A078131 -id:A078131 - cp's OEIS frontend

A122612 Duplicate of A078131.

8, 16, 24, 27, 32, 35, 40, 43, 48, 51, 54, 56, 59, 62, 64, 67, 70, 72, 75, 78, 80, 81, 83, 86, 88, 89, 91, 94, 96, 97, 99, 102, 104, 105, 107, 108, 110, 112, 113, 115, 116, 118, 120, 121, 123, 124, 125, 126, 128, 129, 131, 132, 133, 134, 135, 136, 137, 139, 140, 141

Offset: 1

Jonathan Vos Post, Sep 20 2006

dead

Previous name was: Sums of cubes of primes.
Starts out identical to A078130 (numbers having exactly one representation as sum of cubes>1), until 72. It seems that 154 is the largest integer which cannot be represented as the sum of cubes of primes.
154 is the largest integer that cannot be represented as the sum of cubes of primes. Indeed, every number greater than 154 can be represented as a sum of multiples of 2^3, 3^3, and 5^3. - Giovanni Resta, Jun 16 2016

Cf. A000040 (primes), A030078 (cubes of primes), A078130.

from sympy import primerange, integer_nthroot as iroot
def ok(n):
    cands = [p**3 for p in primerange(2, iroot(n, 3)[0]+1) if p**3 <= n]
    return n in cands or any(ok(n-c) for c in cands)
print(list(filter(ok, range(142)))) # Michael S. Branicky, Aug 16 2021

{A030078} UNION {A030078 + A030078} UNION {A030078 + A030078 + A030078}... = a*8 + b*27 + c*125 + d*343 + e*1331 + f*2197 = a*(p(1))^3 + b*(p(2))^3 + c*(p(3))^3 + d*(p(4))^3 + e*(p(5))^3 + ... where p(i) = A000040(i) and a, b, c, d, e, f, ... are nonnegative integers.

A078128 Number of ways to write n as sum of cubes > 1.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Nov 19 2002

nonn

a(A078129(n))=0; a(A078130(n))=1; a(A078131(n))>0;

Conjecture (lower bound): for all k exists b(k) such that a(n)>k for n>b(k); see b(0)=A078129(83)=154 and b(1)=A078130(63)=218.

			a(160)=4: 160 = 20*2^3 = 4^3+12*2^3 = 2*4^3+4*2^3 = 5^3+3^3+2^3.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of cubes
Eric Weisstein's World of Mathematics, Cubic Number.

Cf. A000578, A003108, A001235, A078132, A078133, A078134.

nmax = 105; CoefficientList[Series[Product[1/(1 - x^(k^3)), {k, 1, nmax}], {x, 0, nmax}], x] // Differences (* Jean-François Alcover, Mar 01 2019, after Vaclav Kotesovec *)

a(n) = 1/n*Sum_{k=1..n} (b(k)-1)*a(n-k), a(0) = 1, where b(k) is sum of cube divisors of k. - Vladeta Jovovic, Nov 20 2002
From Vaclav Kotesovec, Jan 05 2017: (Start)
a(n) = A003108(n) - A003108(n-1).
a(n) ~ exp(4*(Gamma(1/3) * Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) * (Gamma(1/3) * Zeta(4/3))^(3/2) / (8 * 3^(5/2) * Pi^2 * n^2).
(End)

A078137 Numbers which can be written as sum of squares>1.

Original entry on oeis.org

4, 8, 9, 12, 13, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82

Offset: 1

Reinhard Zumkeller, Nov 19 2002

A078134(a(n))>0.
Numbers which can be written as a sum of squares of primes. - Hieronymus Fischer, Nov 11 2007
Equivalently, numbers which can be written as a sum of squares of 2 and 3. Proof for numbers m>=24: if m=4*(k+6), k>=0, then m=(k+6)*2^2; if m=4*(k+6)+1 than m=(k+4)*2^2+3^2; if m=4*(k+6)+2 then m=(k+2)*2^2+2*3^2; if m=4*(k+6)+3 then m=k*2^2+3*3^2. Clearly, the numbers a(n)<24 can also be written as sums of squares of 2 and 3. Explicit representation as a sum of squares of 2 and 3 for numbers m>23: m=c*2^2+d*3^2, where c:=(floor(m/4) - 2*(m mod 4))>=0 and d:=m mod 4. - Hieronymus Fischer, Nov 11 2007

Index entries for sequences related to sums of squares
Eric Weisstein's World of Mathematics, Square Number.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Complement of A078135.

