A143201 Product of distances between prime factors in factorization of n.

1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 6, 3, 1, 1, 2, 1, 4, 5, 10, 1, 2, 1, 12, 1, 6, 1, 6, 1, 1, 9, 16, 3, 2, 1, 18, 11, 4, 1, 10, 1, 10, 3, 22, 1, 2, 1, 4, 15, 12, 1, 2, 7, 6, 17, 28, 1, 6, 1, 30, 5, 1, 9, 18, 1, 16, 21, 12, 1, 2, 1, 36, 3, 18, 5, 22, 1, 4, 1, 40, 1, 10, 13, 42, 27, 10, 1, 6, 7, 22

Offset: 1

Author

Reinhard Zumkeller, Aug 12 2008

Keywords

• nonn

Comments

a(n) is the product of the sum of 1 and first differences of prime factors of n with multiplicity, with a(n) = 1 for n = 1 or prime n. - Michael De Vlieger, Nov 12 2023.

a(A007947(n)) = a(n);

A006093 and A001747 give record values and where they occur:

A006093(n)=a(A001747(n+1)) for n>1.

a(n) = 1 iff n is a prime power: a(A000961(n))=1;

a(n) = 2 iff n has exactly 2 and 3 as prime factors:

a(A033845(n))=2;

a(n) = 3 iff n is in A143202;

a(n) = 4 iff n has exactly 2 and 5 as prime factors:

a(A033846(n))=4;

a(n) = 5 iff n is in A143203;

a(n) = 6 iff n is in A143204;

a(n) = 7 iff n is in A143205;

a(n) <> A006512(k)+1 for k>1.

a(A033849(n))=3; a(A033851(n))=3; a(A033850(n))=5; a(A033847(n))=6; a(A033848(n))=10. [Reinhard Zumkeller, Sep 19 2011]

Examples

			a(86) = a(43*2) = 43-2+1 = 42;
a(138) = a(23*3*2) = (23-3+1)*(3-2+1) = 42;
a(172) = a(43*2*2) = (43-2+1)*(2-2+1) = 42;
a(182) = a(13*7*2) = (13-7+1)*(7-2+1) = 42;
a(276) = a(23*3*2*2) = (23-3+1)*(3-2+1)*(2-2+1) = 42;
a(330) = a(11*5*3*2) = (11-5+1)*(5-3+1)*(3-2+1) = 42.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Index entries for primes, gaps between

Crossrefs

Cf. A000961, A001747, A006093, A020639, A033845, A033846.

Cf. A033847, A033848, A033849, A033850.

Cf. A143203, A143204, A143205.

Cf. A027746.

Programs

Haskell

a143201 1 = 1
a143201 n = product $ map (+ 1) $ zipWith (-) (tail pfs) pfs
   where pfs = a027748_row n
-- Reinhard Zumkeller, Sep 13 2011

Mathematica

Table[Times@@(Differences[Flatten[Table[First[#],{Last[#]}]&/@ FactorInteger[ n]]]+1),{n,100}] (* Harvey P. Dale, Dec 07 2011 *)

Formula

a(n) = f(n,1,1) where f(n,q,y) = if n=1 then y else if q=1 then f(n/p,p,1)) else f(n/p,p,y*(p-q+1)) with p = A020639(n) = smallest prime factor of n.