A263723 Number of representations of the prime P = A182479(n) as P = p^2 + q^2 + r^2, where p < q < r are also primes.

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 4, 2, 2, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1

Offset

1,5

Comments

According to Sierpinski and Schinzel (1988), it is easy to prove that the smallest of p, q, r is always p = 3, and under Schinzel's hypothesis H the sequence is infinite.

References

W. Sierpinski, Elementary Theory of Numbers, 2nd English edition, revised and enlarged by A. Schinzel, Elsevier, 1988; see pp. 220-221.

Links

Table of n, a(n) for n=1..105.

Wikipedia, Schinzel's Hypothesis H

Example

A182479(1) = 83 = 3^2 + 5^2 + 7^2 and A182479(2) = 179 = 3^2 + 7^2 + 11^2 are the only ways to write 83 and 179 as sums of squares of 3 distinct primes, so a(1) = 1 and a(2) = 1.
A182479(5) = 419 = 3^2 + 7^2 + 19^2 = 3^2 + 11^2 + 17^2 are the only such representations of 419, so a(5) = 2.

Mathematica

lst = {}; r = 7; While[r < 132, q = 5; While[q < r, P = 9 + q^2 + r^2; If[PrimeQ@P, AppendTo[lst, P]];
  q = NextPrime@q]; r = NextPrime@r]; Take[Transpose[Tally@Sort@lst][[2]], 105]

Crossrefs

Cf. A085317, A125516, A182479.

Sequence in context: A327804 A056624 A193348 * A354974 A357135 A367987

Adjacent sequences: A263720 A263721 A263722 * A263724 A263725 A263726

Keyword

nonn

Author

Jonathan Sondow and Robert G. Wilson v, Oct 24 2015

Status

approved