A086412 -id:A086412 - cp's OEIS frontend

A112753 Number of distinct prime factors of n-th number of the form 3^i*5^j.

0, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2

Offset: 1

Reinhard Zumkeller, Sep 18 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001221, A003593, A022336, A022337, A086412, A112754, A112758.

With[{nn=120},Take[PrimeNu/@Union[Flatten[{3^#[[1]] 5^#[[2]],5^#[[1]] 3^#[[2]]}&/@Tuples[Range[0,nn],2]]],nn]] (* Harvey P. Dale, Nov 29 2013 *)

a(n) = A001221(A003593(n)) = 2 - 0^A022336(n) + 0^A022337(n);

a(n) <= 2.

A112758 Number of distinct prime factors of n-th 5-smooth number.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 1, 1, 3, 1, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 2, 2, 2, 3, 1, 1, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 3, 1, 2, 1, 3, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 3, 3, 2, 2, 1, 3, 2, 3, 1, 2, 2, 2, 3, 1, 3, 2, 2, 3, 2, 3, 3, 2, 2, 1, 3, 2, 2, 3, 2, 2, 2, 2, 3, 3, 2, 3, 2, 2, 3, 2, 3, 2

Offset: 1

Reinhard Zumkeller, Sep 18 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..750 from G. C. Greubel)

Cf. A001221, A051037, A086412, A112753, A112759, A112760, A112761, A112762.

aa = {}; Do[If[8 n - EulerPhi[30 n] == 0, AppendTo[aa, n]], {n, 1, 100}]; PrimeNu[aa]  (* G. C. Greubel, May 07 2017 *)
PrimeNu[#]&/@Select[Range[2000],Max[FactorInteger[#][[All,1]]]<6&] (* Harvey P. Dale, Apr 12 2020 *)

a(n) = A001221(A051037(n)).
a(n) = 3 - 0^A112760(n) - 0^A112761(n) - 0^A112762(n).
a(n) <= 3.

A086418 Sum of distinct prime factors of 3-smooth numbers.

Original entry on oeis.org

0, 2, 3, 2, 5, 2, 3, 5, 2, 5, 5, 3, 2, 5, 5, 5, 2, 5, 3, 5, 5, 2, 5, 5, 5, 5, 3, 2, 5, 5, 5, 5, 5, 2, 5, 5, 3, 5, 5, 5, 2, 5, 5, 5, 5, 5, 5, 2, 3, 5, 5, 5, 5, 5, 5, 2, 5, 5, 5, 5, 5, 3, 5, 5, 2, 5, 5, 5, 5, 5, 5, 5, 5, 2, 5, 5, 3, 5, 5, 5, 5, 5, 5, 2, 5, 5, 5, 5, 5, 5, 5, 5, 3, 5, 2, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Reinhard Zumkeller, Jul 18 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003586, A008472, A022328, A022329, A086419, A086417, A086412.

s = {}; m = 12; Do[n = 3^k; While[n <= 3^m, AppendTo[s, n]; n*=2], {k, 0, m}]; sopf[1] = 0; sopf[n_] := Plus @@ First@Transpose @ FactorInteger[n]; sopf /@ Union[s] (* Amiram Eldar, Jan 29 2020 *)

a(n) = A008472(A003586(n));

a(n) = 2*0^(0^A022328(n)) + 3*0^(0^A022329(n)).

cp's OEIS Frontend

A112753 Number of distinct prime factors of n-th number of the form 3^i*5^j.

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A112758 Number of distinct prime factors of n-th 5-smooth number.

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A086418 Sum of distinct prime factors of 3-smooth numbers.

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