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A003959
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If n = Product p(k)^e(k) then a(n) = Product (p(k)+1)^e(k), a(1) = 1.
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28
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1, 3, 4, 9, 6, 12, 8, 27, 16, 18, 12, 36, 14, 24, 24, 81, 18, 48, 20, 54, 32, 36, 24, 108, 36, 42, 64, 72, 30, 72, 32, 243, 48, 54, 48, 144, 38, 60, 56, 162, 42, 96, 44, 108, 96, 72, 48, 324, 64, 108, 72, 126, 54, 192, 72, 216, 80, 90, 60, 216, 62, 96, 128, 729, 84, 144, 68
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Completely multiplicative.
Sum of divisors of n with multiplicity. If n = p^m, the number of ways to make p^k as a divisor of n is C(m,k); and sum(C(m,k)*p^k) = (p+1)^k. The rest follows because the function is multiplicative. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jan 25 2010]
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LINKS
| Daniel Forgues, Table of n, a(n) for n=1..100000
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FORMULA
| If n = Product p(k)^e(k) then a(n) = Product (p(k)+1)^e(k), a(1) = 1.
Multiplicative with a(p^e) = (p+1)^e. - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
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MATHEMATICA
| a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]]+1)^fi[[All, 2]])); a /@ Range[67] (* From Jean-François Alcover, Apr 22 2011 *)
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PROG
| (PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1/(1-X-p*X))[n]) (from R. Stephan)
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CROSSREFS
| Apart from initial terms, same as A064478. Cf. A003958.
Cf. A063441.
Cf. A168065, A168066. [From Daniel Forgues (squid(AT)zensearch.com), Dec 01 2009]
Cf. A163407. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jan 25 2010]
Sequence in context: A016656 A084425 A168512 * A168341 A083111 A132065
Adjacent sequences: A003956 A003957 A003958 * A003960 A003961 A003962
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KEYWORD
| nonn,easy,nice,mult
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AUTHOR
| Marc LeBrun (mlb(AT)well.com)
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EXTENSIONS
| Definition reedited (with formula) by Daniel Forgues (squid(AT)zensearch.com), Nov 17 2009
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