A014089 -id:A014089 - cp's OEIS frontend

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

N. J. A. Sloane

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

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).

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

Cf. A064272, A010051, A000290.

a002471 n = sum $ map (a010051 . (n -)) $ takeWhile (< n) a000290_list
-- Reinhard Zumkeller, Jul 23 2013, Sep 30 2011

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):

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 *)

a(n)=if(n>1, sum(k=0,sqrtint(n-2), isprime(n-k^2)), 0) \\ Charles R Greathouse IV, Feb 08 2017

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

Sequence corrected by Paul Zimmermann, Mar 15 1996

A014090 Numbers that are not the sum of a square and a prime.

Original entry on oeis.org

1, 10, 25, 34, 58, 64, 85, 91, 121, 130, 169, 196, 214, 226, 289, 324, 370, 400, 526, 529, 625, 676, 706, 730, 771, 784, 841, 1024, 1089, 1225, 1255, 1351, 1414, 1444, 1521, 1681, 1849, 1906, 1936, 2116, 2209, 2304, 2500, 2809, 2986, 3136, 3364, 3481, 3600

Offset: 1

N. J. A. Sloane, R. K. Guy

Sequence is infinite: if 2n-1 is composite then n^2 is in the sequence. (Proof: If n^2 = x^2 + p with p prime, then p = (n-x)(n+x), so n-x=1 and n+x=p. Hence 2n-1=p is prime, not composite.) - Dean Hickerson, Nov 27 2002
21679 is the last known nonsquare in this sequence. See A020495. - T. D. Noe, Aug 05 2006
A002471(a(n))=0; complement of A014089. - Reinhard Zumkeller, Sep 07 2008
There are no prime numbers in this sequence because at the very least they can be represented as p + 0^2. - Alonso del Arte, May 26 2012
Number of terms <10^k,k=0..8: 1, 8, 27, 75, 223, 719, 2361, 7759, ..., . - Robert G. Wilson v, May 26 2012
So far there are only 21 terms which are not squares and they are the terms of A020495. Those that are squares, their square roots are members of A104275. - Robert G. Wilson v, May 26 2012

			From _Alonso del Arte_, May 26 2012: (Start)
10 is in the sequence because none of 10 - p_i are square (8, 7, 5, 3) and none of 10 - b^2 are prime (10, 9, 6, 1); i goes from 1 to pi(10) or b goes from 0 to floor(sqrt(10)).
11 is not in the sequence because it can be represented as 3^2 + 2 or 0^2 + 11. (End)

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 115 terms from T. D. Noe)

Cf. A020495, A104275.

Cf. A064233 (does not allow 0^2).

t={}; Do[k=0; While[k^2=n, AppendTo[t,n]], {n,25000}]; t (* T. D. Noe, Aug 05 2006 *)
max = 5000; Complement[Range[max], Flatten[Table[Prime[p] + b^2, {p, PrimePi[max]}, {b, 0, Ceiling[Sqrt[max]]}]]] (* Alonso del Arte, May 26 2012 *)
fQ[n_] := Block[{j = Sqrt[n], k}, If[ IntegerQ[j] && !PrimeQ[2j - 1], True, k = Floor[j]; While[k > -1 && !PrimeQ[n - k^2], k--]; If[k == -1, True, False]]]; Select[ Range[3600], fQ] (* Robert G. Wilson v, May 26 2012 *)

A143989 Numbers having a unique representation as a sum of a prime and a square.

Original entry on oeis.org

2, 4, 5, 8, 9, 13, 15, 16, 22, 24, 26, 31, 36, 37, 40, 46, 49, 50, 55, 61, 70, 74, 76, 81, 82, 94, 99, 100, 106, 115, 120, 127, 133, 136, 142, 144, 145, 154, 159, 166, 170, 178, 184, 202, 205, 219, 221, 225, 235, 246, 250, 253, 256, 265, 268, 274, 295, 298, 301, 310

Offset: 1

Reinhard Zumkeller, Sep 07 2008

nonn

A002471(a(n)) = 1; subsequence of A014089.

R. Zumkeller, Table of n, a(n) for n = 1..256

A365127 Numbers that are the sum of a prime number and a fourth power of a nonnegative integer.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 17, 18, 19, 20, 21, 23, 24, 27, 29, 30, 31, 32, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 53, 54, 57, 59, 60, 61, 62, 63, 67, 68, 69, 71, 72, 73, 74, 75, 77, 79, 80, 83, 84, 86, 87, 88, 89, 90, 92, 94, 95, 97, 98, 99, 100

Offset: 1

Ilya Gutkovskiy, Aug 24 2023

nonn

Cf. A014089, A307646, A365126, A365166, A365168.

nmax = 100; f[x_] := Sum[x^Prime[i], {i, 1, nmax}] Sum[x^j^4, {j, 0, nmax^(1/4)}]; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, nmax}]]

A307646 Numbers that are the sum of a prime number and a nonnegative cube.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 23, 24, 25, 27, 29, 30, 31, 32, 34, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 79, 80, 81, 83, 84, 86, 87, 88, 89, 90, 91

Offset: 1

Ilya Gutkovskiy, Apr 19 2019

nonn

Cf. A000040, A000578, A014089, A045911, A302354, A307647.

Exponents in expansion of (Sum_{i>=1} x^prime(i)) * (Sum_{j>=0} x^(j^3)).

A111908 Numbers that are not the sum of a prime and a nonzero triangular number.

Original entry on oeis.org

1, 2, 7, 36, 61, 105, 171, 210, 211, 216, 325, 351, 406, 528, 561, 630, 741, 780, 990, 1081, 1176, 1275, 1596, 1711, 1830, 1953, 2016, 2145, 2346, 2628, 2775, 3003, 3081, 3240, 3321, 3655, 3741, 3916, 4278, 4371, 4465, 4560, 4851, 5253, 5460, 5565, 5886

Offset: 1

Stefan Steinerberger, Nov 25 2005

nonn

Can anybody prove or disprove a(n) = O(n^c) for some constant c?

Jonathan Vos Post has observed that every term in A076768 also occurs in this sequence.

			7 = 1+6 = 2+5 = 3+4; 7 is in the sequence because there is no sum where the first element is a prime and the second one a triangular number.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A064233, A000217, A076768, A020756, A072386, A014089, A014090, A000404, A001481, A002654, A100570, A101181.

lim=6000;plim=PrimePi[lim];tlim=Ceiling[Sqrt[2lim]];Complement[Range[lim],Union[Flatten[Table[Prime[i]+PolygonalNumber[j],{i,plim},{j,tlim}]]]] (* James C. McMahon, Jun 04 2024 *)

a(47) and offset corrected by Donovan Johnson, Feb 09 2013

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A002471 Number of partitions of n into a prime and a square.

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A014090 Numbers that are not the sum of a square and a prime.

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A143989 Numbers having a unique representation as a sum of a prime and a square.

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A365127 Numbers that are the sum of a prime number and a fourth power of a nonnegative integer.

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A307646 Numbers that are the sum of a prime number and a nonnegative cube.

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A111908 Numbers that are not the sum of a prime and a nonzero triangular number.

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