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A273874 Least positive integer k such that k^2 + (k+1)^2 + ... + (k+n-2)^2 + (k+n-1)^2 is the sum of two nonzero squares. a(n) = 0 if no solution exists. 0
5, 1, 2, 0, 2, 0, 0, 0, 0, 2, 5, 1, 12, 0, 3, 0, 3, 0, 0, 0, 0, 0, 53, 1, 1, 1, 2, 0, 4, 0, 0, 0, 5, 2, 0, 0, 2, 0, 3, 0, 5, 0, 0, 5, 0, 0, 73, 1, 3, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Least positive integer k such that sum_{i=0..n-1} (k+i)^2 = n*(6*k^2 +6*k*n -6*k +2*n^2 -3*n +1)/6 is the sum of two nonzero squares. a(n) = 0 if no k exists for corresponding n.

LINKS

Table of n, a(n) for n=1..50.

EXAMPLE

a(1) = 5 because 5^2 = 3^2 + 4^2.

a(3) = 2 because 2^2 + 3^2 + 4^2 = 2^2 + 5^2.

a(11) = 0 because sum_{i=0..11-1} (k+i)^2 = 11*(k^2+10*k+35). Since 11 mod 4 = 3, to be expressed as a sum of two squares the term k^2+10*k+35 must be divisible by 11. This happens if k is congruent to 5 or 7 mod 11, but in both cases the sum, once divided by 11^2, is congruent to 3 mod 4, and thus it cannot be written as the sum of two squares. Similar modular arguments can be used for the other terms equal to 0. - Giovanni Resta, Jun 02 2016

CROSSREFS

Cf. A000404, A034705.

Sequence in context: A174919 A156952 A158748 * A086039 A265824 A097413

Adjacent sequences:  A273871 A273872 A273873 * A273875 A273876 A273877

KEYWORD

nonn,more

AUTHOR

Altug Alkan, Jun 02 2016

EXTENSIONS

a(7)-a(50) from Giovanni Resta, Jun 02 2016

STATUS

approved

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Last modified August 13 19:30 EDT 2020. Contains 336451 sequences. (Running on oeis4.)