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

1,10

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).

Links

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

Formula

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

Sequence in context: A336466 A052125 A214874 * A255811 A131268 A109221

Adjacent sequences: A081057 A081058 A081059 * A081061 A081062 A081063

Keyword

nonn

Author

Reinhard Zumkeller, Mar 04 2003

Status

approved