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

Author

Amarnath Murthy, Nov 11 2002

Keywords

• nonn

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

Examples

			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)
		

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

Crossrefs

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

Programs

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 *)

Formula

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

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

Extensions

Corrected and extended by Ray Chandler, Aug 26 2003

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