A114989 Numbers whose sum of squares of distinct prime factors is prime.

6, 10, 12, 14, 18, 20, 24, 26, 28, 34, 36, 40, 48, 50, 52, 54, 56, 68, 72, 74, 80, 94, 96, 98, 100, 104, 105, 108, 112, 134, 136, 144, 146, 148, 160, 162, 188, 192, 194, 196, 200, 206, 208, 216, 224, 231, 250, 268, 272, 273, 274, 288, 292, 296, 315, 320, 324, 326

Offset

1,1

Comments

A005063 is "sum of squares of primes dividing n." Hence this is the sum of squares of prime factors analog of A114522 "numbers n such that sum of distinct prime divisors of n is prime." Note the distinction between A005063 and A067666 is "sum of squares of prime factors of n (counted with multiplicity)."

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Formula

{k such that A005063(k) is prime}. {k such that A005063(k) is an element of A000040}. {k = (for distinct i, j, ... prime(i)^e_1 * prime(j)^e_2 * ...) such that (prime(i)^2 * prime(j)^2 * ...) is prime}.

Example

a(1) = 6 because 6 = 2 * 3 and 2^2 + 3^2 = 13 is prime.
a(2) = 10 because 10 = 2 * 5 and 2^2 + 5^2 = 29 is prime.
a(3) = 12 because 12 = 2^2 * 3 and 2^2 + 3^2 = 13 is prime (note that we are not counting the prime factors with multiplicity).
a(4) = 14 because 14 = 2 * 7 and 2^2 + 7^2 = 53 is prime.
a(8) = 26 because 26 = 2 * 3 and 2^2 + 13^2 = 173 is prime.
a(10) = 34 because 34 = 2 * 17 and 2^2 + 17^2 = 293 is prime.

Maple

with(numtheory): a:=proc(n) local DPF: DPF:=factorset(n): if isprime(sum(DPF[j]^2, j=1..nops(DPF)))=true then n else fi end: seq(a(n), n=1..400); # Emeric Deutsch, Mar 07 2006

Mathematica

Select[Range[400], PrimeQ[Total[Transpose[FactorInteger[#]][[1]]^2]]&] (* Harvey P. Dale, Jan 16 2016 *)

Prog

(PARI) is(n)=isprime(norml2(factor(n)[, 1]))

Crossrefs

Cf. A005063, A067666, A014612, A014613, A069273, A069279, A069281, A114522.

Sequence in context: A213716 A337486 A373676 * A362012 A334166 A363171

Adjacent sequences: A114986 A114987 A114988 * A114990 A114991 A114992

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 22 2006

Extensions

More terms from Emeric Deutsch, Mar 07 2006

Status

approved