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Search: id:A064608
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| A064608 |
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Partial sums of A034444: sum of number of unitary divisors from 1 to n. |
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+0
3
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| 1, 3, 5, 7, 9, 13, 15, 17, 19, 23, 25, 29, 31, 35, 39, 41, 43, 47, 49, 53, 57, 61, 63, 67, 69, 73, 75, 79, 81, 89, 91, 93, 97, 101, 105, 109, 111, 115, 119, 123, 125, 133, 135, 139, 143, 147, 149, 153, 155, 159, 163, 167, 169, 173, 177, 181, 185, 189, 191, 199, 201
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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a(n)=sum(k<=n,2^omega(k)) where omega(k) is the number of distinct primes in k factorization. - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 16 2002
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REFERENCES
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E. Cohen, The number of unitary divisors of an integer, Am. Math. Mon. 67, 879-880 (1960).
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Leipzig 1909 (Chelsea reprint 1953), p. 594.
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LINKS
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Harry J. Smith, Table of n, a(n) for n=1,...,1000
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.
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FORMULA
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a(n) = a(n-1) + A034444(n) = a(n-1)+2^[A001221(n)] Sum{ud[j]; j=1..n} where ud[j] = A034444(j)=2^A001221(n)
a(n)=n*ln(n)/zeta(2)+O(n) where zeta(2)=Pi^2/6 - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 16 2002
a(n)=sum(k=1, n, mu(k)^2*floor(n/k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 16 2002
Merten's theorem (1874): a(n) = Sum_{k<=n} ud(k) = (n/Zeta(2))*(ln(n)+2*gamma-1-2*Zeta'(2)/Zeta(2)) + O(sqrt(n)*ln(n)), where gamma is the Euler-Mascheroni constant A001620. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002
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PROGRAM
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(PARI) { for (n=1, 1000, a=sum(k=1, n, moebius(k)^2*floor(n/k)); write("b064608.txt", n, " ", a) ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Sep 20 2009]
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CROSSREFS
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Cf. A034444, A064610, A001221.
Sequence in context: A067031 A029740 A063425 this_sequence A024893 A119253 A131437
Adjacent sequences: A064605 A064606 A064607 this_sequence A064609 A064610 A064611
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KEYWORD
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nonn
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AUTHOR
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Labos E. (labos(AT)ana.sote.hu), Sep 24 2001
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