

A140774


Consider the products of all pairs of consecutive (when ordered by size) 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



0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 3, 2, 2, 2, 3, 1, 2, 2, 4, 1, 3, 1, 2, 3, 2, 1, 5, 1, 3, 2, 2, 1, 4, 2, 4, 2, 2, 1, 6, 1, 2, 3, 3, 2, 3, 1, 2, 2, 4, 1, 5, 1, 2, 3, 2, 2, 3, 1, 5, 2, 2, 1, 5, 2, 2, 2, 3, 1, 5, 2, 2, 2, 2, 2, 6, 1, 3, 2, 3, 1, 3, 1, 3, 4
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OFFSET

1,6


COMMENTS

Least number k whose value, a(k)=j beginning with j=0, is: 1, 2, 6, 12, 24, 48, 60, 168, 336, 240, 360, 672, 840, 720, 1512, 1680, 1440, 4320, 2520, 4200, 5040, 6720, 7560, 12480, 13440, 15840, ..., .  Robert G. Wilson v, May 30 2008


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 2 pairs of consecutive divisors whose product divides 20: 1*2=2, 4*5 = 20. Likewise, there are 2 such products that are divisible by 20: 4*5=20, 10*20=200. So a(20) = 2.


MATHEMATICA

f[n_] := Block[{d = Divisors@ n}, Count[n/((Most@d) (Rest@d)), _Integer]]; Array[f, 105]  Robert G. Wilson v, May 30 2008


PROG

(PARI)
\\ Two implementations, after the two different interpretations given by the author of the sequence:
A140774v1(n) = { my(ds = divisors(n), s=0); if(1==n, 0, for(i=1, (#ds)1, if(!(n%(ds[i]*ds[1+i])), s=s+1))); s; }
A140774v2(n) = { my(ds = divisors(n), s=0); if(1==n, 0, for(i=1, (#ds)1, if(!((ds[i]*ds[1+i])%n), s=s+1))); s; }
\\ Antti Karttunen, May 19 2017


CROSSREFS

Cf. A140773.
Differs from A099042 for the first time at n=32, where a(32) = 3, while A099042(32) = 2.
Sequence in context: A241276 A292286 A099042 * A056924 A316364 A318357
Adjacent sequences: A140771 A140772 A140773 * A140775 A140776 A140777


KEYWORD

nonn


AUTHOR

Leroy Quet, May 29 2008


EXTENSIONS

More terms from Robert G. Wilson v, May 30 2008


STATUS

approved



