A068916 Smallest positive integer that is equal to the sum of the n-th powers of its prime factors (counted with multiplicity).

2, 16, 1096744, 3125, 256, 823543, 19683

Offset: 1

Author

Dean Hickerson, Mar 07 2002

Keywords

• nonn
• hard
• more

Comments

Does a(n) exist for all n?

a(12)=65536, a(27)=4294967296. a(n) exists for all n of the form n=p^i-i, where p is prime and i > 0, since p^p^i is an example (see A067688 and A081177). - Jud McCranie, Mar 16 2003

a(23) <= 298023223876953125. a(24) <= 7625597484987. - Jud McCranie, Jan 18 2016

a(10) = 285311670611. - Jud McCranie, Jan 25 2016

a(24) = 7625597484987. - Jud McCranie, Jan 30 2016

Examples

			a(3) = 1096744 = 2^3*11^3*103; the sum of the cubes of the prime factors is 3*2^3 + 3*11^3 + 103^3 = 1096744.
		

Links

• S. P. Hurd and J. S. McCranie, Integers that are Sums of Uniform Powers of all their Prime Factors: the sequence A068916, J. of Int. Seq., vol 22, article 19.3.4.

Crossrefs

Cf. A067688, A268036.

Cf. A081177, A000325, A024024, A024050.

Programs

Mathematica

a[n_] := For[x=1, True, x++, If[x==Plus@@(#[[2]]#[[1]]^n&/@FactorInteger[x]), Return[x]]]

PARI

isok(k, n) = {my(f=factor(k)); sum(j=1, #f~, f[j,2]*f[j,1]^n) == k;}
a(n) = {my(k = 1); while(! isok(k,n), k++); k;} \\ Michel Marcus, Jan 25 2016

Python

from sympy import factorint
def a(n):
  k = 1
  while True:
    f = factorint(k)
    if k == sum(f[d]*d**n for d in f): return k
    k += 1
for n in range(1, 8):
  print(a(n), end=", ") # Michael S. Branicky, Feb 16 2021