A078129 -id:A078129 - cp's OEIS frontend

A078135 Numbers which cannot be written as a sum of squares > 1.

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 19, 23

Offset: 1

Reinhard Zumkeller, Nov 19 2002

Numbers such that A078134(n)=0.
"Numbers which cannot be written as sum of squares > 1" is equivalent to "Numbers which cannot be written as sum of squares of primes." Equivalently, numbers which can be written as the sum of nonzero squares can also be written as sum of the squares of primes." cf. A090677 = number of ways to partition n into sums of squares of primes. - Jonathan Vos Post, Sep 20 2006
The sequence is finite with a(12)=23 as last member. Proof: When k=a^2+b^2+..., k+4 = 2^2+a^2+b^2+... If k can be written as sum of the squares of primes, k+4 also has this property. As 24,25,26,27 have the property, by induction, all numbers > 23 can be written as sum of squares>1. - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Apr 07 2007
Also, numbers which cannot be written as sum of squares of 2 and 3 (see A078137 for the proof). Explicit representation as sum of squares of primes, or rather of squares of 2 and 3, for numbers m>23: we have m=c*2^2+d*3^2, where c:=(floor(m/4) - 2*(m mod 4))>=0, d:=m mod 4. For that, the finiteness of the sequence is proved constructively. - Hieronymus Fischer, Nov 11 2007
Also numbers n such that every integer partition of n contains a squarefree number. For example, 21 does not belong to the sequence because there are integer partitions of 21 containing no squarefree numbers, namely: (12,9), (9,8,4), (9,4,4,4). - Gus Wiseman, Dec 14 2018

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

Cf. A000290, A001422, A078137, A078139, A078136, A078129.
Cf. A090677.
Cf. A078134, A078139, A090677, A078137, A134754, A134755.
Cf. A005117, A101417, A320322, A322526, A322546, A322547.

nn=100;
ser=Product[If[SquareFreeQ[n],1,1/(1-x^n)],{n,nn}];
Join@@Position[CoefficientList[Series[ser,{x,0,nn}],x],0]-1 (* Gus Wiseman, Dec 14 2018 *)

A090677(a(n)) = 0. - Jonathan Vos Post, Sep 20 2006 [corrected by Joerg Arndt, Dec 16 2018]

A033183(a(n)) = 0. - Reinhard Zumkeller, Nov 07 2009

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)

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

A078131 Numbers which can be written as sum of cubes > 1.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Nov 19 2002

nonn

A078128(a(n)) > 0.

			35 is in the sequence because 35 = 27 + 8.

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

Cf. A000578, A078129 (complement of this sequence), A078130, A078137.

a:=proc(n) local h, hser: h:=1/product(1-x^(j^3),j=2..30): hser:=series(h,x=0,150): if coeff(hser,x^n)>0 then n else fi end: seq(a(n),n=1..140); # Emeric Deutsch, Mar 30 2006

terms = 60;
(Exponent[#, x]& /@ List @@ Normal[1/Product[1-x^j^3, {j, 2, Ceiling[(3 terms)^(1/3)]}] + O[x]^(3 terms)])[[2 ;; terms+1]] (* Jean-François Alcover, Aug 04 2018, after Emeric Deutsch *)

a(n) = n + 83 for n >= 72. - Robert Israel, Apr 05 2020

A078133 Primes which cannot be written as sum of cubes>1.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 61, 71, 73, 79, 101, 103, 109, 127

Offset: 1

Reinhard Zumkeller, Nov 19 2002

Sequence is finite, see A078129.

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

Cf. A000578, A078128, A078132, A000040, A078139.

A333821 Numbers k that can be represented in the form k = p^3 - q^3 - r^3, where p, q, r are positive integers satisfying p = q + r.

Original entry on oeis.org

6, 18, 36, 48, 60, 90, 126, 144, 162, 168, 210, 216, 252, 270, 288, 330, 360, 378, 384, 396, 468, 480, 486, 540, 546, 594, 630, 720, 750, 792, 816, 858, 918, 924, 972, 990, 1008, 1026, 1140, 1152, 1170, 1260, 1296, 1344, 1386, 1404, 1518, 1530, 1560, 1620, 1638, 1656, 1680, 1728, 1800

Offset: 1

Antonio Roldán, Apr 06 2020

nonn

An alternative representation of k is k = 3*q*r*(q+r), with q, r positive integers, then k is a multiple of 6.

			60 is in the sequence because 60 = 5^3 - 4^3 - 1^3, with 5 = 4 + 1.

Cf. A000578, A003072, A024975, A078129, A255018, A275154.

ok(n) = {my(i=1, a=0, m=0, j); if(n%6==0, while(a<=n&&m==0, j=1; while(j

a(n) = 6 * A121741(n).

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A078135 Numbers which cannot be written as a sum of squares > 1.

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

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

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A078131 Numbers which can be written as sum of cubes > 1.

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A078133 Primes which cannot be written as sum of cubes>1.

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A333821 Numbers k that can be represented in the form k = p^3 - q^3 - r^3, where p, q, r are positive integers satisfying p = q + r.

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