A002471 Number of partitions of n into a prime and a square. (Formerly M0073 N0025)

0, 1, 2, 1, 1, 2, 2, 1, 1, 0, 3, 2, 1, 2, 1, 1, 2, 2, 2, 2, 2, 1, 3, 1, 0, 1, 3, 2, 2, 2, 1, 3, 2, 0, 2, 1, 1, 4, 2, 1, 3, 2, 2, 2, 2, 1, 4, 2, 1, 1, 2, 2, 3, 3, 1, 3, 2, 0, 3, 2, 1, 4, 2, 0, 2, 3, 3, 4, 2, 1, 3, 3, 2, 1, 3, 1, 4, 2, 2, 3, 1

Offset

1,3

Comments

a(A014090(n))=0; a(A014089(n))>0; a(A143989(n))=1. - Reinhard Zumkeller, Sep 07 2008

References

Selmer, Ernst S.; Eine numerische Untersuchung ueber die Darstellung der natuerlichen Zahlen als Summe einer Primzahl und einer Quadratzahl. Arch. Math. Naturvid. 47, (1943). no. 2, 21-39.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Formula

G.f.: (Sum_{i>=0} x^(i^2))*(Sum_{j>=1} x^prime(j)). - Ilya Gutkovskiy, Feb 07 2017

Maple

n->nops(select(isprime, [ seq(n-i^2, i=0..trunc(sqrt(n))) ])):
with(combstruct): specM0073 := {N=Prod(P, S), P=Set(Z, card>=1), S=Set(Z, card>=0)}: `combstruct/compile`(specM0073, unlabeled): `combstruct/Count`[ specM0073, unlabeled ][ P ] := proc(p) option remember; if isprime(p) then 1 else 0 fi end: `combstruct/Count`[ specM0073, unlabeled ][ S ] := proc(p) option remember; if type(sqrt(p), integer) then 1 else 0 fi end: M0073 := n->count([ N, specM0073, unlabeled ], size=n):

Mathematica

a[n_] := Count[p /. {ToRules[ Reduce[ p > 1 && q >= 0 && n == p + q^2, {p, q}, Integers]]}, _?PrimeQ]; Table[ a[n], {n, 1, 81}] (* from Jean-François Alcover, Sep 30 2011 *)

Prog

(Haskell)
a002471 n = sum $ map (a010051 . (n -)) $ takeWhile (< n) a000290_list
-- Reinhard Zumkeller, Jul 23 2013, Sep 30 2011
(PARI) a(n)=if(n>1, sum(k=0, sqrtint(n-2), isprime(n-k^2)), 0) \\ Charles R Greathouse IV, Feb 08 2017

Crossrefs

Cf. A064272, A010051, A000290.

Sequence in context: A242998 A140885 A064286 * A218622 A337621 A091243

Adjacent sequences: A002468 A002469 A002470 * A002472 A002473 A002474

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

Sequence corrected by Paul Zimmermann, Mar 15 1996

Status

approved