A263979 - OEIS

A263979

Least prime p of the form p = a^2 + b^2 with a > n and b > n.

2, 13, 41, 41, 61, 113, 113, 181, 181, 269, 313, 313, 421, 421, 613, 613, 613, 761, 761, 929, 1013, 1013, 1201, 1201, 1301, 1637, 1741, 1741, 1741, 1861, 2113, 2113, 2381, 2381, 2521, 2969, 2969, 3121, 3121, 3449, 3613, 3613, 4153, 4337, 4513, 4513, 4513, 5101, 5101, 5101, 5737, 5953, 6173, 6389, 6389, 6857, 7321, 7321, 7321, 7321

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OFFSET

0,1

COMMENTS

a(n) exists for every n; see Sierpinski (1988), p. 221.

The distinct primes in the sequence form A263980.

Conjecture: a(n) <= 2*(2n+1)^2 for all n >= 0.

REFERENCES

W. Sierpinski, Elementary Theory of Numbers, 2nd English edition, revised and enlarged by A. Schinzel, Elsevier, 1988.

LINKS

Table of n, a(n) for n=0..59.

FORMULA

a(n) == 1 or 2 mod 4.

EXAMPLE

The smallest prime of the form a^2 + b^2 with a > 2 and b > 2 is 41 = 4^2 + 5^2, so a(2) = 41 and a(3) = 41.

MATHEMATICA

Table[ Min[ Select[ Union[ Flatten[ With[{n = k}, Array[#1^2 + #2^2 &, {2n + 1, 2n + 1}, {n + 1, n + 1}] ]]], PrimeQ]], {k, 0, 59}] (* This assumes the Conjecture above. *)

CROSSREFS

Cf. A002144, A002313, A263980.

Sequence in context: A011919 A323684 A264529 * A042795 A361071 A179925

Adjacent sequences: A263976 A263977 A263978 * A263980 A263981 A263982

KEYWORD

nonn

AUTHOR

Jonathan Sondow, Nov 09 2015

STATUS

approved