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A258618
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a(n) = (4*n+9)*n^2.
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1
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0, 13, 68, 189, 400, 725, 1188, 1813, 2624, 3645, 4900, 6413, 8208, 10309, 12740, 15525, 18688, 22253, 26244, 30685, 35600, 41013, 46948, 53429, 60480, 68125, 76388, 85293, 94864, 105125, 116100, 127813, 140288, 153549, 167620, 182525, 198288, 214933
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OFFSET
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0,2
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COMMENTS
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Consider a natural number r such that r has 19 proper divisors and 6 prime factors. (Note that these prime factors do not have to be distinct.) The difference between these two values, say d(r), is in this case 13. Where n is a positive integer, d(r^n)=(4*n+9)*n^2.
The integers that satisfy the proper-divisor-prime-factor requirement are those of A179644.
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LINKS
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FORMULA
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EXAMPLE
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The smallest integer that satisfies this is 240: It has 19 proper divisors (1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120) and 6 prime factors (2, 2, 2, 2, 3, 5), so d(240)=13. The square of 240, 57600, we would expect to have a difference of 68 between the number of its proper divisors and prime factors, and with respectively 80 and 12, d(57600)=68 indeed. Checking this with further integer powers of 240 will continue to generate terms in this sequence.
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MATHEMATICA
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LinearRecurrence[{4, -6, 4, -1}, {0, 13, 68, 189}, 40] (* Harvey P. Dale, Sep 12 2020 *)
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PROG
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(PARI) vector(50, n, n--; (4*n+9)*n^2) \\ Derek Orr, Jun 21 2015
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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