A353162 -id:A353162 - cp's OEIS frontend

A353163 Expansion of e.g.f. exp(Sum_{p prime} x^p / (p-1)!).

1, 0, 2, 3, 12, 65, 210, 1477, 7560, 45864, 338310, 2176031, 17657640, 139280869, 1150004856, 10572694860, 94834041120, 931995595457, 9384294360168, 96974005210273, 1066116104926500, 11838081891521760, 137785102884102366, 1652584041236345933

Offset: 0

Seiichi Manyama, Apr 28 2022

nonn

Cf. A000040, A000248, A190476, A353162.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, isprime(k)*x^k/(k-1)!))))

a(n) = if(n==0, 1, sum(k=1, n, isprime(k)*k*binomial(n-1, k-1)*a(n-k)));

a(0) = 1; a(n) = Sum_{p<=n, p prime} p * binomial(n-1,p-1) * a(n-p).

A353164 Expansion of 1/(1 - Sum_{p prime} p * x^p).

Original entry on oeis.org

1, 0, 2, 3, 4, 17, 17, 63, 100, 211, 495, 846, 2057, 3831, 8181, 17078, 33788, 72705, 144801, 303452, 623115, 1274365, 2652052, 5408046, 11207927, 23020231, 47378495, 97774736, 200819019, 414365805, 852285510, 1755453858, 3616014678, 7441523271, 15332278869

Offset: 0

Seiichi Manyama, Apr 28 2022

nonn

Cf. A000040, A023360, A088305, A353162, A353165.

my(N=40, x='x+O('x^N)); Vec(1/(1-sum(k=1, N, isprime(k)*k*x^k)))

a(n) = if(n==0, 1, sum(k=1, n, isprime(k)*k*a(n-k)));

a(0) = 1; a(n) = Sum_{p<=n, p prime} p * a(n-p).

A352914 Expansion of e.g.f. exp(Sum_{k>=1} prime(k)*x^k).

Original entry on oeis.org

1, 2, 10, 74, 676, 7592, 97024, 1416200, 23015248, 412777952, 8090869984, 171435904928, 3908548404160, 95264270043776, 2470715015425024, 67913132377486208, 1971038886452490496, 60212661838223997440, 1930529043247940342272, 64801071784954698480128

Offset: 0

Seiichi Manyama, Apr 28 2022

nonn

Cf. A000040, A007446, A033286, A353162, A353166.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      ithprime(j)*j*binomial(n, j)*j!, j=1..n)/n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 28 2022

a[0] = 1; a[n_] := a[n] = (n-1)! Sum[k Prime[k] a[n-k]/(n-k)!, {k, 1, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 28 2022 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, prime(k)*x^k))))

a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*prime(k)*a(n-k)/(n-k)!));

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} A033286(k) * a(n-k)/(n-k)!.

cp's OEIS Frontend

A353163 Expansion of e.g.f. exp(Sum_{p prime} x^p / (p-1)!).

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A353164 Expansion of 1/(1 - Sum_{p prime} p * x^p).

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A352914 Expansion of e.g.f. exp(Sum_{k>=1} prime(k)*x^k).

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