A071837 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.

4, 27, 72, 108, 800, 3125, 12500, 247808, 823543, 22579200, 37879808, 190512000, 266716800, 428652000, 529200000, 600112800, 868020300, 1190700000, 1234800000, 1452124800, 2420208000, 2679075000, 3267280800, 3307500000, 4984012800, 6994132992, 7351381800, 7441875000, 7717500000, 9376762500

Offset: 1

Author

Reinhard Zumkeller, Jun 08 2002

Keywords

• nonn
• nice

Examples

			800 is a term as 800 = 2^5 * 5^2, 2+5 = 5+2 = 7, and 7,5,2 are primes.
		

Crossrefs

Cf. A054411, A008472, A001222, A056166, A070215.

A240983 and A051674 are subsequences. - Zak Seidov, Aug 21 2014

Programs

Mathematica

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 *)

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

Python

from sympy import factorint, isprime
A071837 = []
for n in range(1,10**5):
    f = factorint(n)
    fp, fe = list(f.keys()),list(f.values())
    if sum(fp) == sum(fe) and isprime(sum(fe)) and all([isprime(e) for e in fe]):
        A071837.append(n)
# Chai Wah Wu, Aug 27 2014

Extensions

Missing terms inserted by Sean A. Irvine, Aug 17 2024