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

1,2

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.

Sequence in context: A117204 A354356 A129127 * A073328 A119988 A087005

Adjacent sequences: A078132 A078133 A078134 * A078136 A078137 A078138

Keyword

nonn,fini,full

Author

Reinhard Zumkeller, Nov 19 2002

Status

approved