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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

%I

%S 4,16,27,256,3125,19683,65536,823543,1096744,2836295,4294967296,

%T 4473671462,23040925705,285311670611,7625597484987,13579716377989,

%U 119429556097859,302875106592253

%N 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).

%C Every prime is the sum of the first powers of its prime factors, so only composite numbers have been considered in this sequence.

%C 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.

%C 4473671462 = 2*13*179*593*1621 is also not a prime power.

%C a(15) <= 7625597484987. a(16) <= 302875106592253. - _Donovan Johnson_, May 17 2010

%C a(16) <= 13579716377989, a(17) <= 119429556097859, a(18) <= 302875106592253. - _Jud McCranie_, Feb 09 2016

%C a(19) <= 298023223876953125. - _Jud McCranie_, Apr 25 2016

%H S. P. Hurd and J. S. McCranie, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Hurd/hurd1.html">Integers that are the Uniform Sum of Uniform Powers of all their Prime Factors</a>, J. of Int. Seq., vol 22 (2019), article 19.3.4.

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

%t 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}]]]]]

%o (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

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

%K nonn

%O 1,1

%A _Joseph L. Pe_, Feb 04 2002

%E Edited by _Dean Hickerson_, Mar 07 2002

%E More terms from _Jud McCranie_, Mar 10 2003

%E a(13)-a(14) from _Donovan Johnson_, May 17 2010

%E a(15) confirmed by _Jud McCranie_, Jan 30 2016

%E a(16) from _Jud McCranie_, Feb 13 2016

%E a(17) from _Jud McCranie_, Mar 20 2016

%E a(18) from _Jud McCranie_, Apr 23 2016

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Last modified July 23 14:11 EDT 2019. Contains 325254 sequences. (Running on oeis4.)