A257118 Smallest of three consecutive prime numbers each of which is the sum of two squares.

89, 389, 397, 449, 661, 757, 761, 929, 997, 1193, 1201, 1669, 2213, 2269, 2293, 2593, 2609, 2617, 2741, 3037, 3041, 3209, 3217, 3413, 3433, 3449, 3697, 3877, 4397, 4801, 5189, 5233, 5237, 5569, 5689, 5717, 6101, 6217, 6389, 6469, 6733, 6829, 6833, 6997, 7529

Offset

1,1

Comments

This sequence is a subsequence of A257117.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..200

Example

389 = 10^2 + 17^2, 397 = 6^2 + 19^2, and 401 = 1^2 + 20^2, so 389 is a term.
397 = 6^2 + 19^2, 401 = 1^2 + 20^2, and 409 = 3^2 + 20^2, so 397 is a term.

Mathematica

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 *)

Prog

(Python)
# a(1) is not displayed.
import sympy
def sumpow(sn0, n, p):
    af=0; bf=0; an=1
    sn1=sn0+n
    if n!=0:
        sn1=sympy.nextprime(sn0, n)
    while an**p<sn1:
        bnsq=sn1-(an**p)
        bn=sympy.ntheory.perfect_power(bnsq)
        if bn!=False and list(bn)[1]==p:
            af=an
            bf=list(bn)[0]
            an=sn1+100
        an=an+1
    return(af, bf)
s0=1; pw=2
while s0>0:
    a0, b0=sumpow(s0, 0, pw)
    a1, b1=sumpow(s0, 1, pw)
    a2, b2=sumpow(s0, 2, pw)
    if a0!=0 and a1!=0 and a2!=0:
        print(s0)
    s0=sympy.nextprime(s0)

Crossrefs

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

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

Sequence in context: A143828 A253140 A142448 * A244777 A107192 A061372

Adjacent sequences: A257115 A257116 A257117 * A257119 A257120 A257121

Keyword

nonn,easy

Author

Abhiram R Devesh, Apr 25 2015

Extensions

Corrected and extended by and prior b-file replaced by Harvey P. Dale, Jul 08 2018

Status

approved