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).

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

Offset: 1

Author

Joseph L. Pe, Feb 04 2002

Keywords

• nonn

Comments

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

Examples

			The sum of the cubes of the prime factors of 1096744 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 (2019), article 19.3.4.

Crossrefs

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

Programs

Mathematica

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}]]]]]

PARI

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

Extensions

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