A002471 Number of partitions of n into a prime and a square.
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
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
Programs
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Haskell
a002471 n = sum $ map (a010051 . (n -)) $ takeWhile (< n) a000290_list -- Reinhard Zumkeller, Jul 23 2013, Sep 30 2011
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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 *) -
PARI
a(n)=if(n>1, sum(k=0,sqrtint(n-2), isprime(n-k^2)), 0) \\ Charles R Greathouse IV, Feb 08 2017
Formula
G.f.: (Sum_{i>=0} x^(i^2))*(Sum_{j>=1} x^prime(j)). - Ilya Gutkovskiy, Feb 07 2017
Extensions
Sequence corrected by Paul Zimmermann, Mar 15 1996
Comments