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A067688
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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).
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3
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4, 16, 27, 256, 3125, 19683, 65536, 823543, 1096744, 2836295, 4294967296, 4473671462, 23040925705, 285311670611, 7625597484987, 13579716377989, 119429556097859, 302875106592253
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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The sum of the cubes of the prime factors of 1096744 is 3*2^3 + 3*11^3 + 103^3 = 1096744.
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MATHEMATICA
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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}]]]]]
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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