A081060 Product of differences of distinct prime factors of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 2, 1, 1, 1, 1, 3, 4, 9, 1, 1, 1, 11, 1, 5, 1, 6, 1, 1, 8, 15, 2, 1, 1, 17, 10, 3, 1, 20, 1, 9, 2, 21, 1, 1, 1, 3, 14, 11, 1, 1, 6, 5, 16, 27, 1, 6, 1, 29, 4, 1, 8, 72, 1, 15, 20, 30, 1, 1, 1, 35, 2, 17, 4, 110, 1, 3, 1, 39, 1, 20, 12, 41, 26, 9, 1, 6, 6, 21, 28

Offset: 1

Author

Reinhard Zumkeller, Mar 04 2003

Keywords

• nonn

Comments

a(n)=1 iff n is 3-smooth (A003586) or n is a prime power (A000961), see A081061;

a(A006881(k)) > 1 for k > 1; if a(n) > 1 then A079275(n) > 0.

From Robert G. Wilson v, Aug 06 2018: (Start)

First occurrence of k, k=1,2,3,... or 0 if impossible: 1, 15, 10, 21, 14, 30, 0, 33, 22, 39, 26, 85, 0, 51, 34, 57, 38, 115, ..., ;

Impossible values: 7, 13, 19, 23, 25, 31, 33, 37, 43, 47, 49, 53, 55, 61, 63, 67, 73, 75, 79, 83, 85, 89, 91, 93, 97, ..., ;

Records: 1, 3, 5, 9, 11, 15, 17, 20, 21, 27, 29, 72, 110, 210, 272, 420, 540, 702, 812, 1190, 1482, 1640, 1980, 2262, 2550, 2592, 3192, 3422, 5280, 5760, 5852, ..., .

(End).

Examples

			a(42) = a(2*3*7) = |2-3|*|2-7|*|3-7| = 1*5*4 = 20.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Programs

Mathematica

a[n_] := Times @@ Flatten[Differences@# & /@ Subsets[First@# & /@ FactorInteger@n, {2}]]; Array[a, 90] (* Robert G. Wilson v, Aug 06 2018 *)

PARI

A081060(n) = if(omega(n)<=1,1,my(ps = factor(n)[, 1]~, m=1); for(i=1,(#ps)-1,for(j=i+1,#ps, m *= (ps[j]-ps[i]))); (m)); \\ Antti Karttunen, Aug 06 2018

Formula

a(n) = Product(abs(p-q): p, q distinct prime factors of n).