

A230476


a(n) = Sum_{i=1..n} d(8*i+1)  Sum_{i=1..n} d(2*i+1), where d(n) = A000005(n) is the number of divisors of n.


2



1, 1, 2, 3, 3, 4, 4, 6, 6, 7, 7, 6, 10, 10, 11, 11, 9, 11, 13, 15, 16, 14, 16, 15, 15, 17, 17, 22, 22, 22, 20, 18, 20, 24, 24, 25, 27, 27, 27, 26, 28, 26, 30, 30, 29, 31, 31, 37, 35, 35, 35, 31, 35, 35, 40, 40, 38, 40, 40, 41, 41, 41, 43, 47, 47, 46, 42, 44, 46, 50, 48, 46, 52, 52, 52, 54, 52, 55, 55, 53, 55, 53, 59, 58, 56, 58
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OFFSET

1,3


COMMENTS

Cimadevilla proved that a(n) >= 0. That is surprising because d(8*i+1)  d(2*i+1) < 0 for i = 12, 17, 22, 24, 31, 32, 40, 42, 45, 49, 52, 57, 66, 67, 71, 72, 77, 80, 82, 84, 85, ...


LINKS

Table of n, a(n) for n=1..86.
Jorge Luis Cimadevilla Villacorta, Certain inequalities associated with the divisor function, Amer. Math. Monthly, 120 (2013), 832837. See inequalities (1.5).


FORMULA

a(n) = Sum_{i=1..n} (d(8*i+1)  d(2*i+1)) = A230293(n) + A230294(n).


EXAMPLE

The divisors of 8*1+1=9 are 1, 3, 9 and those of 2*1+1=3 are 1, 3, so a(1)=d(9)d(3)=32=1.


MATHEMATICA

Sum[ DivisorSigma[0, 8 i + 1]  DivisorSigma[0, 2 i + 1], {i, n}]


PROG

(PARI) a(n) = sum(i=1, n, numdiv(8*i+1)  numdiv(2*i+1)); \\ Michel Marcus, Jun 19 2015


CROSSREFS

Cf. A000005, A230290, A230291, A230292, A230293, A230295, A230296.
Sequence in context: A344870 A318283 A105677 * A103297 A274017 A267597
Adjacent sequences: A230473 A230474 A230475 * A230477 A230478 A230479


KEYWORD

nonn


AUTHOR

Jonathan Sondow, Oct 20 2013


STATUS

approved



