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A143201 Product of distances between prime factors in factorization of n. 16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

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]

LINKS

R. Zumkeller, Table of n, a(n) for n = 1..10000

Index entries for primes, gaps between

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.

EXAMPLE

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.

MATHEMATICA

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

PROG

(Haskell)

a143201 1 = 1

a143201 n = product $ map (+ 1) $ zipWith (-) (tail pfs) pfs

   where pfs = a027748_row n

-- Reinhard Zumkeller, Sep 13 2011

CROSSREFS

Cf. A109313.

Cf. A027748.

Sequence in context: A174204 A118106 A324574 * A340082 A254048 A306671

Adjacent sequences:  A143198 A143199 A143200 * A143202 A143203 A143204

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Aug 12 2008

STATUS

approved

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Last modified April 10 14:07 EDT 2021. Contains 342845 sequences. (Running on oeis4.)