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 A307219 a(n) is the number of partitions of (prime(n)^2 + 2)/3 into 3 squares. 0
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,3 COMMENTS If p >= 5 is a prime number it can be written as p = 6m-1 or p = 6m+1. The identities ((6m-1)^2 + 2)/3 = (2m)^2 + (2m)^2 + (2m-1)^2 and ((6m+1)^2 + 2)/3 = (2m)^2 + (2m)^2 + (2m+1)^2 show that the number (p^2 + 2)/3 can be written as a sum of 3 squares of integers in at least one way. 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). LINKS EXAMPLE 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. PROG (MAGMA) [#RestrictedPartitions(Floor((p*p+2)/3), 3, {d*d:d in [1..p]}): p in PrimesInInterval(5, 500) ]; (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 CROSSREFS Cf. A000040, A000378, A024795, A025427. Sequence in context: A075445 A216612 A194447 * A236573 A333252 A293375 Adjacent sequences:  A307216 A307217 A307218 * A307220 A307221 A307222 KEYWORD nonn AUTHOR Marius A. Burtea, Mar 29 2019 STATUS approved

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Last modified June 22 12:56 EDT 2021. Contains 345380 sequences. (Running on oeis4.)