

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
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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*(pq+1)) with p = A020639(n) = smallest prime factor of n.


EXAMPLE

a(86) = a(43*2) = 432+1 = 42;
a(138) = a(23*3*2) = (233+1)*(32+1) = 42;
a(172) = a(43*2*2) = (432+1)*(22+1) = 42;
a(182) = a(13*7*2) = (137+1)*(72+1) = 42;
a(276) = a(23*3*2*2) = (233+1)*(32+1)*(22+1) = 42;
a(330) = a(11*5*3*2) = (115+1)*(53+1)*(32+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 * A254048 A306671 A308210
Adjacent sequences: A143198 A143199 A143200 * A143202 A143203 A143204


KEYWORD

nonn


AUTHOR

Reinhard Zumkeller, Aug 12 2008


STATUS

approved



