1, 18, 1, 4, 1, 2, 1, 96, 1, 2, 1, 100

Offset: 1

Paul Curtz, Dec 20 2012

nonn

a(n) is the product over all prime powers p^e, where p^e is the highest power of p dividing n and p-1 divides n. - Peter Luschny, Mar 12 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

using Nemo
function A193267(n) P = 1
    for (p, e) in factor(ZZ(n))
        divisible(ZZ(n), p - 1) && (P *= p^e) end
P end
[A193267(n) for n in 1:100] |> println # Peter Luschny, Mar 12 2018

[Denominator(Bernoulli(n)/n)/Denominator(Bernoulli(n)): n in [1..100]]; // Vincenzo Librandi, Mar 12 2018

with(numtheory); a := proc(n) divisors(n); map(i->i+1, %); select(isprime, %);
mul(k^padic[ordp](n,k),k=%) end: seq(a(n), n=1..100); # Peter Luschny, Mar 12 2018
# Alternatively:
A193267 := proc(n) local P, F, f, divides; divides := (a,b) -> is(irem(b,a) = 0):
P := 1; F := ifactors(n)[2]; for f in F do if divides(f[1]-1, n) then
P := P*f[1]^f[2] fi od; P end: seq(A193267(n), n=1..100); # Peter Luschny, Mar 12 2018

a[n_] := If[OddQ[n], 1, Denominator[ BernoulliB[n]/n ] / Denominator[ BernoulliB[n]] ]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 21 2012 *)

a(n+1) = A185633(n+1)/A027760(n+1).

a(n+1) = c(n+2)/c(n+1).

A225481 a(n) = product{ p primes <= n+1 such that p divides n+1 or p-1 divides n }.

Original entry on oeis.org

1, 2, 6, 2, 30, 6, 42, 2, 30, 10, 66, 6, 2730, 14, 30, 2, 510, 6, 798, 10, 2310, 22, 138, 6, 2730, 26, 6, 14, 870, 30, 14322, 2, 5610, 34, 210, 6, 1919190, 38, 78, 10, 13530, 42, 1806, 22, 690, 46, 282, 6, 46410, 10, 1122, 26, 1590, 6, 43890, 14, 16530, 58

Offset: 0

Peter Luschny, May 29 2013

nonn

a(n) is the product over the primes <= n+1 which satisfy the weak Clausen condition. The weak Clausen condition relaxes the Clausen condition (p-1)|n by logical disjunction with p|(n+1).

			a(20) = 2310 = 2*3*5*7*11, because {3, 7} are divisors of 21 and {2, 5, 11} meet the Clausen condition 'p-1 divides n'.

Peter Luschny, Table of n, a(n) for n = 0..100
Peter Luschny, Generalized Bernoulli numbers.

A027760, A160014, A226040.

Cf. A000040.

a225481 n = product [p | p <- takeWhile (<= n + 1) a000040_list,
                         mod n (p - 1) == 0 || mod (n + 1) p == 0]
-- Reinhard Zumkeller, Jun 10 2013

divides := (a, b) -> b mod a = 0; primes := n -> select(isprime, [$2..n]);
A225481 := n -> mul(k,k in select(p -> divides(p,n+1) or divides(p-1,n), primes(n+1))); seq(A225481(n), n = 0..57);

a[n_] := Product[ If[ Divisible[n+1, p] || Divisible[n, p-1], p, 1], {p, Prime /@ Range @ PrimePi[n+1]}]; Table[a[n], {n, 0, 57}] (* Jean-François Alcover, Jun 07 2013 *)

def divides(a, b): return b % a == 0
def A225481(n):
    return mul(filter(lambda p: divides(p,n+1) or divides(p-1,n), primes(n+2)))
[A225481(n) for n in (0..57)]

a(n) / A027760(n) = A226040(n) for n > 0.

A141047 Numerators of A091137(n)T(n,n)/n! where T(i,j)=Integral (x= i.. i+1) x(x-1)(x-2) .. *(x-j+1) dx.

Original entry on oeis.org

1, 3, 23, 55, 1901, 4277, 198721, 434241, 14097247, 30277247, 2132509567, 4527766399, 13064406523627, 27511554976875, 173233498598849, 362555126427073, 192996103681340479, 401972381695456831, 333374427829017307697, 691668239157222107697, 236387355420350878139797

Offset: 0

Paul Curtz, Jul 31 2008

nonn

Numerators of the main diagonal of the array A091137(j)*T(i,j)/j! where T(i,j)=Integral (x= i.. i+1) x*(x-1)*(x-2)* .. *(x-j+1) dx.
The reduced fractions of the array T(i,j) are shown in A140825, which also describes how the integrand is a generating function of Stirling numbers.
The sequence A027760 plays a role i) in relating to A091137 as described there and
ii) in a(n+1)-A027760(n+1)*a(n)= A002657(n+1), numerators of the diagonal T(n,n+1).

