A123120 Numbers k>0 such that m+k is not the sum of m nonzero squares for any m>5.

1, 2, 4, 5, 7, 10, 13

Offset: 1

Author

Alexander Adamchuk, Sep 28 2006

Keywords

• nonn
• fini
• full

Comments

Also, numbers which are not the sum of "squares-minus-1" (cf. A005563). - Benoit Jubin, Apr 14 2010

Conjecture: All but (n+6) positive numbers are equal to the sum of n>5 nonzero squares. For all n>5 the only (n+6) positive numbers that are not equal to the sum of n nonzero squares are {1,2,3,...,n-3,n-2,n-1,n+1,n+2,n+4,n+5,n+7,n+10,n+13}.

Numbers that are not squares (n=1): A000037 = {2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28,...}

Numbers that are not the sum of 2 nonzero squares (n=2): A018825 = {1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 16, 19, 21, 22, 23, 24, 27, 28, 30, 31, 33, 35, 36,...}.

Numbers that are not the sum of three nonzero squares (n=3): A004214 = {1, 2, 4, 5, 7, 8, 10, 13, 15, 16, 20, 23, 25, 28, 31, 32, 37, 39, 40, 47, 52, 55, 58,...}.

Numbers that are not the sum of 4 nonzero squares (n=4): A000534 ={1, 2, 3, 5, 6, 8, 9, 11, 14, 17, 24, 29, 32, 41, 56, 96, 128, 224, 384, 512,...}.

Numbers that are not the sum of 5 nonzero squares (n=5): A047701 = {1, 2, 3, 4, 6, 7, 9, 10, 12, 15, 18, 33}.

Numbers that are not the sum of 6 nonzero squares (n=6): {1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 16, 19}.

Numbers that are not the sum of 7 nonzero squares (n=7): {1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 17, 20}.

Numbers that are not the sum of 8 nonzero squares (n=8): {1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 15, 18, 21}.

Numbers that are not the sum of 9 nonzero squares (n=9): {1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 16, 19, 22}.

The above conjecture appears as Theorem 2 on p. 73 in the Grosswald reference, where it is attributed to E. Dubouis (1911). - Wolfdieter Lang, Mar 27 2013

References

• E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, pp. 73-74.

Crossrefs

Cf. A000037, A018825, A004214, A000534, A047701.

Extensions

Edited by M. F. Hasler, Feb 23 2018