A002471 -id:A002471 - cp's OEIS frontend

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

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 *)

A064272 Number of representations of n as the sum of a prime number and a nonzero square.

Original entry on oeis.org

0, 1, 1, 0, 2, 1, 1, 1, 0, 2, 2, 0, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 0, 1, 3, 2, 1, 2, 0, 3, 2, 0, 2, 1, 0, 4, 2, 1, 2, 2, 1, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 3, 1, 3, 2, 0, 2, 2, 0, 4, 2, 0, 2, 3, 2, 4, 2, 1, 2, 3, 1, 1, 3, 1, 4, 2, 1, 3, 1, 1, 5, 3, 0, 3, 3, 2, 2, 2, 0, 4, 2, 1, 3, 2, 1, 4, 1, 1, 2, 3, 2, 3, 4, 1

Offset: 2

Vladeta Jovovic, Sep 23 2001

nonn

a(A064233(n))=0.
A002471(n) - 1 <= a(n) <= A002471(n). [Reinhard Zumkeller, Sep 30 2011]
A224076(n) <= a(A214583(n)+1) for n such that A214583 is defined; a(A064283(n)) = n and a(m) <> n for m < A064283(n). - Reinhard Zumkeller, Mar 31 2013

			6=2+4=5+1, thus a(6)=2.

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

Cf. A064233.

Cf. A000290.

a064272 n = sum $
   map (a010051 . (n -)) $ takeWhile (< n) $ tail a000290_list
-- Reinhard Zumkeller, Jul 23 2013, Sep 30 2011

a(n) = SUM(A010051(k)*A010052(n-k+1): 1<=k<=n). [From Reinhard Zumkeller, Nov 05 2009]

G.f.: (Sum_{k>=1} x^prime(k))*(Sum_{k>=1} x^(k^2)). - Ilya Gutkovskiy, Feb 05 2017

A014089 Sum of a square and a prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62

Offset: 1

N. J. A. Sloane

A002471(a(n))>0; complement of A014090; A143989 is a subsequence. [From Reinhard Zumkeller, Sep 07 2008]

Cf. A058654.

isok(n) = for (i = 0, sqrtint(n), if (isprime(n - i^2), return (1))); 0 \\ Michel Marcus, Sep 04 2013

A302354 Expansion of (Sum_{i>=1} x^prime(i))*(Sum_{j>=0} x^(j^3)).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 06 2018

nonn

Number of representations of n as the sum of a prime number and a nonnegative cube.

			a(11) = 2 because 11 = 3 + 2^3 = 11 + 0^3.

Cf. A002471, A010051, A010057, A045911, A064272, A283760.

nmax = 120; Rest[CoefficientList[Series[Sum[x^Prime[i], {i, 1, nmax}] Sum[x^j^3, {j, 0, nmax}], {x, 0, nmax}], x]]

G.f.: (Sum_{i>=1} x^prime(i))*(Sum_{j>=0} x^(j^3)).

A365126 Number of representations of n as the sum of a prime number and a fourth power of a nonnegative integer.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Aug 24 2023

nonn

Cf. A002471, A302354, A365127, A365166, A365167.

nmax = 120; CoefficientList[Series[Sum[x^Prime[i], {i, 1, nmax}] Sum[x^j^4, {j, 0, nmax^(1/4)}], {x, 0, nmax}], x] // Rest

G.f.: (Sum_{i>=1} x^prime(i)) * (Sum_{j>=0} x^(j^4)).

A133364 Number of ways of writing n as a sum of a prime and a square-full number.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 1, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 2, 2, 2, 3, 1, 2, 1, 1, 1, 4, 2, 2, 3, 1, 4, 2, 2, 3, 1, 2, 5, 4, 2, 2, 2, 2, 3, 4, 2, 3, 2, 3, 2, 4, 2, 2, 3, 3, 4, 2, 1, 2, 2, 2, 4, 3, 1, 2, 3, 3, 5, 4, 2, 2, 3, 2, 3, 6, 1, 5, 2, 4, 4, 2, 1, 6, 3, 2, 4, 4, 3

