A001480 Let p = A007645(n) be the n-th generalized cuban prime and write p = x^2 + 3*y^2; a(n) = y.

1, 1, 2, 1, 3, 2, 3, 2, 1, 4, 5, 4, 1, 6, 3, 5, 7, 6, 7, 2, 8, 1, 7, 3, 6, 8, 5, 6, 3, 9, 8, 5, 4, 10, 11, 2, 11, 6, 4, 10, 12, 9, 12, 11, 1, 9, 13, 2, 7, 13, 4, 12, 13, 14, 11, 7, 9, 10, 4, 15, 14, 9, 6, 15, 5, 14, 16, 1, 3, 7, 10, 2, 5, 14, 17, 13, 9, 16, 17

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

a(n) = A000196((A007645(n) - A000290(A001479(n))) / 3). - Reinhard Zumkeller, Jul 11 2013

References

• A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• B. van der Pol and P. Speziali, The primes in k(rho). Nederl. Akad. Wetensch. Proc. Ser. A. {54} = Indagationes Math. 13, (1951). 9-15 (1 plate).

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904 [Annotated scans of selected pages]
• S. R. Finch, Powers of Euler's q-Series, (arXiv:math.NT/0701251).
• B. van der Pol and P. Speziali, The primes in k(rho) (annotated and scanned copy)

Crossrefs

Cf. A001479, A007645.

Programs

Haskell

a001480 n = a000196 $ (`div` 3) $ (a007645 n) - (a001479 n) ^ 2
-- Reinhard Zumkeller, Jul 11 2013

Mathematica

nmax = 63; nextCuban[p_] := If[p1 = NextPrime[p]; Mod[p1, 3] > 1, nextCuban[p1], p1]; cubanPrimes = NestList[ nextCuban, 3, nmax ]; f[p_] := y /. ToRules[ Reduce[x > 0 && y > 0 && p == x^2 + 3*y^2, {x, y}, Integers]]; a[1] = 1; a[n_] := f[cubanPrimes[[n]]]; Table[ a[n] , {n, 1, nmax}] (* Jean-François Alcover, Oct 19 2011 *)

PARI

do(lim)=my(v=List(), q=Qfb(1,0,3)); forprime(p=2,lim, if(p%3==2,next); listput(v, qfbsolve(q,p)[2])); Vec(v) \\ Charles R Greathouse IV, Feb 07 2017

Extensions

Definition revised by N. J. A. Sloane, Jan 29 2013