P. Curtz, Integration numerique des systemes differentiels a conditions initiales. Note 12, Centre de Calcul Scientifique de l' Armement, Arcueil (1969), p. 36.

Cf. A140811, A140825, A141045.

T := proc(i,j) local var,k ; var := x ; for k from 1 to j-1 do var := var*(x-k) ; od: int(var,x=i..i+1) ; simplify(A091137(j)*%/j!) ; numer(%) ; end:
A141047 := proc(n) T(n,n) ; end: for n from 0 to 20 do printf("%a,",A141047(n) ) ; od: # R. J. Mathar, Feb 23 2009

b[n_] := b[n] = (* A091137 *) If[n==0, 1, Product[d, {d, Select[Divisors[n] + 1, PrimeQ]}]*b[n-1]]; T[i_, j_] := Integrate[Product[x-k, {k, 0, j-1}], {x, i, i+1}]; a[n_] := b[n]*T[n, n]/n!; Table[a[n] // Numerator, {n, 0, 20}] (* Jean-François Alcover, Jan 10 2016 *)

a(n) = numerator( A091137(n)*T(n,n)/n!) where T(n,n) = sum_{k=0..n} A048994(n,k)*((n+1)^(k+1)-n^(k+1))/(k+1).

Edited and extended by R. J. Mathar, Feb 23 2009

A157411 a(n) = 30n^4 - 120n^3 + 120*n^2 - 19.

Original entry on oeis.org

-19, 11, -19, 251, 1901, 6731, 17261, 36731, 69101, 119051, 191981, 294011, 431981, 613451, 846701, 1140731, 1505261, 1950731, 2488301, 3129851, 3887981, 4776011, 5807981, 6998651, 8363501, 9918731, 11681261, 13668731, 15899501, 18392651

Offset: 0

Paul Curtz, Feb 28 2009

These are the numerators in column j=4 of the array in A140825 (reference p. 36).
The other columns in A140825 are represented by A000012, A005408, A140811 and A141530.
The link between these columns is given by the first differences: a(n+1) - a(n) = 30*A141530(n), where 30 = A027760(4) = A027760(3) = A027642(4) = A002445(2), then for j=3, A141530(n+1) - A141530(n) = A140070(2)*A140811(n).

P. Curtz, Integration numerique des systemes differentiels a conditions initiales, Centre de Calcul Scientifique de l'Armement, Note 12, Arcueil (1969).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

[30*n^4 - 120*n^3 + 120*n^2 - 19: n in [0..50]]; // Vincenzo Librandi, Aug 07 2011

Table[30n^4-120n^3+120n^2-19,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{-19,11,-19,251,1901},40] (* Harvey P. Dale, Mar 08 2015 *)

a(n)=30*n^4-120*n^3+120*n^2-19 \\ Charles R Greathouse IV, Oct 16 2015

a(n)= 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: (-19 + 106*x - 264*x^2 + 646*x^3 + 251*x^4)/(1-x)^5.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 720. Fourth differences are constant, 720.

Edited, one index corrected and extended by R. J. Mathar, Sep 17 2009

cp's OEIS Frontend

A067513 Number of divisors d of n such that d+1 is prime.

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A141459 a(n) = Product_{p-1 divides n} p, where p is an odd prime.

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A141417 (-1)^(n+1)*A091137(n)*a(0,n), where a(i,j) = Integral_{x=i..i+1} x*(x-1)*(x-2)*...*(x-j+1)/j! dx.

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A176591 Bernoulli denominators A141056(n), with the exception a(1)=1.

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A027762 Denominator of Sum_{p prime, p-1 divides 2*n} 1/p.

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A185633 For odd n, a(n) = 2; for even n, a(n) = denominator of Bernoulli(n)/n; The number 2 alternating with the elements of A006953.

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A141417 (-1)^(n+1)A091137(n)a(0,n), where a(i,j) = Integral_{x=i..i+1} x(x-1)(x-2)...(x-j+1)/j! dx.

A141047 Numerators of A091137(n)T(n,n)/n! where T(i,j)=Integral (x= i.. i+1) x(x-1)(x-2) .. *(x-j+1) dx.

A157411 a(n) = 30n^4 - 120n^3 + 120*n^2 - 19.