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A002471
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Number of partitions of n into a prime and a square.
(Formerly M0073 N0025)
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10
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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
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OFFSET
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1,3
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COMMENTS
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a(A014090(n))=0; a(A014089(n))>0; a(A143989(n))=1. - Reinhard Zumkeller, Sep 07 2008
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REFERENCES
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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).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
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FORMULA
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G.f.: (Sum_{i>=0} x^(i^2))*(Sum_{j>=1} x^prime(j)). - Ilya Gutkovskiy, Feb 07 2017
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MAPLE
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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):
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A064272, A010051, A000290.
Sequence in context: A242998 A140885 A064286 * A218622 A337621 A091243
Adjacent sequences: A002468 A002469 A002470 * A002472 A002473 A002474
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Sequence corrected by Paul Zimmermann, Mar 15 1996
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STATUS
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approved
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