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%I M0073 N0025 #35 Jan 29 2022 01:06:25
%S 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,
%T 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,
%U 2,1,3,3,2,1,3,1,4,2,2,3,1
%N Number of partitions of n into a prime and a square.
%C a(A014090(n))=0; a(A014089(n))>0; a(A143989(n))=1. - _Reinhard Zumkeller_, Sep 07 2008
%D 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.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A002471/b002471.txt">Table of n, a(n) for n = 1..10000</a>
%F G.f.: (Sum_{i>=0} x^(i^2))*(Sum_{j>=1} x^prime(j)). - _Ilya Gutkovskiy_, Feb 07 2017
%p n->nops(select(isprime,[ seq(n-i^2,i=0..trunc(sqrt(n))) ])):
%p 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):
%t 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 *)
%o (Haskell)
%o a002471 n = sum $ map (a010051 . (n -)) $ takeWhile (< n) a000290_list
%o -- _Reinhard Zumkeller_, Jul 23 2013, Sep 30 2011
%o (PARI) a(n)=if(n>1, sum(k=0,sqrtint(n-2), isprime(n-k^2)), 0) \\ _Charles R Greathouse IV_, Feb 08 2017
%Y Cf. A064272, A010051, A000290.
%K nonn,nice
%O 1,3
%A _N. J. A. Sloane_
%E Sequence corrected by _Paul Zimmermann_, Mar 15 1996
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