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A227680
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Numbers whose sum of semiprime divisors is a prime number.
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2
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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
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OFFSET
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1,1
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COMMENTS
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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, ...
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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semipSigma[n_] := DivisorSum[n, # &, PrimeOmega[#] == 2 &]; Select[Range[500], PrimeQ @ semipSigma[#] &] (* Amiram Eldar, May 10 2020 *)
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CROSSREFS
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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).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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