A276420 Number of palindromic compositions of n into prime parts.

1, 0, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 5, 6, 8, 7, 12, 14, 16, 17, 26, 27, 36, 40, 55, 56, 81, 88, 118, 124, 177, 189, 257, 275, 384, 404, 564, 605, 833, 880, 1233, 1314, 1813, 1929, 2685, 2850, 3956, 4215, 5845, 6203, 8629, 9185, 12731, 13531, 18807, 19994

Offset: 0

Author

Emeric Deutsch, Sep 02 2016

Keywords

• nonn

Examples

			a(9) = 2 because we have [2,5,2] and [3,3,3].
a(12) = 5 because we have [5,2,5], [2,3,2,3,2], [3,2,2,2,3], [3,3,3,3], and [2,2,2,2,2,2].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

Crossrefs

Cf. A016116, A023360.

Programs

Maple

F := sum(z^ithprime(j),j=1..90): F2:=sum(z^(2*ithprime(j)),j=1..90): g:= (1+F)/(1-F2): gser:=series(g,z=0,55): seq(coeff(gser,z,n),n=0..50);
# second Maple program:
a:= proc(n) option remember; `if`(n=0 or isprime(n), 1, 0)+
      add(`if`(isprime(j), a(n-2*j), 0), j=1..n/2)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 02 2016

Mathematica

Table[Count[Flatten[Map[Permutations, Select[IntegerPartitions@ n, Times @@ Boole@ Map[PrimeQ, #] > 0 &]], 1], w_ /; Reverse@ w == w], {n, 0, 40}] (* Michael De Vlieger, Sep 02 2016 *)
a[n_] := a[n] = If[n == 0 || PrimeQ[n], 1, 0] +
     Sum[If[PrimeQ[j], a[n - 2*j], 0], {j, 1, n/2}];
a /@ Range[0, 60] (* Jean-François Alcover, Jun 01 2021, after Alois P. Heinz *)

Formula

G.f.: g(z) = (1+F(z))/(1-F(z^2)), where F(z) = Sum_{p prime} z^p = z^2 + z^3 + z^5 + z^7 + ... .