A078129 -id:A078129

Search: a078129 -id:a078129

Displaying 1-6 of 6 results found.

page 1

A078135

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

+10
20

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..12.

Eric Weisstein's World of Mathematics, Square Number.

Index entries for sequences related to sums of squares

FORMULA

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]

MATHEMATICA

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 *)

CROSSREFS

Cf. A000290, A001422, A078137, A078139, A078136, A078129.

Cf. A090677.

Cf. A078134, A078139, A090677, A078137, A134754, A134755.

Cf. A005117, A101417, A320322, A322526, A322546, A322547.

KEYWORD

nonn,fini,full

AUTHOR

Reinhard Zumkeller, Nov 19 2002

STATUS

approved

A078128

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

+10
13

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,64

COMMENTS

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.

LINKS

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.

FORMULA

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)

EXAMPLE

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

MATHEMATICA

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 *)

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Nov 19 2002

STATUS

approved

A078130

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

A078128(a(n))=1.

Conjecture: the sequence is finite; is a(63)=218 the last entry?

LINKS

Table of n, a(n) for n=1..57.

Index entries for sequences related to sums of cubes

Eric Weisstein's World of Mathematics, Cubic Number.

EXAMPLE

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

CROSSREFS

Cf. A000578, A078131, A078129, A078136.

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Nov 19 2002

STATUS

approved

A078131

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

A078128(a(n)) > 0.

LINKS

Table of n, a(n) for n=1..59.

Eric Weisstein's World of Mathematics, Cubic Number.

Index entries for sequences related to sums of cubes

FORMULA

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

EXAMPLE

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

MAPLE

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

MATHEMATICA

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 *)

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Nov 19 2002

STATUS

approved

A078133

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Sequence is finite, see A078129.

LINKS

Table of n, a(n) for n=1..23.

Eric Weisstein's World of Mathematics, Cubic Number.

Index entries for sequences related to sums of cubes

CROSSREFS

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

KEYWORD

nonn,fini,full

AUTHOR

Reinhard Zumkeller, Nov 19 2002

STATUS

approved

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.

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..55.

FORMULA

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

EXAMPLE

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

PROG

(PARI) ok(n) = {my(i=1, a=0, m=0, j); if(n%6==0, while(a<=n&&m==0, j=1; while(j<i&&m==0, a=3*i*j*(i-j); if(a==n, m=1); j+=1); i+=1)); m}

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Antonio Roldán, Apr 06 2020

STATUS

approved

page 1

Search completed in 0.005 seconds