A225481 -id:A225481 - cp's OEIS frontend

A226157 a(n) = BS2(n) * W(n) where BS2 = sum_{k=0..n} ((-1)^k*k!/(k+1)) S_{2}(n, k) and S_{2}(n, k) are the Stirling-Frobenius subset numbers A039755(n, k). W(n) = product{p primes <= n+1 such that p divides n+1 or p-1 divides n} = A225481(n).

1, 1, -2, -2, 14, 33, -62, -132, 254, 14585, -5110, -313266, 2828954, 38669001, -573370, -404801672, 237036478, 117650567067, -11499383114, -24255028327410, 1281647882998, 8203584532193105, -3584085584926, -418397193140056356, 3965530936622474, 405233976502715850633

Offset: 0

Peter Luschny, May 30 2013

a(n)/A225481(n) is case m = 2 of the scaled generalized Bernoulli numbers defined as Sum_{k=0..n} ((-1)^k*k!/(k+1)) S_{m}(n, k) where S_{m}(n, k) are Stirling-Frobenius subset numbers. A225481(n) can be seen as an analog of the Clausen numbers A141056(n).

			The numerators of 1/1, 1/2, -2/6, -2/2, 14/30, 33/6, -62/42, -132/2, 254/30, 14585/10, -5110/66, ...(the denominators are A225481(n)).

Peter Luschny, Stirling-Frobenius numbers
Peter Luschny, Generalized Bernoulli numbers.

Cf. A225481, A226156.

EulerianNumber[n_, k_, m_] := EulerianNumber[n, k, m] = If[n == 0, If[k == 0, 1 , 0], (m*(n-k) + m - 1)*EulerianNumber[n-1, k-1, m] + (m*k + 1)* EulerianNumber[n-1, k, m]];
BS[n_, m_] := Sum[Sum[EulerianNumber[n, j, m]*Binomial[j, n-k], {j, 0, n}]/ ((-m)^k*(k+1)), {k, 0, n}]
a[n_] := Product[If[Divisible[n+1, p] || Divisible[n, p-1], p, 1], {p, Prime /@ Range @ PrimePi[n+1]}] * BS[n, 2];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 27 2019, from Sage *)

@CachedFunction
def EulerianNumber(n, k, m) :   # The Eulerian numbers
    if n == 0: return 1 if k == 0 else 0
    return ((m*(n-k)+m-1)*EulerianNumber(n-1, k-1, m) +
           (m*k+1)*EulerianNumber(n-1, k, m))
@CachedFunction
def BS(n, m):   # The generalized scaled Bernoulli numbers
    return (add(add(EulerianNumber(n, j, m)*binomial(j, n - k)
           for j in (0..n))/((-m)^k*(k+1)) for k in (0..n)))
def A226157(n):   # The numerators of BS(n, 2) relative to A225481
    C = mul(filter(lambda p: ((n+1)%p == 0) or (n%(p-1) == 0), primes(n+2)))
    return C*BS(n, 2)
[A226157(n) for n in (0..25)]

A226156 a(n) = BS(n) * W(n) where BS = Sum_{k=0..n} ((-1)^k*k!/(k+1)) S(n, k) and S(n, k) the Stirling subset numbers A048993(n, k). W(n) = Product_{ p primes <= n+1 such that p divides n+1 or p-1 divides n } = A225481(n).

Original entry on oeis.org

1, -1, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 35, 0, -3617, 0, 43867, 0, -1222277, 0, 854513, 0, -236364091, 0, 8553103, 0, -23749461029, 0, 8615841276005, 0, -84802531453387, 0, 90219075042845, 0, -26315271553053477373, 0, 38089920879940267

Offset: 0

Peter Luschny, May 30 2013

a(n)/A225481(n) is a representation of the Bernoulli numbers. This is case m = 1 of the scaled generalized Bernoulli numbers defined as Sum_{k=0..n} ((-1)^k*k!/(k+1)) S_{m}(n,k) where S_{m}(n,k) are generalized Stirling subset numbers. A225481(n) can be seen as an analog of the Clausen numbers A141056(n). Reduced to lowest terms a(n)/A225481(n) becomes A027641(n)/A027642(n).

			The numerators of 1/1, -1/2, 1/6, 0/2, -1/30, 0/6, 1/42, 0/2, -1/30, 0/10, 5/66, 0/6, -691/2730, 0/14, 35/30, 0/2, -3617/510, 0/6, 43867/798, ... (the denominators are A225481(n)).

Peter Luschny, Stirling-Frobenius numbers
Peter Luschny, Generalized Bernoulli numbers.

Cf. A225481, A226157.

