This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A085082 #33 Aug 24 2024 17:11:14
%S A085082 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 A085082 3,6,2,3,3,7,2,4,2,5,5,3,2,9,3,5,3,5,2,7,3,7,3,3,2,7,2,3,5,7,3,4,2,5,
%U A085082 3,4,2,9,2,3,5,5,3,4,2,9,5,3,2,7,3,3,3,7,2,7,3,5,3,3,3,11,2,5,5,6,2,4,2,7,4
%N A085082 Number of distinct prime signatures arising among the divisors of n.
%C A085082 For a squarefree number n with k distinct prime divisors, a(n) = k+1.
%C A085082 If n = p^r then a(n) = tau(n) = r+1.
%C A085082 Question: Find a(n) in the following cases:
%C A085082 1. n = m^k where m is a squarefree number with r distinct prime divisors.
%C A085082 2. n = Product_{i=1..r} (p_i)^i, where p_i is the i-th distinct prime divisor of n.
%C A085082 Answers: 1. (r+k)!/(r!k!). 2. A000108(r+1). - _David Wasserman_, Jan 20 2005
%C A085082 I have submitted comments for A000108 and A016098 that each include a combinatorial statement equivalent to the second problem and its solution. - _Matthew Vandermast_, Nov 22 2010
%H A085082 Alois P. Heinz, <a href="/A085082/b085082.txt">Table of n, a(n) for n = 1..10000</a>
%e A085082 a(30) = 4 and the divisors with distinct prime signatures are 1, 2, 6 and 30. The divisors 3 and 5 with the same prime signature as of 2 and the divisors 10 and 15 with the same prime signature as that of 6 are not counted.
%e A085082 The divisors of 36 are 1, 2, 3, 4, 6, 9, 12 and 36. We can group them as (1), (2, 3), (6), (4, 9), (12, 18), (36) so that every group contains divisors with the same prime signature and we have a(36) = 6.
%p A085082 with(numtheory):
%p A085082 a:= n-> nops({seq(sort(map(x->x[2], ifactors(d)[2])), d=divisors(n))}):
%p A085082 seq(a(n), n=1..120); # _Alois P. Heinz_, Jun 12 2012
%t A085082 ps[1] = {}; ps[n_] := FactorInteger[n][[All, 2]] // Sort; a[n_] := ps /@ Divisors[n] // Union // Length; Array[a, 120] (* _Jean-François Alcover_, Jun 10 2015 *)
%o A085082 (PARI) a(n)=my(f=vecsort(factor(n)[,2]),v=[1],s); for(i=1,#f, s=0; v=vector(f[i]+1,i, if(i<=#v, s+=v[i]); s)); vecsum(v) \\ _Charles R Greathouse IV_, Feb 03 2017
%Y A085082 Cf. A000108.
%Y A085082 The second problem describes A076954(i). See also A006939.
%K A085082 easy,nonn
%O A085082 1,2
%A A085082 _Amarnath Murthy_, Jul 01 2003
%E A085082 More terms from _David Wasserman_, Jan 20 2005