A309955 -id:A309955 - cp's OEIS frontend

A340990 a(n) is the (2n)-th term of the n-fold self-convolution of the primes.

1, 3, 29, 291, 3121, 34123, 379853, 4280251, 48681569, 557686227, 6425630909, 74384480019, 864461820049, 10079577033243, 117859582680813, 1381492094548651, 16227770995740865, 190979248798795427, 2251327736286726749, 26579050506578504195, 314212180691846338801

Offset: 0

Alois P. Heinz, Feb 01 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..924

Cf. A000040, A309955, A340991.

b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, ithprime(n+1),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..23);

b[n_, k_] := b[n, k] = If[k == 0, 1, If[k == 1, Prime[n + 1], With[{q = Quotient[k, 2]}, Sum[b[j, q] b[n - j, k - q], {j, 0, n}]]]];
a[n_] := b[n, n];
a /@ Range[0, 23] (* Jean-François Alcover, Feb 04 2021, after Alois P. Heinz *)

a(n) = [x^(2n)] (Sum_{j>=1} prime(j)*x^j)^n.

a(n) = A340991(2n,n).

A309950 G.f.: Product_{j>=1} (1 + p(x^j)), where p(x) is the g.f. of A000040.

Original entry on oeis.org

1, 2, 5, 11, 22, 43, 78, 140, 238, 405, 665, 1077, 1710, 2685, 4140, 6336, 9551, 14280, 21117, 30994, 45051, 65046, 93170, 132600, 187439, 263449, 367999, 511409, 706833, 972257, 1330929, 1813846, 2461090, 3325803, 4476276, 6002036, 8018216, 10674307, 14161656

Offset: 0

Alois P. Heinz, Aug 24 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000040, A006171, A008578, A162891, A309955.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1, ithprime(n),
      add(b(j, 1)*(t-> b(t, min(t, i-1)))(n-i*j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);

b[n_, i_] := b[n, i] = If[n==0, 1, If[i==1,
   Prime[n], Sum[b[j, 1]*Function[t,
   b[t, Min[t, i-1]]][n-i*j], {j, 0, n/i}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Sep 15 2022, after Alois P. Heinz *)

cp's OEIS Frontend

A340990 a(n) is the (2n)-th term of the n-fold self-convolution of the primes.

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