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A181796 a(n) = number of divisors of n whose canonical prime factorizations contain no repeated positive exponents (cf. A130091).

%I A181796 #37 Feb 16 2025 08:33:13
%S A181796 1,2,2,3,2,3,2,4,3,3,2,5,2,3,3,5,2,5,2,5,3,3,2,7,3,3,4,5,2,4,2,6,3,3,
%T A181796 3,7,2,3,3,7,2,4,2,5,5,3,2,9,3,5,3,5,2,7,3,7,3,3,2,7,2,3,5,7,3,4,2,5,
%U A181796 3,4,2,10,2,3,5,5,3,4,2,9,5,3,2,7,3,3,3,7,2,7,3,5,3,3,3,11,2,5,5,7,2,4,2,7,4
%N A181796 a(n) = number of divisors of n whose canonical prime factorizations contain no repeated positive exponents (cf. A130091).
%C A181796 The canonical factorization of n into prime powers can be written as Product p(i)^e(i), for example. A host of equivalent notations can also be used (for another example, see Weisstein link). a(n) depends only on prime signature of n (cf. A025487).
%C A181796 a(n) >= A085082(n). (A085082(n) equals the number of members of A025487 that divide A046523(n), and each member of A025487 is divisible by at least one member of A130091 that divides no smaller member of A025487.) a(n) > A085082(n) iff n has in its canonical prime factorization at least two exponents greater than 1.
%C A181796 a(n) = number of such divisors of n that in their prime factorization all exponents are unique. - _Antti Karttunen_, May 27 2017
%C A181796 First differs from A335549 at a(90) = 7, A335549(90) = 8. First differs from A335516 at a(180) = 9, A335516(180) = 10. - _Gus Wiseman_, Jun 28 2020
%H A181796 Antti Karttunen, <a href="/A181796/b181796.txt">Table of n, a(n) for n = 1..10000</a>
%H A181796 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeFactorization.html">Prime Factorization</a>.
%H A181796 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vSX9dPMGJhxB8rOknCGvOs6PiyhupdWNpqLsnphdgU6MEVqFBnWugAXidDhwHeKqZe_YnUqYeGOXsOk/pub">Sequences counting and encoding certain classes of multisets</a>
%F A181796 a(A000079(n)) = a(A002110(n)) = n+1.
%F A181796 a(A006939(n)) = A000110(n+1).
%F A181796 a(A181555(n)) = A002720(n).
%e A181796 12 has a total of six divisors (1, 2, 3, 4, 6 and 12). Of those divisors, the number 1 has no prime factors, hence, no positive exponents at all (and no repeated positive exponents) in its canonical prime factorization. The lists of positive exponents for 2, 3, 4, 6 and 12 are (1), (1), (2), (1,1) and (2,1) respectively (cf. A124010). Of all six divisors, only the number 6 (2^1*3^1) has at least one positive exponent repeated (namely, 1). The other five do not; hence, a(12) = 5.
%e A181796 For n = 90 = 2 * 3^2 * 5, the divisors that satisfy the condition are: 1, 2, 3, 3^2, 5, 2 * 3^2, 3^2 * 5, altogether 7, (but for example 90 itself is not included), thus a(90) = 7.
%t A181796 Table[DivisorSum[n, 1 &, Length@ Union@ # == Length@ # &@ FactorInteger[#][[All, -1]] &], {n, 105}] (* _Michael De Vlieger_, May 28 2017 *)
%o A181796 (PARI)
%o A181796 no_repeated_exponents(n) = { my(es = factor(n)[, 2]); if(length(Set(es)) == length(es),1,0); }
%o A181796 A181796(n) = sumdiv(n,d,no_repeated_exponents(d)); \\ _Antti Karttunen_, May 27 2017
%o A181796 (Python)
%o A181796 from sympy import factorint, divisors
%o A181796 def ok(n):
%o A181796     f=factorint(n)
%o A181796     ex=[f[i] for i in f]
%o A181796     for i in ex:
%o A181796         if ex.count(i)>1: return 0
%o A181796     return 1
%o A181796 def a(n): return sum([1 for i in divisors(n) if ok(i)]) # _Indranil Ghosh_, May 27 2017
%Y A181796 Diverges from A088873 at n=24 and from A085082 at n=36. a(36) = 7, while A085082(36) = 6.
%Y A181796 Partitions with distinct multiplicities are A098859.
%Y A181796 Sorted prime signature is A118914.
%Y A181796 Unsorted prime signature is A124010.
%Y A181796 a(n) is the number of divisors of n in A130091.
%Y A181796 Factorizations with distinct multiplicities are A255231.
%Y A181796 The largest of the counted divisors is A327498.
%Y A181796 Factorizations using the counted divisors are A327523.
%Y A181796 Cf. A000005, A007916, A045778, A056239, A212168, A327498, A335519, A335549.
%K A181796 nonn
%O A181796 1,2
%A A181796 _Matthew Vandermast_, Nov 22 2010