

A140773


Consider the products of all pairs of (not necessarily distinct) positive divisors of n. a(n) = the number of these products that divide n. a(n) also = the number of the products that are divisible by n.


2



1, 2, 2, 4, 2, 5, 2, 6, 4, 5, 2, 10, 2, 5, 5, 9, 2, 10, 2, 10, 5, 5, 2, 16, 4, 5, 6, 10, 2, 14, 2, 12, 5, 5, 5, 20, 2, 5, 5, 16, 2, 14, 2, 10, 10, 5, 2, 24, 4, 10, 5, 10, 2, 16, 5, 16, 5, 5, 2, 28, 2, 5, 10, 16, 5, 14, 2, 10, 5, 14, 2, 32, 2, 5, 10, 10, 5, 14, 2, 24, 9, 5, 2, 28, 5, 5, 5, 16, 2, 28, 5
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OFFSET

1,2


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000


EXAMPLE

The divisors of 20 are 1,2,4,5,10,20. There are 10 pairs of divisors whose product divides 20: 1*1=1, 1*2=2, 1*4=4, 1*5=5, 1*10=10, 1*20=20, 2*2=4, 2*5=10, 2*10=20, 4*5 = 20. Likewise, there are 10 products that are divisible by 20: 4*5=20, 2*10=20, 4*10=40, 10*10=100, 1*20=20, 2*20=40, 4*20=80, 5*20=100, 10*20=200, 20*20=400. So a(20) = 10.


MATHEMATICA

(* first do *) Needs["Combinatorica`"] (* then *) f[n_] := Count[ n/Times @@@ Union[Sort /@ Tuples[Divisors@ n, 2]], _Integer]; Array[f, 91]  Robert G. Wilson v, May 31 2008


PROG

(PARI)
\\ Two implementations, after the two different interpretations given by the author of the sequence:
A140773v1(n) = { my(ds = divisors(n), s=0); for(i=1, #ds, for(j=i, #ds, if(!(n%(ds[i]*ds[j])), s=s+1))); s; }
A140773v2(n) = { my(ds = divisors(n), s=0); for(i=1, #ds, for(j=i, #ds, if(!((ds[i]*ds[j])%n), s=s+1))); s; }
\\ Antti Karttunen, May 19 2017


CROSSREFS

Cf. A140774.
Sequence in context: A054134 A319822 A005127 * A133911 A069932 A056148
Adjacent sequences: A140770 A140771 A140772 * A140774 A140775 A140776


KEYWORD

nonn


AUTHOR

Leroy Quet, May 29 2008


EXTENSIONS

Corrected and extended by Robert G. Wilson v, May 31 2008


STATUS

approved



