This site is supported by donations to The OEIS Foundation.



Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A078135 Numbers which cannot be written as a sum of squares > 1. 13
1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 19, 23 (list; graph; refs; listen; history; text; internal format)



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

A033183(a(n)) = 0. [From Reinhard Zumkeller, Nov 07 2009]


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


A090677(n) = 0. - Jonathan Vos Post, Sep 20 2006


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

Cf. A090677.

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

Sequence in context: A117204 A117344 A129127 * A073328 A119988 A087005

Adjacent sequences:  A078132 A078133 A078134 * A078136 A078137 A078138




Reinhard Zumkeller, Nov 19 2002



Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified November 20 08:12 EST 2017. Contains 294962 sequences.