cp's OEIS Frontend

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

%I A078135 #25 Apr 05 2025 06:31:47
%S A078135 1,2,3,5,6,7,10,11,14,15,19,23
%N A078135 Numbers which cannot be written as a sum of squares > 1.
%C A078135 Numbers such that A078134(n)=0.
%C A078135 "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
%C A078135 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
%C A078135 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
%C A078135 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
%H A078135 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquareNumber.html">Square Number</a>.
%H A078135 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A078135 A090677(a(n)) = 0. - _Jonathan Vos Post_, Sep 20 2006 [corrected by _Joerg Arndt_, Dec 16 2018]
%F A078135 A033183(a(n)) = 0. - _Reinhard Zumkeller_, Nov 07 2009
%t A078135 nn=100;
%t A078135 ser=Product[If[SquareFreeQ[n],1,1/(1-x^n)],{n,nn}];
%t A078135 Join@@Position[CoefficientList[Series[ser,{x,0,nn}],x],0]-1 (* _Gus Wiseman_, Dec 14 2018 *)
%Y A078135 Cf. A000290, A001422, A078137, A078139, A078136, A078129.
%Y A078135 Cf. A090677.
%Y A078135 Cf. A078134, A078139, A090677, A078137, A134754, A134755.
%Y A078135 Cf. A005117, A101417, A320322, A322526, A322546, A322547.
%K A078135 nonn,fini,full
%O A078135 1,2
%A A078135 _Reinhard Zumkeller_, Nov 19 2002