A257118 - OEIS

%I #37 Jan 15 2025 17:48:44

%S 89,389,397,449,661,757,761,929,997,1193,1201,1669,2213,2269,2293,

%T 2593,2609,2617,2741,3037,3041,3209,3217,3413,3433,3449,3697,3877,

%U 4397,4801,5189,5233,5237,5569,5689,5717,6101,6217,6389,6469,6733,6829,6833,6997,7529

%N Smallest of three consecutive prime numbers each of which is the sum of two squares.

%C This sequence is a subsequence of A257117.

%H Harvey P. Dale, <a href="/A257118/b257118.txt">Table of n, a(n) for n = 1..200</a>

%e 389 = 10^2 + 17^2, 397 = 6^2 + 19^2, and 401 = 1^2 + 20^2, so 389 is a term.

%e 397 = 6^2 + 19^2, 401 = 1^2 + 20^2, and 409 = 3^2 + 20^2, so 397 is a term.

%t Prime/@SequencePosition[Table[If[Count[IntegerPartitions[n,{2}],_?(AllTrue[ Sqrt[#],IntegerQ]&)]>0,1,0],{n,Prime[Range[3000]]}],{1,1,1},Overlaps-> All] [[All,1]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jul 08 2018 *)

%o (Python)

%o # a(1) is not displayed.

%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     a2, b2=sumpow(s0, 2, pw)

%o     if a0!=0 and a1!=0 and a2!=0:

%o         print(s0)

%o     s0=sympy.nextprime(s0)

%Y Cf. A064716 (Smallest member of three consecutive numbers).

%Y Cf. A257117 (Smallest member of two consecutive prime numbers).

%K nonn,easy

%O 1,1

%A _Abhiram R Devesh_, Apr 25 2015

%E Corrected and extended by and prior b-file replaced by _Harvey P. Dale_, Jul 08 2018