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Search: id:A137377
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| A137377 |
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a(1)=0; for n >= 2, a(n) = a(n-1) + |d(n)-d(n-1)|, where d(n) is the number of positive divisors of n. |
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+0
1
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| 0, 1, 1, 2, 3, 5, 7, 9, 10, 11, 13, 17, 21, 23, 23, 24, 27, 31, 35, 39, 41, 41, 43, 49, 54, 55, 55, 57, 61, 67, 73, 77, 79, 79, 79, 84, 91, 93, 93, 97, 103, 109, 115, 119, 119, 121, 123, 131, 138, 141, 143, 145, 149, 155, 159, 163, 167, 167, 169, 179, 189
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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For any given n >= 2, a(n)/(n-1) is the average of the |d(k)-d(k-1)| over all k with 2 <= k <= n.
Partial sums of | A051950 | ("starting with n=1", i.e. more precisely, a(n) = sum(i=2..n, | d(i)-d(i-1) | ) = sum(i=1..n-1, | d(i+1)-d(i) | ) = sum(i=1..n-1, | A051950(i) | ) - M. F. Hasler, Apr 21 2008
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FORMULA
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The following is an empirical formula which is a very good fit for the range n >= 10290 out to about n = 500000000: a(n) ~= n*log(n)+(log(n)*0.122-1)*(n*log(log(n))). - Jack Brennan (jb(AT)brennen.net), Apr 21 2008. The constant 0.122 is an empirical guess analogous to Legendre's constant B in Pi(n) ~ n/(log(n)+B).
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PROGRAM
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(PARI) a(n)=sum(i=2, n, abs(numdiv(i)-numdiv(i-1))) - M. F. Hasler, Apr 21 2008
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CROSSREFS
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Cf. A000005, A051950.
Sequence in context: A035061 A096738 A117284 this_sequence A026277 A076355 A083033
Adjacent sequences: A137374 A137375 A137376 this_sequence A137378 A137379 A137380
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KEYWORD
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nonn
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AUTHOR
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Leroy Quet (qq-quet(AT)mindspring.com), Apr 21 2008
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EXTENSIONS
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More terms from M. F. Hasler, Apr 21 2008
Edited by njas, Apr 26 2008
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