Offset: 1

Jonathan Vos Post, Oct 26 2007

This is to square-full numbers A001694 as A098983 is to squarefree numbers A005117 and as A002471 is to squares A000290. Asymptotics of this should relate to A098983.

			a(3) = 1 because 3=2+1 where 2 is prime and 1 is square-full.
a(4) = 1 because 4=3+1 where 3 is prime and 1 is square-full.
a(5) = 0 because there is no positive solution to 5 = prime+(square-full).
a(6) = 2 because 6=5+1=2+4.
a(7) = 1 because 7=3+4.
a(8) = 1 because 8=7+1.
a(9) = 1 because 9=5+4.
a(10) = 1 because 10=2+8.
a(11) = 3 because 11=2+9=3+8=7+4.
a(12) = 2 because 12=3+9=11+1.
a(13) = 1 because 13=5+8.
a(14) = 2 because 14=5+9=13+1.
a(15) = 2 because 15=7+8=11+4.
a(16) = 1 because 16=7+9.
a(17) = 1 because 17=13+4.
a(18) = 2 because 18=2+16=17+1.
a(19) = 2 because 19=3+16=11+8.
a(20) = 2 because 20=19+1=11+9.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A000040, A000290, A001694, A002471, A005117, A098983.

isA001694 := proc(n) local digs,i ; digs := ifactors(n)[2] ; for i in digs do if op(2,i) = 1 then RETURN(false) ; fi ; od: RETURN(true) ; end: A133364 := proc(n) local a,p ; a := 0 ; p := 2 ; while p < n do if isA001694(n-p) then a := a+1 ; fi ; p := nextprime(p) ; od: RETURN(a) ; end: seq(A133364(n),n=3..90) ; # R. J. Mathar, Nov 09 2007

a = {}; For[n = 3, n < 100, n++, c = 0; For[j = 1, Prime[j] < n, j++, d = 1; b = FactorInteger[n - Prime[j]]; For[m = 1, m < Length[b] + 1, m++, If[b[[m, 2]] < 2, d = 0]]; If[d == 1, c++ ]]; AppendTo[a, c]]; a (* Stefan Steinerberger, Oct 29 2007 *)

a(n) = Card{n = i + j where i is in A000040 and j is in A001694}.

Corrected and extended by Stefan Steinerberger, Oct 29 2007 and by R. J. Mathar, Nov 09 2007

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

A365288 a(n) is the least positive integer that can be expressed as the sum of a prime number and a square of a nonnegative integer in exactly n ways.

Original entry on oeis.org

1, 2, 3, 11, 38, 107, 83, 167, 293, 227, 398, 677, 668, 902, 908, 1238, 1487, 1448, 1748, 1592, 2273, 2672, 3167, 3947, 4457, 3632, 3713, 6047, 5843, 7052, 8123, 5792, 5297, 9602, 9092, 6368, 9908, 13268, 8153, 9833, 8777, 16112, 15923, 14528, 14852, 18233

Offset: 0

Ilya Gutkovskiy, Aug 31 2023

nonn

			For n = 3: 11 = 2 + 3^2 = 7 + 2^2 = 11 + 0^2.

Cf. A000040, A000290, A002471, A064283, A365289, A365291.

A059400 a(n) is the least odd number of the form p + k^2 with p prime and k > 0 which can be represented in exactly n different ways.

Original entry on oeis.org

1, 3, 11, 27, 77, 83, 167, 293, 227, 503, 437, 887, 923, 1007, 1133, 1487, 2243, 2147, 2477, 2273, 2537, 3167, 3947, 4457, 4703, 3737, 3713, 5843, 6233, 8123, 8333, 5297, 11513, 10127, 9407, 10853, 10577, 13187, 8153, 12473, 8777, 15923, 16463, 17513

Offset: 0

Robert G. Wilson v, Mar 15 2001

nonn

Note that A002471 allows for k to equal zero.

			a(3) = 27 because 27 = 23+2^2 = 11+4^2 = 2+5^2 and is the least odd number to exhibit this property of 3 representations.

David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, Penguin Books, London, 1997, page 63.

