

A137377


a(1)=0; for n >= 2, a(n) = a(n1) + d(n)d(n1), 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
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OFFSET

1,4


COMMENTS

For any given n >= 2, a(n)/(n1) is the average of the d(k)d(k1) over all k with 2 <= k <= n.
Partial sums of A051950, i.e., a(n) = Sum_{i=2..n} d(i)d(i1) = 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)*0.1221)*(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(i1)))  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