BS[n_] := Sum[((-1)^k*k!/(k + 1)) StirlingS2[n, k], {k, 0, n}];
W[n_] := Product[If[Divisible[n + 1, p] || Divisible[n, p - 1], p, 1], {p, Prime /@ Range[PrimePi[n + 1]]}];
a[n_] := BS[n] W[n];
Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Jul 08 2019 *)

@CachedFunction
def EulerianNumber(n, k, m) :   # -- The Eulerian numbers --
    if n == 0: return 1 if k == 0 else 0
    return (m*(n-k)+m-1)*EulerianNumber(n-1, k-1, m) + \
           (m*k+1)*EulerianNumber(n-1, k, m)
@CachedFunction
def SF_BS(n, m):   # -- The scaled Stirling-Frobenius Bernoulli numbers --
    return add(add(EulerianNumber(n, j, m)*binomial(j, n - k) \
           for j in (0..n))/((-m)^k*(k+1)) for k in (0..n))
def A226156(n):    # -- The numerators of SF_BS(n, 1) relative to A225481 --
    C = mul(filter(lambda p: ((n+1)%p == 0) or (n%(p-1) == 0), primes(n+2)))
    return C*SF_BS(n, 1)
[A226156(n) for n in (0..25)]

A226040 a(n) = product{ p prime such that p divides n + 1 and p - 1 does not divide n }.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 7, 5, 1, 1, 3, 1, 5, 7, 11, 1, 3, 1, 13, 1, 7, 1, 15, 1, 1, 11, 17, 35, 3, 1, 19, 13, 5, 1, 21, 1, 11, 1, 23, 1, 3, 1, 5, 17, 13, 1, 3, 55, 7, 19, 29, 1, 15, 1, 31, 7, 1, 13, 33, 1, 17, 23, 35, 1, 3, 1, 37, 5, 19, 77, 39

Offset: 0

Peter Luschny, May 26 2013

nonn

			a(41) = 21 = 3*7 = product({2,3,7} setminus {2}).

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Peter Luschny, Generalized Bernoulli numbers.

Cf. A160014, A226038, A226039, A225481, A141056.

s:= (p, n) -> ((n+1) mod p = 0) and (n mod (p-1) <> 0);
A226040 := n -> mul(z, z = select(p->s(p,n), select('isprime', [$2..n])));
seq(A226040(n), n=0..77);

a[n_] := Times @@ Select[ FactorInteger[n+1][[All, 1]], !Divisible[n, #-1] &]; a[0] = 1; Table[a[n], {n, 0, 77}] (* Jean-François Alcover, Jun 27 2013, after Maple *)

a(n)=my(f=factor(n+1)[,1],s=1);prod(i=1,#f,if(n%(f[i]-1),f[i],1)) \\ Charles R Greathouse IV, Jun 27 2013

def A226040(n):
    F = filter(lambda p: ((n+1) % p == 0) and (n % (p-1)), primes(n))
    return mul(F)
[A226040(n) for n in (0..77)]

a(n) = A225481(n) / A141056(n).

A226038 Numbers k such that there are no primes p which divide k+1 and p-1 does not divide k.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 10, 12, 15, 16, 18, 22, 24, 26, 28, 30, 31, 36, 40, 42, 44, 46, 48, 52, 58, 60, 63, 66, 70, 72, 78, 80, 82, 88, 96, 100, 102, 106, 108, 112, 120, 124, 126, 127, 130, 136, 138, 148, 150, 156, 162, 166, 168, 172, 178, 180, 190, 192, 196, 198

Offset: 1

Peter Luschny, May 27 2013

These are the numbers which satisfy the weak Clausen condition but not the Clausen condition.

			A counterexample is n = 14. 5 divides 15 but 4 does not divide 14.

Ivan Neretin, Table of n, a(n) for n = 1..10000
Peter Luschny, Generalized Bernoulli numbers.

Cf. A226039, A226040, A225481.

s := (p,n) -> ((n+1) mod p = 0) and (n mod (p-1) <> 0);
F := n -> select(p -> s(p,n), select('isprime', [$2..n]));
A226038_list := n -> select(k -> [] = F(k), [$0..n]);
A226038_list(200);

s[p_, n_] := Mod[n+1, p] == 0 && Mod[n, p-1] != 0; f[n_] := Select[ Select[ Range[n], PrimeQ], s[#, n] &]; A226038 = Select[ Range[0, 200], f[#] == {} &] (* Jean-François Alcover, Jul 29 2013, after Maple *)
Join[{0}, Select[Range[200], And @@ Divisible[#, FactorInteger[# + 1][[All, 1]] - 1] &]] (* Ivan Neretin, Aug 04 2016 *)

def F(n): return filter(lambda p: ((n+1) % p == 0) and (n % (p-1) != 0), primes(n))
def A226038_list(n): return list(filter(lambda n: not list(F(n)), (0..n)))
A226038_list(200)

cp's OEIS Frontend

A226157 a(n) = BS2(n) * W(n) where BS2 = sum_{k=0..n} ((-1)^k*k!/(k+1)) S_{2}(n, k) and S_{2}(n, k) are the Stirling-Frobenius subset numbers A039755(n, k). W(n) = product{p primes <= n+1 such that p divides n+1 or p-1 divides n} = A225481(n).

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A226156 a(n) = BS(n) * W(n) where BS = Sum_{k=0..n} ((-1)^k*k!/(k+1)) S(n, k) and S(n, k) the Stirling subset numbers A048993(n, k). W(n) = Product_{ p primes <= n+1 such that p divides n+1 or p-1 divides n } = A225481(n).

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A226040 a(n) = product{ p prime such that p divides n + 1 and p - 1 does not divide n }.

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A226038 Numbers k such that there are no primes p which divide k+1 and p-1 does not divide k.

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Maple

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Sage