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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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2
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4, 16, 27, 256, 3125, 19683, 65536, 823543, 1096744, 2836295, 4294967296, 4473671462, 23040925705, 285311670611
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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. [From Donovan Johnson (donovan.johnson(AT)yahoo.com), May 17 2010]
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EXAMPLE
| 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
| 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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CROSSREFS
| Cf. A068916, A081177 (for values of r).
Sequence in context: A072653 A008478 A111260 * A046358 A046366 A097374
Adjacent sequences: A067685 A067686 A067687 * A067689 A067690 A067691
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KEYWORD
| nonn
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AUTHOR
| Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Feb 04 2002
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EXTENSIONS
| Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Mar 07 2002
More terms from Jud McCranie (JudMcCranie(AT)ugaalum.uga.edu), Mar 10 2003
a(13)-a(14) from Donovan Johnson (donovan.johnson(AT)yahoo.com), May 17 2010
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