The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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). 3
 4, 16, 27, 256, 3125, 19683, 65536, 823543, 1096744, 2836295, 4294967296, 4473671462, 23040925705, 285311670611, 7625597484987, 13579716377989, 119429556097859, 302875106592253 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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 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. EXAMPLE The sum of the cubes of the prime factors of 1096744 is 3*2^3 + 3*11^3 + 103^3 = 1096744. 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}]]]]] PROG (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 CROSSREFS Cf. A068916, A081177 (for values of r), A268036 (for a subsequence). Sequence in context: A008478 A201009 A111260 * A097374 A257309 A271936 Adjacent sequences: A067685 A067686 A067687 * A067689 A067690 A067691 KEYWORD nonn AUTHOR Joseph L. Pe, Feb 04 2002 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 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 1 18:36 EST 2022. Contains 358475 sequences. (Running on oeis4.)