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A071837 Numbers n with property that in the prime factorization of n all prime exponents are prime, their sum is also prime and equals the sum of distinct prime factors of n. 2

%I

%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 n with property that in the prime factorization of n all prime exponents are prime, their sum is also prime and equals the sum of distinct prime factors of n.

%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-Fran├žois 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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Last modified September 16 10:23 EDT 2019. Contains 327094 sequences. (Running on oeis4.)