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A137377
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.
1
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
OFFSET
1,4
COMMENTS
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|, i.e., a(n) = Sum_{i=2..n} |d(i)-d(i-1)| = Sum_{i=2..n} |A051950(i)|. - M. F. Hasler, Apr 21 2008
LINKS
FORMULA
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 Brennen, Apr 21 2008. The constant 0.122 is an empirical guess analogous to Legendre's constant B in Pi(n) ~ n/(log(n)+B).
MATHEMATICA
nxt[{n_, a_}]:={n+1, a+Abs[DivisorSigma[0, n+1]-DivisorSigma[0, n]]}; NestList[ nxt, {1, 0}, 60][[All, 2]] (* Harvey P. Dale, Nov 05 2019 *)
PROG
(PARI) a(n)=sum(i=2, n, abs(numdiv(i)-numdiv(i-1))) \\ M. F. Hasler, Apr 21 2008
CROSSREFS
Sequence in context: A096738 A167857 A117284 * A274793 A339238 A168543
KEYWORD
nonn
AUTHOR
Leroy Quet, Apr 21 2008
EXTENSIONS
More terms from M. F. Hasler, Apr 21 2008
Edited by N. J. A. Sloane, Apr 26 2008
STATUS
approved