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%I #45 Jan 16 2018 02:39:55
%S 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,
%T 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,
%U 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
%N Number of factorizations of n where each factor has a different prime signature.
%C 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
%D Amarnath Murthy, Generalization of partition function. Introducing Smarandache Factor Partition. Smarandache Notions Journal, Vol. 11, 1-2-3,2000.
%H David A. Corneth, <a href="/A077565/b077565.txt">Table of n, a(n) for n = 1..10000</a> (first 6143 terms from Antti Karttunen, computed with the given Scheme-program)
%H Antti Karttunen, <a href="/A077565/a077565.txt">Scheme-program for computing this sequence</a>
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F a(n) <= A001055(n). - _Antti Karttunen_, Nov 24 2017
%F a(p^e) = A000009(p^e). - _David A. Corneth_, Nov 24 2017
%e a(24) = 4, 24 = 12*2 = 8*3 = 6*4. The factorizations 2*3*4, 2*2*2*3 etc. are not counted.
%e From _Antti Karttunen_, Nov 24 2017: (Start)
%e For n = 30 the solutions are 30, 2*15, 3*10, 5*6, thus a(30) = 4.
%e For n = 36 the solutions are 36, 2*18, 3*12, thus a(36) = 3.
%e For n = 60 the solutions are 60, 2*30, 3*20, 4*15, 5*12, thus a(60) = 5.
%e For n = 72 the solutions are 72, 2*36, 3*24, 4*18, 6*12, 8*9, 3*4*6, thus a(72) = 7.
%e (End)
%t 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 *)
%Y Cf. A000009, A001055, A025487, A077563, A077564, A077566, A046523, A101296.
%K nonn
%O 1,8
%A _Amarnath Murthy_, Nov 11 2002
%E Corrected and extended by _Ray Chandler_, Aug 26 2003
%E Name improved by _Antti Karttunen_ and _David A. Corneth_, Nov 24 2017
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