A211167 -id:A211167 - cp's OEIS frontend

A045911 Neither a cube nor the sum of a nonnegative cube and a prime.

9, 16, 22, 26, 28, 33, 35, 36, 52, 57, 63, 65, 76, 78, 82, 85, 92, 96, 99, 112, 118, 119, 120, 122, 126, 129, 133, 141, 146, 155, 160, 169, 170, 183, 185, 188, 202, 209, 210, 217, 225, 236, 244, 246, 248, 267, 273, 280, 286, 300, 302, 309, 326

Offset: 1

John Robertson (Jpr2718(AT)aol.com)

nonn

Numbers of the form 1 + k^3, as {9, 28, 65, 126, 217, 344, 513, 730, 1001, 1332, 1729, ...}, are allowed unless they can also be expressed as p + j^3 for some prime p (thus excluding {344, 513, 1001, 1729, ...}). - Daniel Forgues, Feb 13 2013
From Daniel Forgues, Feb 15 2013: (Start)
The graph seems to suggest either that (is there a conjecture?):
  * the sequence grows extremely fast (fewer and fewer integers survive),
  * the sequence is finite (at some point, no more integers survive).
If the sequence is not finite, what then is the asymptotic behavior?
Growth pattern (why is there an exponential growth interlude?):
  * up to about n = 2000 the growth is subexponential (from slightly superlinear, progressing towards exponential growth),
  * from about n = 2000 to 5000 the growth is nearly exponential,
  * above n = 5000 the growth becomes superexponential (taking off from exponential growth) (there might be a last finite integer term!). (End)

Computed by James Van Buskirk, who finds 6195 solutions between 0 and 3000000000.

D. Wilson, Table of n, a(n) for n = 1..6195

Cf. A211167.

isA045911(n) = {if (ispower(n, 3), return (0)); forprime(p=2, n, if (ispower(n-p, 3), return (0));); return (1);} \\ Michel Marcus, May 19 2013

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)).

A307647 Numbers that are the sum of a prime number and a positive cube.

Original entry on oeis.org

3, 4, 6, 8, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 24, 25, 27, 29, 30, 31, 32, 34, 37, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 54, 55, 56, 58, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 74, 75, 77, 79, 80, 81, 83, 84, 86, 87, 88, 90, 91, 93, 94, 95, 97, 98, 100, 101, 102, 104, 105

Offset: 1

Ilya Gutkovskiy, Apr 19 2019

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A000578, A058654, A211167, A283760, A307646.

N:= 200: # for terms <= N
P:= select(isprime,[2,seq(i,i=3..N-1,2)]):
C:= [seq(j^3,j=1..floor((N-2)^(1/3)))]:
sort(convert(select(`<=`,{seq(seq(p+c,p=P),c=C)},N),list)); # Robert Israel, Apr 22 2019

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

A365169 Numbers that are not the sum of a prime number and a fourth power of a positive integer.

Original entry on oeis.org

1, 2, 5, 7, 9, 10, 11, 13, 15, 16, 17, 22, 25, 26, 28, 31, 34, 36, 37, 40, 41, 43, 46, 49, 50, 51, 52, 55, 56, 58, 61, 64, 65, 66, 67, 70, 71, 73, 76, 78, 79, 81, 82, 85, 91, 93, 96, 97, 101, 103, 106, 107, 109, 111, 115, 116, 120, 121, 126, 127, 130, 131, 133, 135

Offset: 1

Ilya Gutkovskiy, Aug 24 2023

nonn

Cf. A064233, A211167, A365166, A365167, A365168.

nmax = 138; Position[CoefficientList[Series[Sum[x^Prime[i], {i, 1, nmax}] Sum[x^j^4, {j, 1, nmax^(1/4)}], {x, 0, nmax}], x] // Rest, 0] // Flatten

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A045911 Neither a cube nor the sum of a nonnegative cube and a prime.

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

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A307647 Numbers that are the sum of a prime number and a positive cube.

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A365169 Numbers that are not the sum of a prime number and a fourth power of a positive integer.

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