A030015 -id:A030015 - cp's OEIS frontend

A300632 Expansion of e.g.f. exp(x + Sum_{k>=2} prime(k-1)*x^k/k!).

1, 1, 3, 10, 42, 203, 1119, 6839, 45895, 334142, 2619052, 21946647, 195537777, 1843619725, 18321431155, 191242913022, 2090436115146, 23864653888881, 283865214366771, 3510656353388517, 45056394441558593, 599057016471131604, 8238406603745152620, 117020080948487107289

Offset: 0

Ilya Gutkovskiy, Mar 10 2018

nonn

Exponential transform of A008578.

			E.g.f.: A(x) = 1 + x/1! + 3*x^2/2! + 10*x^3/3! + 42*x^4/4! + 203*x^5/5! + 1119*x^6/6! + 6839*x^7/7! + ..

Alois P. Heinz, Table of n, a(n) for n = 0..534
N. J. A. Sloane, Transforms

Cf. A000040, A007446, A008578, A023626, A030011, A030012, A030013, A030014, A030015, A030016.

a:= proc(n) option remember; (p-> `if`(n=0, 1, add(a(n-j)*p(j)*
      binomial(n-1, j-1), j=1..n)))(t-> `if`(t=1, 1, ithprime(t-1)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 10 2018

nmax = 23; CoefficientList[Series[Exp[x + Sum[Prime[k - 1] x^k/k!, {k, 2, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
p[1] = 1; p[n_] := p[n] = Prime[n - 1]; a[n_] := a[n] = Sum[p[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]

E.g.f.: exp(Sum_{k>=1} A008578(k)*x^k/k!).

A300662 Expansion of 1/(1 - x - Sum_{k>=2} prime(k-1)*x^k).

Original entry on oeis.org

1, 1, 3, 8, 22, 59, 160, 429, 1155, 3105, 8354, 22474, 60457, 162636, 437509, 1176941, 3166097, 8517138, 22912002, 61635707, 165806564, 446037175, 1199887133, 3227823181, 8683185454, 23358686444, 62837334885, 169039070970, 454732963567, 1223279724439, 3290751724917

Offset: 0

Ilya Gutkovskiy, Mar 10 2018

nonn

Invert transform of A008578.

Alois P. Heinz, Table of n, a(n) for n = 0..2327
N. J. A. Sloane, Transforms

Cf. A000040, A008578, A023626, A030011, A030012, A030013, A030014, A030015, A030016, A030017, A030018, A292744, A300632.

a:= proc(n) option remember; `if`(n=0, 1, add(
     `if`(j=1, 1, ithprime(j-1))*a(n-j), j=1..n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Mar 10 2018

nmax = 30; CoefficientList[Series[1/(1 - x - Sum[Prime[k - 1] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]
p[1] = 1; p[n_] := p[n] = Prime[n - 1]; a[n_] := a[n] = Sum[p[k] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 30}]

G.f.: 1/(1 - Sum_{k>=1} A008578(k)*x^k).

A050513 a(n) = (-1)^n * Sum_{i=0..n} binomial(n+1,i+1)*prime(i+1).

Original entry on oeis.org

2, -7, 20, -53, 136, -341, 836, -2005, 4712, -10881, 24770, -55763, 124464, -275933, 608282, -1334119, 2911870, -6325091, 13674120, -29425307, 63042232, -134517425, 285984130, -606056545, 1280778342, -2700105565, 5680099084, -11925792491

Offset: 0

N. J. A. Sloane, Dec 28 1999

sign

Cf. A030015.

Table[(-1)^(n-1)*Sum[Binomial[n, k]*Prime[k], {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Mar 01 2016 *)

A333176 a(n) = Sum_{k=1..n} (binomial(n,k) mod 2) * prime(k).

Original entry on oeis.org

2, 3, 10, 7, 20, 23, 58, 19, 44, 51, 112, 63, 140, 151, 328, 53, 114, 117, 250, 131, 276, 287, 604, 161, 342, 355, 742, 383, 798, 825, 1720, 131, 270, 273, 566, 289, 596, 607, 1252, 323, 664, 675, 1392, 711, 1458, 1481, 3046, 407, 832, 839, 1718, 875, 1782

Offset: 1

Ilya Gutkovskiy, Mar 10 2020

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A007443, A007504, A010060, A030015, A050513.

N:= 200: # for a(1) .. a(N)
P:= [seq(ithprime(i),i=1..N)]:
B:= [1,1]: R:= 2:
for n from 2 to N do
  B:= [1,op(B[2..-1]+B[1..-2] mod 2),1];
  R:= R, convert(P[select(t -> B[t+1] = 1,[$1..n])],`+`);
od:
R; # Robert Israel, Jan 29 2025

Table[Sum[Mod[Binomial[n, k], 2] Prime[k], {k, 1, n}], {n, 1, 53}]

a(n) = sum(k=1, n, if (binomial(n, k) % 2, prime(k))); \\ Michel Marcus, Mar 10 2020

from sympy import prime
def A333176(n): return sum(prime(k) for k in range(1,n+1) if not ~n&k) # Chai Wah Wu, Jul 22 2025

Sum_{k=1..n} (-1)^A010060(n-k) * (binomial(n,k) mod 2) * a(k) = prime(n).

cp's OEIS Frontend

A300632 Expansion of e.g.f. exp(x + Sum_{k>=2} prime(k-1)*x^k/k!).

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A300662 Expansion of 1/(1 - x - Sum_{k>=2} prime(k-1)*x^k).

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A050513 a(n) = (-1)^n * Sum_{i=0..n} binomial(n+1,i+1)*prime(i+1).

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A333176 a(n) = Sum_{k=1..n} (binomial(n,k) mod 2) * prime(k).

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