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A114989 Numbers whose sum of squares of distinct prime factors is prime. 1
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 (list; graph; refs; listen; history; text; internal format)
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: A100367 A213716 A337486 * A334166 A336323 A175352

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

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Last modified February 3 20:51 EST 2023. Contains 360045 sequences. (Running on oeis4.)