A268036 -id:A268036 - cp's OEIS frontend

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

Dean Hickerson, Mar 07 2002

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

			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.

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.

Cf. A067688, A268036.

Cf. A081177, A000325, A024024, A024050.

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

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

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

A067688 Composite n such that for some integer r, n equals the sum of the r-th powers of the prime factors of n (counted with multiplicity).

Original entry on oeis.org

4, 16, 27, 256, 3125, 19683, 65536, 823543, 1096744, 2836295, 4294967296, 4473671462, 23040925705, 285311670611, 7625597484987, 13579716377989, 119429556097859, 302875106592253

Offset: 1

Joseph L. Pe, Feb 04 2002

nonn

Every prime is the sum of the first powers of its prime factors, so only composite numbers have been considered in this sequence.
Every integer of the form p^p^k with p prime and k>0 is in the sequence, since it equals the sum of the (p^k - k)-th powers of its prime factors. The first 8 terms of the sequence are of this form, but 1096744 = 2^3*11^3*103 and 2836295 = 5*7*11*53*139 are not.
4473671462 = 2*13*179*593*1621 is also not a prime power.
a(15) <= 7625597484987. a(16) <= 302875106592253. - Donovan Johnson, May 17 2010
a(16) <= 13579716377989, a(17) <= 119429556097859, a(18) <= 302875106592253. - Jud McCranie, Feb 09 2016
a(19) <= 298023223876953125. - Jud McCranie, Apr 25 2016

			The sum of the cubes of the prime factors of 1096744 is 3*2^3 + 3*11^3 + 103^3 = 1096744.

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 (2019), article 19.3.4.

Cf. A068916, A081177 (for values of r), A268036 (for a subsequence).

For[n=2, True, n++, If[ !PrimeQ[n], For[r=1; fn=FactorInteger[n]; s=0, s<=n, r++, s=Plus@@((#[[2]]#[[1]]^r)&/@fn); If[s==n, Print[{n, r}]]]]]

is(n)=if(isprime(n)||n<4, return(0)); my(f=factor(n),t=#f~); for(r=1,logint(n\f[t,2],f[t,1]), if(sum(i=1,t,f[i,2]*f[i,1]^r)==n, return(1))); 0 \\ Charles R Greathouse IV, Jan 30 2016

Edited by Dean Hickerson, Mar 07 2002
More terms from Jud McCranie, Mar 10 2003
a(13)-a(14) from Donovan Johnson, May 17 2010
a(15) confirmed by Jud McCranie, Jan 30 2016
a(16) from Jud McCranie, Feb 13 2016
a(17) from Jud McCranie, Mar 20 2016
a(18) from Jud McCranie, Apr 23 2016

cp's OEIS Frontend

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

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