A307219 a(n) is the number of partitions of (prime(n)^2 + 2)/3 into 3 squares.
1, 1, 2, 2, 1, 2, 2, 5, 6, 2, 6, 3, 6, 5, 14, 8, 6, 5, 6, 15, 10, 6, 14, 24, 14, 6, 12, 12, 6, 16, 19, 18, 18, 36, 18, 10, 16, 20, 20, 12, 28, 18, 8, 24, 38, 27, 40, 20, 17, 30, 52, 18, 22, 26, 29, 21, 42, 31, 26, 26, 18, 44, 38, 40, 46, 26, 30, 44, 38, 36, 52, 28, 27, 38, 103, 22, 38, 78
Offset: 3
Keywords
Examples
For n = 3, p = prime(3) = 5, (5^2+2)/3 = 9 = 2^2 + 2^2 + 1^2, so a(3) = 1. For n = 9, p = prime(9) = 23, (23^2+2)/3 = 177 = 13^2 + 2^2 + 2^2 = 8^2 + 8^2 + 7^2, so a(9) = 2. For n = 17, p = prime(17) = 59, (59^2+2)/3 = 1161 = 34^2 + 2^2 + 1^2 = 33^2 + 6^2 + 6^2 = 24^2 + 11^2 + 4^2 = 31^2 + 14^2 + 2^2 = 31^2 + 10^2 + 10^2 = 30^2 + 15^2 + 6^2 = 29^2 + 16^2 + 8^2 = 28^2 + 19^2 + 4^2 = 28^2 + 16^2 + 11^2 = 26^2 + 22^2 + 1^2 = 26^2 + 17^2 + 14^2 = 24^2 + 24^2 + 3^2 = 24^2 + 21^2 + 12^2 = 20^2 + 20^2 + 19^2, so a(17) = 14.
References
- Ion Cucurezeanu, Pătrate și cuburi perfecte de numere întregi (Squares and perfect cubes of integer numbers), Ed. Gil., Zalău, 2007, ch. 1, p. 21, pr. 166.
- Laurențiu Panaitopol, Dinu Șerbănescu, Number theory and combinatorial problems for juniors, Ed. Gil, Zalău, (2003), ch. 1, p. 5, pr. 4. (in Romanian).
Programs
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Magma
[#RestrictedPartitions(Floor((p*p+2)/3),3,{d*d:d in [1..p]}): p in PrimesInInterval(5,500) ];
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PARI
a(n)={k=(prime(n+2)^2+2)/3;sum(a=1, k, sum(b=1, a, sum(c=1, b, a^2+b^2+c^2==k)));} \\ Jinyuan Wang, Mar 30 2019
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