A077565 Number of factorizations of n where each factor has a different prime signature.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, 4, 1, 1, 2, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 1, 4, 1, 4, 1, 3, 3, 1, 1, 6, 1, 3, 1, 3, 1, 4, 1, 4, 1, 1, 1, 5, 1, 1, 3, 4, 1, 4, 1, 3, 1, 4, 1, 7, 1, 1, 3, 3, 1, 4, 1, 6, 2, 1, 1, 5, 1, 1, 1, 4, 1, 5, 1, 3, 1, 1, 1, 9, 1, 3, 3, 3, 1, 4, 1, 4, 4

Offset

1,8

Comments

In contrast to A001055 this sequence excludes from the count all such factorizations of n that include two such factors, f and g, for which it would hold that A046523(f) = A046523(g), or equally A101296(f) = A101296(g). - Antti Karttunen, Nov 24 2017

References

Amarnath Murthy, Generalization of partition function. Introducing Smarandache Factor Partition. Smarandache Notions Journal, Vol. 11, 1-2-3,2000.

Links

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 6143 terms from Antti Karttunen, computed with the given Scheme-program)

Antti Karttunen, Scheme-program for computing this sequence

Index entries for sequences computed from exponents in factorization of n

Formula

a(n) <= A001055(n). - Antti Karttunen, Nov 24 2017

a(p^e) = A000009(p^e). - David A. Corneth, Nov 24 2017

Example

a(24) = 4, 24 = 12*2 = 8*3 = 6*4. The factorizations 2*3*4, 2*2*2*3 etc. are not counted.
From Antti Karttunen, Nov 24 2017: (Start)
For n = 30 the solutions are 30, 2*15, 3*10, 5*6, thus a(30) = 4.
For n = 36 the solutions are 36, 2*18, 3*12, thus a(36) = 3.
For n = 60 the solutions are 60, 2*30, 3*20, 4*15, 5*12, thus a(60) = 5.
For n = 72 the solutions are 72, 2*36, 3*24, 4*18, 6*12, 8*9, 3*4*6, thus a(72) = 7.
(End)

Mathematica

Table[1 + Count[Subsets[Rest@ Divisors@ n, {2, Infinity}], _?(And[Times @@ # == n, UnsameQ @@ Map[Sort[FactorInteger[#][[All, -1]], Greater] &, #]] &)], {n, 105}] (* Michael De Vlieger, Nov 24 2017 *)

Crossrefs

Cf. A000009, A001055, A025487, A077563, A077564, A077566, A046523, A101296.

Sequence in context: A160267 A115621 A326514 * A316979 A331024 A115561

Adjacent sequences: A077562 A077563 A077564 * A077566 A077567 A077568

Keyword

nonn

Author

Amarnath Murthy, Nov 11 2002

Extensions

Corrected and extended by Ray Chandler, Aug 26 2003

Name improved by Antti Karttunen and David A. Corneth, Nov 24 2017

Status

approved