A112761 -id:A112761 - cp's OEIS frontend

A022336 Exponent of 3 (value of i) in n-th number of form 3^i*5^j.

0, 1, 0, 2, 1, 0, 3, 2, 1, 4, 0, 3, 2, 5, 1, 4, 0, 3, 6, 2, 5, 1, 4, 7, 0, 3, 6, 2, 5, 8, 1, 4, 7, 0, 3, 6, 9, 2, 5, 8, 1, 4, 7, 10, 0, 3, 6, 9, 2, 5, 8, 11, 1, 4, 7, 10, 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, 10, 13, 0, 3, 6, 9, 12, 2, 5, 8, 11, 14, 1, 4, 7, 10, 13, 0, 3, 6, 9, 12, 15, 2, 5, 8, 11, 14, 1, 4, 7

Offset: 1

Clark Kimberling

nonn

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

Cf. A003593, A007949, A022329, A022337, A112754, A112761.

s = {}; m = 12; Do[n = 5^k; While[n <= 5^m, AppendTo[s, n]; n *= 3], {k, 0, m}]; IntegerExponent[#, 3] & /@ Union[s] (* Amiram Eldar, Feb 06 2020 *)

a(n) = A007949(A003593(n)) = A112754(n) - A022337(n). - Reinhard Zumkeller, Sep 18 2005

A112759 Total number of prime factors of n-th 5-smooth number.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 2, 2, 3, 2, 4, 3, 3, 4, 2, 3, 3, 5, 4, 4, 3, 5, 3, 4, 4, 6, 5, 3, 5, 4, 4, 6, 4, 5, 5, 3, 7, 4, 6, 4, 6, 5, 5, 7, 5, 6, 4, 6, 5, 4, 8, 5, 7, 5, 7, 6, 6, 4, 8, 6, 5, 7, 5, 7, 6, 5, 9, 6, 8, 6, 4, 8, 7, 5, 7, 6, 5, 9, 7, 6, 8, 6, 8, 7, 6, 10, 7, 5, 9, 7, 6, 5, 9, 8, 6, 8, 7, 6, 10, 8, 7, 9, 7

Offset: 1

Reinhard Zumkeller, Sep 18 2005

nonn

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

Cf. A001222, A051037, A069352, A112754, A112758, A112760, A112761, A112762.

PrimeOmega @ Select[Range[3000], Last @ Map[First, FactorInteger[#]] <= 5 &] (* Amiram Eldar, Feb 07 2020 *)

a(n) = A001222(A051037(n));

a(n) = A112760(n) + A112761(n) + A112762(n).

A112762 Exponent of 5 (value of k) in n-th number of the form 2^i3^j5^k.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 1, 0, 2, 0, 1, 0, 0, 2, 1, 0, 1, 0, 2, 0, 1, 3, 0, 1, 0, 2, 1, 0, 1, 0, 2, 0, 2, 1, 0, 3, 0, 1, 0, 2, 1, 0, 1, 3, 0, 2, 1, 0, 2, 1, 0, 3, 0, 1, 0, 2, 4, 1, 0, 2, 1, 0, 3, 0, 2, 1, 0, 2, 1, 0, 3, 0, 1, 3, 0, 2, 1, 4, 1, 0, 2, 1, 0, 3, 0, 2, 1, 0, 2, 4

Offset: 1

Reinhard Zumkeller, Sep 18 2005

nonn

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

Cf. A022337, A051037, A112765, A112760, A112761.

IntegerExponent[#, 5] & /@ Select[Range[3000], Last @ Map[First, FactorInteger[#]] <= 5 &] (* Amiram Eldar, Feb 07 2020 *)

a(n) = A112765(A051037(n));

a(n) = A112759(n) - A112760(n) - A112761(n).

A112760 Exponent of 2 (value of i) in n-th number of the form 2^i3^j5^k.

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 3, 0, 1, 2, 0, 4, 1, 2, 3, 0, 0, 1, 5, 2, 3, 0, 4, 1, 1, 2, 6, 3, 0, 4, 0, 1, 5, 2, 2, 3, 0, 7, 0, 4, 1, 5, 1, 2, 6, 3, 3, 0, 4, 0, 1, 8, 1, 5, 2, 6, 2, 3, 0, 7, 4, 0, 4, 1, 5, 1, 2, 9, 2, 6, 3, 0, 7, 3, 0, 4, 0, 1, 8, 5, 1, 5, 2, 6, 2, 3, 10, 3, 0, 7, 4, 0, 1, 8, 4, 1, 5, 1, 2, 9, 6, 2, 6, 3, 0

Offset: 1

Reinhard Zumkeller, Sep 18 2005

nonn

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

Cf. A007814, A022328, A051037, A112759, A112761, A112762.

IntegerExponent[#, 2] & /@ Select[Range[3000], Last @ Map[First, FactorInteger[#]] <= 5 &] (* Amiram Eldar, Feb 07 2020 *)

a(n) = A007814(A051037(n));

a(n) = A112759(n) - A112761(n) - A112762(n).

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.

cp's OEIS Frontend

A022336 Exponent of 3 (value of i) in n-th number of form 3^i*5^j.

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A112759 Total number of prime factors of n-th 5-smooth number.

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A112762 Exponent of 5 (value of k) in n-th number of the form 2^i*3^j*5^k.

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A112760 Exponent of 2 (value of i) in n-th number of the form 2^i*3^j*5^k.

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

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A112762 Exponent of 5 (value of k) in n-th number of the form 2^i3^j5^k.

A112760 Exponent of 2 (value of i) in n-th number of the form 2^i3^j5^k.