Donovan Johnson, Table of n, a(n) for n = 0..1000

Cf. A002471.

a = Table[ 0, {55} ]; Do[ c = 0; k = 1; While[ n - k^2 > 1, If[ PrimeQ[ n - k^2], c++ ]; k++ ]; If[ a[[c]] == 0, a[[c]] = n], { n, 1, 30500, 2} ]; a

Name clarified by Donovan Johnson, Nov 24 2012

A253238 Number of ways to write n as a sum of a perfect power (>1) and a prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 0, 1, 1, 4, 2, 2, 2, 1, 3, 2, 2, 3, 1, 2, 4, 4, 2, 2, 1, 2, 2, 4, 2, 3, 1, 3, 2, 4, 2, 2, 2, 3, 4, 2, 1, 2, 1, 2, 3, 3, 1, 2, 3, 3, 4, 4, 2, 2, 2, 2, 1, 5, 1, 4, 2, 3, 3, 2, 1, 5, 2, 1, 4, 4, 3, 2, 1, 2, 4, 3, 2, 3, 2, 2, 4, 2, 2, 2, 2, 3, 2, 6, 2, 4, 2, 2, 4, 5, 2, 3, 1, 3, 3, 5, 2, 3, 1, 2, 4, 4, 3, 3, 2, 1, 6

Offset: 1

Eric Chen, May 17 2015

In this sequence, "perfect power" does not include 0 or 1, "prime" does not include 1. Both "perfect power" and "prime" must be positive.
In the past, I conjectured that a(n) > 0 for all n>24, but this is not true. My PARI program found that a(1549) = 0.
I also asked which a(n) are 1. For example, 331 is a de Polignac number (A006285), so it cannot be written as 2^n+p with p prime, and 331-6^n must divisible by 5, 331-10^n must divisible by 3, ..., 331-18^2 = 331-324 = 7 is prime (and it is the only prime of the form 331-m^n, with m, n natural numbers, m>1, n>1), so a(331) = 1. Similarly, a(3319) = 1. Conjecture: a(n) > 1 for all n > 3319.
This conjecture is not true: a(1771561) = 0. (See A119748)
Another conjecture: For every number m>=0, there is a number k such that a(n)>=m for all n>=k.
Another conjecture: Except for k=2, first occurrence of k must be earlier then first occurrence of k+1.
For n such that a(n) = 0, see A119748.
For n such that a(n) = 1, see the following a-file of this sequence.

Eric Chen and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 3000 terms from Chen)
Eric Chen, The unique way to write n as a sum of a perfect power (>1) and a prime for those n such a(n)=1

Cf. A196228, A119748, A109925, A118954, A064272, A064233, A002471, A014090.

nn = 128; pwrs = Flatten[Table[Range[2, Floor[nn^(1/ex)]]^ex, {ex, 2, Floor[Log[2, nn]]}]]; pp = Prime[Range[PrimePi[nn]]]; t = Table[0, {nn}]; Do[ t[[i[[1]]]] = i[[2]], {i, Tally[Sort[Select[Flatten[Outer[Plus, pwrs, pp]], # <= nn &]]]}]; t

a(n) = sum(k=1, n-1, ispower(k) && isprime(n-k))

a(n)=sum(e=2,log(n)\log(2),sum(b=2,sqrtnint(n,e),isprime(n-b^e)&&!ispower(b))) \\ Charles R Greathouse IV, May 28 2015

cp's OEIS Frontend

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

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A064272 Number of representations of n as the sum of a prime number and a nonzero square.

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A014089 Sum of a square and a prime.

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A302354 Expansion of (Sum_{i>=1} x^prime(i))*(Sum_{j>=0} x^(j^3)).

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A365126 Number of representations of n as the sum of a prime number and a fourth power of a nonnegative integer.

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A133364 Number of ways of writing n as a sum of a prime and a square-full number.

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

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A365288 a(n) is the least positive integer that can be expressed as the sum of a prime number and a square of a nonnegative integer in exactly n ways.

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A059400 a(n) is the least odd number of the form p + k^2 with p prime and k > 0 which can be represented in exactly n different ways.

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