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Smaller of two consecutive primes each of which is the sum of two squares.
3

%I #33 Oct 11 2024 16:11:01

%S 37,109,193,229,277,313,349,389,397,401,449,457,509,613,661,673,701,

%T 757,761,769,797,853,929,937,997,1009,1093,1109,1193,1201,1213,1237,

%U 1373,1429,1489,1549,1597,1609,1637,1669

%N Smaller of two consecutive primes each of which is the sum of two squares.

%C This sequence is a subsequence of A002313 (Primes of form x^2 + y^2).

%H Abhiram R Devesh, <a href="/A257117/b257117.txt">Table of n, a(n) for n = 1..1000</a>

%e 37 = 1^2 + 6^2 and 41 = 4^2 + 5^2, so 37 is a term.

%e 109 = 3^2 + 10^2 and 113 = 7^2 + 8^2, so 109 is a term.

%o (Python)

%o import sympy

%o def sumpow(sn0, n, p):

%o af=0; bf=0; an=1

%o sn1=sn0+n

%o if n!=0:

%o sn1=sympy.nextprime(sn0, n)

%o while an**p<sn1:

%o bnsq=sn1-(an**p)

%o bn=sympy.ntheory.perfect_power(bnsq)

%o if bn!=False and list(bn)[1]==p:

%o af=an

%o bf=list(bn)[0]

%o an=sn1+100

%o an=an+1

%o return(af, bf)

%o s0=1; pw=2

%o while s0>0:

%o a0, b0=sumpow(s0, 0, pw)

%o a1, b1=sumpow(s0, 1, pw)

%o if a0!=0 and a1!=0:

%o print(s0)

%o s0=sympy.nextprime(s0)

%Y Cf. A002313 (Primes of form x^2 + y^2).

%K nonn,easy

%O 1,1

%A _Abhiram R Devesh_, Apr 25 2015