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A123120 Numbers k>0 such that m+k is not the sum of m nonzero squares for any m>5. 2
1, 2, 4, 5, 7, 10, 13 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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}. {2,3,5,6,7,8,10,11,12,13,14,15,17,18,19,20,21,22,23,24,26,27,28,...} = A000037 Numbers that are not squares (n=1). {1,3,4,6,7,9,11,12,14,15,16,19,21,22,23,24,27,28,30,31,33,35,36,...} = A018825 Numbers that are not the sum of 2 nonzero squares (n=2). {1,2,4,5,7,8,10,13,15,16,20,23,25,28,31,32,37,39,40,47,52,55,58,...} = A004214 Numbers that are not the sum of three nonzero squares (n=3). {1,2,3,5,6,8,9,11,14,17,24,29,32,41,56,96,128,224,384,512,...} = A000534 Numbers that are not the sum of 4 nonzero squares (n=4). {1,2,3,4,6,7,9,10,12,15,18,33} = A047701 Numbers that are not the sum of 5 nonzero squares (n=5). {1,2,3,4,5,7,8,10,11,13,16,19} Numbers that are not the sum of 6 nonzero squares (n=6). {1,2,3,4,5,6,8,9,11,12,14,17,20} Numbers that are not the sum of 7 nonzero squares (n=7). {1,2,3,4,5,6,7,9,10,12,13,15,18,21} Numbers that are not the sum of 8 nonzero squares (n=8). {1,2,3,4,5,6,7,8,10,11,13,14,16,19,22} Numbers that are not the sum of 9 nonzero squares (n=9).

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.

LINKS

Table of n, a(n) for n=1..7.

CROSSREFS

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

Sequence in context: A247589 A058212 A007997 * A194462 A163267 A036559

Adjacent sequences:  A123117 A123118 A123119 * A123121 A123122 A123123

KEYWORD

fini,nonn

AUTHOR

Alexander Adamchuk, Sep 28 2006

STATUS

approved

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Last modified August 17 15:19 EDT 2017. Contains 290635 sequences.