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

1,1

Links

Table of n, a(n) for n=1..30.

Example

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

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 *)

Prog

(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

Crossrefs

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

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

Sequence in context: A122405 A122406 A276372 * A334633 A266011 A015238

Adjacent sequences: A071834 A071835 A071836 * A071838 A071839 A071840

Keyword

nonn,nice

Author

Reinhard Zumkeller, Jun 08 2002

Extensions

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

Status

approved