A112754 -id:A112754 - cp's OEIS frontend

A022337 Exponent of 5 (value of j) in n-th number of form 3^i*5^j.

0, 0, 1, 0, 1, 2, 0, 1, 2, 0, 3, 1, 2, 0, 3, 1, 4, 2, 0, 3, 1, 4, 2, 0, 5, 3, 1, 4, 2, 0, 5, 3, 1, 6, 4, 2, 0, 5, 3, 1, 6, 4, 2, 0, 7, 5, 3, 1, 6, 4, 2, 0, 7, 5, 3, 1, 8, 6, 4, 2, 0, 7, 5, 3, 1, 8, 6, 4, 2, 0, 9, 7, 5, 3, 1, 8, 6, 4, 2, 0, 9, 7, 5, 3, 1, 10, 8, 6, 4, 2, 0, 9, 7, 5, 3, 1, 10, 8, 6, 4, 2, 0, 11, 9, 7, 5

Offset: 1

Clark Kimberling

nonn

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

Cf. A003593, A022336, A112754, A112762, A112765.

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

a(n) = A112765(A003593(n)) = A112754(n) - A022336(n). - Reinhard Zumkeller, Sep 18 2005

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

Original entry on oeis.org

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).

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

Original entry on oeis.org

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.

A356242 a(n) is the number of Fermat numbers dividing n, counted with multiplicity.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jul 30 2022

nonn

The multiplicity of a divisor d (not necessarily a prime) of n is defined in A169594 (see also the first formula).
A000244(n) is the least number k such that a(k) = n.
The asymptotic density of occurrences of 0 is 1/2.
The asymptotic density of occurrences of 1 is (1/2) * Sum_{k>=0} 1/(2^(2^k)+1) = (1/2) * A051158 = 0.2980315860... .

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fermat Number.
Wikipedia, Fermat number.

Cf. A000244, A000215, A007404, A051158, A051179, A169594, A356241.

Cf. A080307 (positions of nonzeros), A080308 (positions of 0's).

f = Table[(2^(2^n) + 1), {n, 0, 5}]; a[n_] := Total[IntegerExponent[n, f]]; Array[a, 100]

a(n) = Sum_{k>=1} v(A000215(k), n), where v(m, n) is the exponent of the largest power of m that divides n.
a(A000215(n)) = 1.
a(A000244(n)) = a(A000351(n)) = a(A001026(n)) = n.
a(A003593(n)) = A112754(n).
a(n) >= A356241(n).
a(A051179(n)) = n.
a(A080307(n)) > 0 and a(A080308(n)) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=0} 1/(2^(2^k)) = 0.8164215090... (A007404).

cp's OEIS Frontend

A022337 Exponent of 5 (value of j) in n-th number of form 3^i*5^j.

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

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A356242 a(n) is the number of Fermat numbers dividing n, counted with multiplicity.

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