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.

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

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