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%I #24 Jul 16 2021 13:17:37
%S 4,27,72,108,800,3125,12500,247808,823543,37879808,190512000,
%T 266716800,428652000,529200000,600112800,1234800000,1452124800,
%U 2420208000,2679075000,3307500000,7351381800,7441875000,7717500000,9376762500
%N Numbers k with the property that in the prime factorization of k all prime exponents are prime, their sum is also prime and equals the sum of distinct prime factors of k.
%e 800 is a term as 800 = 2^5 * 5^2, 2+5 = 5+2 = 7, and 7,5,2 are primes.
%t terms = 24; fromFactors[s_List] := (Times @@ (s^#)&) /@ Permutations[s]; Clear[f]; f[n_] := f[n] = (ssp = Select[Subsets[Prime[Range[n]]] // Rest, PrimeQ[Total[#]]&]; fromFactors /@ ssp // Flatten // Union // PadRight[#, terms]& ); f[2]; f[n = 4]; While[Print["n = ", n]; f[n] != f[n-2], n = n+2]; f[n] (* _Jean-François Alcover_, Jul 20 2015 *)
%o (PARI) isok(n) = {f = factor(n); for (i=1, #f~, if (! isprime(f[i, 2]), return (0));); isprime(se = sum(i=1, #f~, f[i, 2])) && (se == sum(i=1, #f~, f[i, 1]));} \\ _Michel Marcus_, Aug 21 2014
%o (Python)
%o from sympy import factorint, isprime
%o A071837 = []
%o for n in range(1,10**5):
%o ....f = factorint(n)
%o ....fp, fe = list(f.keys()),list(f.values())
%o ....if sum(fp) == sum(fe) and isprime(sum(fe)) and all([isprime(e) for e in fe]):
%o ........A071837.append(n)
%o # _Chai Wah Wu_, Aug 27 2014
%Y Cf. A054411, A008472, A001222, A056166, A070215.
%Y A240983 and A051674 are subsequences. - _Zak Seidov_, Aug 21 2014
%K nonn,nice
%O 1,1
%A _Reinhard Zumkeller_, Jun 08 2002
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