Cf. A000290, A078136, A078131, A001597, A025475, A078134, A078135, A078139, A090677, A134600, A134605, A134608, A134612, A134616, A134618, A134620.

Join[{4, 8, 9, 12, 13, 16, 17, 18, 20, 21, 22}, Range[24, 82]] (* Jean-François Alcover, Aug 01 2018 *)

a(n)=if(n>11,n+12,[4,8,9,12,13,16,17,18,20,21,22][n]) \\ Charles R Greathouse IV, Aug 21 2011

a(n)=n + 12 for n >= 12. - Hieronymus Fischer, Nov 11 2007

Edited by N. J. A. Sloane, Oct 17 2009 at the suggestion of R. J. Mathar.

A078129 Numbers which cannot be written as sum of cubes > 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 36, 37, 38, 39, 41, 42, 44, 45, 46, 47, 49, 50, 52, 53, 55, 57, 58, 60, 61, 63, 65, 66, 68, 69, 71, 73, 74, 76, 77, 79, 82, 84, 85, 87, 90, 92, 93, 95, 98, 100, 101, 103, 106, 109, 111, 114, 117, 119, 122, 127, 130, 138, 146, 154

Offset: 1

Reinhard Zumkeller, Nov 19 2002

A078128(a(n))=0.
The sequence is finite because every number greater than 181 can be represented using just 8 and 27. - Franklin T. Adams-Watters, Apr 21 2006
More generally, the numbers which are not the sum of k-th powers larger than 1 are exactly those in [1, 6^k - 3^k - 2^k] but not of the form 2^k*a + 3^k*b + 5^k*c with a,b,c nonnegative. This relies on the following fact applied to m=2^k and n=3^k: if m and n are relatively prime, then the largest number which is not a linear combination of m and n with positive integer coefficients is mn - m - n. - Benoit Jubin, Jun 29 2010

			181 is not in the list since 181 = 7*2^3 + 5^3.

Eric Weisstein's World of Mathematics, Cubic Number.
Index entries for sequences related to sums of cubes

Cf. A000578, A078131, A078133, A078130, A078135.

terms = 83; A078131 = (Exponent[#, x]& /@ List @@ Normal[1/Product[1-x^j^3, {j, 2, Ceiling[(3 terms)^(1/3)]}] + O[x]^(3 terms)])[[2 ;; terms+1]];
Complement[Range[Max[A078131]], A078131] (* Jean-François Alcover, Aug 04 2018 *)

Sequence completed by Franklin T. Adams-Watters, Apr 21 2006

Edited by R. J. Mathar and N. J. A. Sloane, Jul 06 2010

A078130 Numbers having exactly one representation as sum of cubes > 1.

Original entry on oeis.org

8, 16, 24, 27, 32, 35, 40, 43, 48, 51, 54, 56, 59, 62, 67, 70, 75, 78, 81, 83, 86, 89, 94, 97, 102, 105, 108, 110, 113, 116, 121, 124, 125, 129, 132, 133, 135, 137, 140, 141, 143, 148, 149, 151, 156, 157, 159, 162, 164, 165, 167, 170, 173, 175, 178, 181, 183, 186, 191, 194, 202, 210, 218

Offset: 1

Reinhard Zumkeller, Nov 19 2002

A078128(a(n))=1.
Conjecture: the sequence is finite; is a(63)=218 the last entry?
Yes. An argument similar to that in A078136 can be made, based on the identity m = 8*k = k*2^3 = 4^3 + (k-8)*2^3 which enables trading 4^3 for 8 repeats of 2^3. Then, the remaining residue classes m = 8*k+r for r=1..7 can be handled by known representations for m = 145, 226, 91, 172, 189, 118, and 199, respectively. - Sean A. Irvine, Jun 17 2025

			72 is not a term, as 72 = 8+8+8+8+8+8+8+8+8 = 8+64.

Eric Weisstein's World of Mathematics, Cubic Number.
Index entries for sequences related to sums of cubes

Cf. A000578, A078131, A078129, A078136.

More terms from Sean A. Irvine, Jun 17 2025

cp's OEIS Frontend

A122612 Duplicate of A078131.

Views

Author

Keywords

Comments

Crossrefs

Programs

Python

Formula

A078128 Number of ways to write n as sum of cubes > 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A078137 Numbers which can be written as sum of squares>1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A078129 Numbers which cannot be written as sum of cubes > 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A078130 Numbers having exactly one representation as sum of cubes > 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions