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 A227680 Numbers whose sum of semiprime divisors is a prime number. 2
 30, 36, 42, 66, 70, 72, 78, 105, 108, 114, 130, 144, 154, 165, 174, 182, 196, 210, 216, 222, 231, 238, 246, 255, 273, 282, 285, 286, 288, 310, 318, 324, 345, 357, 366, 370, 385, 392, 399, 418, 430, 432, 434, 441, 442, 455, 462, 465, 474, 483, 494, 498, 518 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS There exists a subsequence of infinite squares {36, 144, 196, 324, 441, 576, 676, 784, 1089, 1225, 1296, 1764,...} because the numbers of the form n = (p*q)^2 with p and q primes are in the sequence if p^2 + p*q + q^2 is prime (subsequence of A007645), and the numbers p^2, p*q and q^2 are the three possible semiprime divisors of n. This numbers of the sequence are 6^2, 14^2, 21^2, 26^2, 33^2, 35^2, 51^2, 69^2,... The numbers of the form n = (p^a*q^v)^2 are also in the sequence => the sequence is infinite. There exists a subsequence of numbers having three distinct prime divisors p, q and r such that p*q+q*r+r*p is prime (see A087054). This numbers are 30, 42, 66, 70, 78, 105, 114, ... LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 EXAMPLE 30 is in the sequence because the semiprime divisors of 30 are 2*3, 2*5 and 3*5 and the sum 6+10+15 = 31 is a prime number. MAPLE with(numtheory):for n from 2 to 600 do:x:=divisors(n):n1:=nops(x): y:=factorset(n):n2:=nops(y):s1:=0:s2:=0:for i from 1 to n1 do: if bigomega(x[i])=2 then s1:=s1+x[i]:else fi:od: s2:=sum('y[i]', 'i'=1..n2):if type(s1, prime)=true then printf(`%d, `, n):else fi:od: MATHEMATICA semipSigma[n_] := DivisorSum[n, # &, PrimeOmega[#] == 2 &]; Select[Range, PrimeQ @ semipSigma[#] &] (* Amiram Eldar, May 10 2020 *) CROSSREFS Cf. A007645 (primes of the form x^2 + xy + y^2). Cf. A087054 (primes of the form p*q + q*r + r*p where p, q and r are distinct prime numbers). Sequence in context: A357874 A325264 A345382 * A109426 A167325 A051657 Adjacent sequences: A227677 A227678 A227679 * A227681 A227682 A227683 KEYWORD nonn AUTHOR Michel Lagneau, Jul 19 2013 STATUS approved

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Last modified December 5 10:40 EST 2023. Contains 367589 sequences. (Running on oeis4.)