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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 (list; graph; refs; listen; history; text; internal format)
 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 Harvey P. Dale, Table of n, a(n) for n = 1..1000 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 Cf. A000005, A051950. Sequence in context: A096738 A167857 A117284 * A274793 A339238 A168543 Adjacent sequences:  A137374 A137375 A137376 * A137378 A137379 A137380 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

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Last modified October 5 21:37 EDT 2022. Contains 357261 sequences. (Running on oeis4.)