A064272 -id:A064272 - 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

A064283 Least number which may be expressed as the sum of a prime number and a nonzero square in exactly n different ways.

Original entry on oeis.org

3, 6, 27, 38, 83, 167, 248, 227, 488, 398, 887, 668, 902, 908, 1238, 2012, 1448, 1748, 1592, 2537, 2672, 3902, 4457, 4703, 3632, 3713, 5843, 6233, 7052, 8333, 5297, 8888, 9602, 9092, 6368, 9908, 13187, 8153, 12473, 8777, 15923, 16463, 14528, 14852, 20807

Offset: 1

Robert G. Wilson v, Sep 24 2001

nonn

A064272(a(n)) = n and A064272(m) <> n for m < a(n). - Reinhard Zumkeller, Mar 31 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..100

Cf. A000040, A000290, A064233, A064272.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a064283 = (+ 2) . fromJust . (`elemIndex` a064272_list)
-- Reinhard Zumkeller, Mar 31 2013

More terms from Vladeta Jovovic, Sep 25 2001

More terms from Sean A. Irvine, Jun 25 2023

A214583 Numbers m such that for all k with gcd(m, k) = 1 and m > k^2, m - k^2 is prime.

Original entry on oeis.org

3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 48, 54, 60, 62, 68, 72, 80, 84, 90, 98, 108, 110, 132, 138, 140, 150, 180, 182, 198, 252, 318, 360, 398, 468, 570, 572, 930, 1722

Offset: 1

Hans Ruegg, Jul 21 2012

No further terms < 10^10.
This sequence is based on a remark in a paper distributed over the Internet (see the Leo Moser link) under the heading "Unsolved Problems and Conjectures" (page 84):
"Is 968 the largest number n such that for all k with (n, k) = 1 and n > k^2, n - k^2 is prime? (Erdős)"
The statement by Moser contains an error: 968 does NOT have this property (968-25*25 = 343 = 7*7*7), and the largest such number (1722) is larger than 968.
A224076(n) <= A064272(a(n)+1). - Reinhard Zumkeller, Mar 31 2013

			For example, the number 20 is part of this sequence because 20-1*1 = 19 (prime), and 20-3*3 = 11 (prime). Not considered are 20-2*2 and 20-4*4, because 2 and 4 are not relative primes to 20.

Leo Moser, An Introduction to the Theory of Numbers, The Trillia Group 2004, page 84.

Cf. A065428.
Cf. A057828, A000290, A010051.
Cf. A224075; subsequence of A008864.

a214583 n = a214583_list !! (n-1)
a214583_list = filter (p 3 1) [2..] where
   p i k2 x = x <= k2 || (gcd k2 x > 1 || a010051' (x - k2) == 1) &&
                         p (i + 2) (k2 + i) x
-- Reinhard Zumkeller, Mar 31 2013, Jul 22 2012

Reap[For[p = 2, p < 2000, p = NextPrime[p], n = p+1; q = True; k = 1; While[k*k < n, If[GCD[k, n] == 1, If[! PrimeQ[n - k^2], q = False; Break[]]]; k += 1]; If[q, Sow[n]]]] [[2, 1]] (* Jean-François Alcover, Oct 11 2013, after Joerg Arndt's Pari program *)
gQ[n_]:=AllTrue[n-#^2&/@Select[Range[Floor[Sqrt[n]]],CoprimeQ[ #, n]&], PrimeQ]; Select[Range[2000],gQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 02 2018 *)

N=10^10;
default(primelimit,N);
{ forprime (p=2, N,
    n = p + 1;
    q = 1;
    k = 1;
    while ( k*k < n,
        if ( gcd(k,n)==1,
            if ( ! isprime(n-k^2),  q=0; break() );
        );
        k += 1;
    );
    if ( q, print1(n,", ") );
); }
/* Joerg Arndt, Jul 21 2012 */

A283760 Expansion of (Sum_{i>=1} x^prime(i))*(Sum_{j>=1} x^(j^3)).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Mar 16 2017

nonn

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

			a(32) = 2 because 32 = 31 + 1^3 = 5 + 3^3.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Ilya Gutkovskiy, Extended graphical example

Cf. A000040, A000578, A064272, A211167.

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

concat([0,0], Vec((sum(i=1, 120, x^prime(i)) * sum(j=1, 120, x^(j^3))) + O(x^121))) \\ Indranil Ghosh, Mar 16 2017

(define (A283760 n) (cond ((< n 2) 0) (else (let loop ((k (A048766 n)) (s 0)) (if (< k 1) s (loop (- k 1) (+ s (A010051 (- n (expt k 3)))))))))) ;; Antti Karttunen, Aug 18 2017

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

A224076 Row lengths of triangle in A224075.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 3, 2, 2, 3, 2, 4, 4, 3, 3, 2, 2, 4, 3, 4, 3, 4, 4, 3, 4, 5, 4, 4, 6, 5, 10, 6, 6, 10, 8, 11

Offset: 1

Reinhard Zumkeller, Mar 31 2013

Cf. A064272, A214583, A224075, A224076.

Haskell
```
a224076 = length . a224075_row
```

a(n) <= A064272(A214583(n)+1) for n such that A214583 is defined.

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)).

A365167 Number of representations of n as the sum of a prime number and a fourth power of a positive integer.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Aug 24 2023

nonn

Cf. A064272, A283760, A365126, A365168, A365169.

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

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

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

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

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A064283 Least number which may be expressed as the sum of a prime number and a nonzero square in exactly n different ways.

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A214583 Numbers m such that for all k with gcd(m, k) = 1 and m > k^2, m - k^2 is prime.

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A283760 Expansion of (Sum_{i>=1} x^prime(i))*(Sum_{j>=1} x^(j^3)).

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A224076 Row lengths of triangle in A224075.

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

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A365167 Number of representations of n as the sum of a prime number and a fourth power of a positive integer.

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

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