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.

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

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: A292286 A341596 A099042 * A345345 A056924 A342083

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