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Search: id:A032741
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| A032741 |
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a(0) = 0; for n > 0, a(n) = number of proper divisors of n. |
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+0
68
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| 0, 0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 5, 1, 3, 3, 4, 1, 5, 1, 5, 3, 3, 1, 7, 2, 3, 3, 5, 1, 7, 1, 5, 3, 3, 3, 8, 1, 3, 3, 7, 1, 7, 1, 5, 5, 3, 1, 9, 2, 5, 3, 5, 1, 7, 3, 7, 3, 3, 1, 11, 1, 3, 5, 6, 3, 7, 1, 5, 3, 7, 1, 11, 1, 3, 5, 5, 3, 7, 1, 9, 4, 3, 1, 11, 3, 3, 3, 7, 1, 11, 3, 5, 3, 3, 3, 11, 1, 5, 5
(list; graph; listen)
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OFFSET
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0,5
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COMMENT
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Number of d < n which divide n.
Call an integer k between 1 and n a "semi-divisor" of n if n leaves a remainder of 1 when divided by k, i.e. n == 1 (mod k). a(n) gives the number of semi-divisors of n+1. - Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Sep 11 2002
a(n+1) is also the number of k, 0<=k<=n-1, such that C(n,k) divides C(n,k+1) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 17 2002
Also the number of factors of the n-th degree polynomial x^n + x^(n-1) + x^(n-2) + ... + x^2 + x + 1.
Also the number of factors of the n-th Fibonacci polynomial. - T. D. Noe (noe(AT)sspectra.com), Mar 09 2006
Number of partitions of n into 2 parts with the second dividing the first. - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Sep 20 2006
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LINKS
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Eric Weisstein's World of Mathematics, Proper divisors
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FORMULA
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G.f.: Sum_{n>0} x^(2n)/(1-x^n). - Michael Somos, Apr 29, 2003
G.f.: sum(i=1, oo, (1-x^i+x^(2*i))/(1-x^i)) - Jon Perry (perry(AT)globalnet.co.uk), Jul 03 2004
a(n) = SUM(A051731(n-k,k): 1 <= k <= n/2). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 01 2009]
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EXAMPLE
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a(6) = 3 since the proper divisors of 6 are 1, 2, 3.
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PROGRAM
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(PARI) a(n) = if(n<1, 0, numdiv(n)-1)
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CROSSREFS
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Equals tau(n) - 1 = A000005(n) - 1. Cf. A039653.
Column 2 of A122934.
Cf. A003238, A001065, A027751 (list of proper divisors).
Sequence in context: A128199 A036459 A079167 this_sequence A046051 A025812 A109698
Adjacent sequences: A032738 A032739 A032740 this_sequence A032742 A032743 A032744
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KEYWORD
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nonn,easy
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AUTHOR
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Patrick De Geest (pdg(AT)worldofnumbers.com), May 15 1998.
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EXTENSIONS
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Typos in definition corrected by Omar E. Pol, Dec 13 2008
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