A086415 -id:A086415 - cp's OEIS frontend

A069352 Total number of prime factors of 3-smooth numbers.

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

Offset: 1

Reinhard Zumkeller, Mar 18 2002

nonn

a(n) = A001222(A003586(n));
a(n) = A022328(n) + A022329(n);
A086414(n) <= A086415(n) <= a(n).

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A003586, A001222, A003586, A022328, A022329, A086414, A086415.

a069352 = a001222 . a003586  -- Reinhard Zumkeller, May 16 2015

smoothNumbers[p_, max_] := Module[{a, aa, k, pp, iter}, k = PrimePi[p]; aa = Array[a, k]; pp = Prime[Range[k]]; iter = Table[{a[j], 0, PowerExpand @ Log[pp[[j]], max/Times @@ (Take[pp, j-1]^Take[aa, j-1])]}, {j, 1, k}]; Table[Times @@ (pp^aa), Sequence @@ iter // Evaluate] // Flatten // Sort]; PrimeOmega /@ smoothNumbers[3, 10^5] (* Jean-François Alcover, Nov 11 2016 *)

a(n) = i+j for 3-smooth numbers n = 2^i*3^j (A003586).

a(n) = A001222(A033845(n))-2. - Enrique Pérez Herrero, Jan 04 2012

Edited by N. J. A. Sloane, Oct 27 2008 at the suggestion of R. J. Mathar.

A086414 Minimal exponent in prime factorization of 3-smooth numbers.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 1, 4, 1, 1, 3, 5, 2, 1, 1, 6, 2, 4, 1, 2, 7, 2, 1, 1, 3, 5, 8, 2, 2, 1, 3, 1, 9, 2, 3, 6, 1, 3, 2, 10, 2, 4, 1, 1, 3, 3, 11, 7, 2, 4, 2, 1, 3, 4, 12, 1, 2, 4, 3, 1, 8, 3, 5, 13, 2, 2, 4, 4, 1, 1, 3, 5, 14, 3, 2, 9, 4, 5, 1, 2, 3, 5, 15, 4, 2, 1, 4, 6, 1, 3, 3, 10, 5, 16, 5, 2, 2

Offset: 1

Reinhard Zumkeller, Jul 18 2003

nonn

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

Cf. A003586, A051904, A069352, A086415.

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

a(n) = A051904(A003586(n));

a(n) <= A086415(n) <= A069352(n).

A086416 Number of divisors of 3-smooth numbers.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 3, 6, 5, 6, 8, 4, 6, 9, 10, 8, 7, 12, 5, 12, 12, 8, 15, 10, 14, 16, 6, 9, 18, 15, 16, 20, 12, 10, 21, 20, 7, 18, 24, 18, 11, 24, 25, 14, 20, 28, 24, 12, 8, 27, 30, 21, 22, 32, 30, 13, 16, 30, 35, 28, 24, 9, 36, 36, 14, 24, 33, 40, 35, 26, 18, 40, 42, 15, 32

Offset: 1

Reinhard Zumkeller, Jul 18 2003

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A086414, A086415, A086417.

Cf. A000005, A003586, A022328, A022329.

DivisorSigma[0, Select[Range[10000], # == 2^IntegerExponent[#, 2] * 3^IntegerExponent[#, 3] &]] (* Amiram Eldar, Apr 15 2024 *)

a(n) = A000005(A003586(n)).
a(n) = if A086414(n) = A086415(n) then A086414(n)+1 else (A086414(n)+1)*(A086415(n)+1).
a(n) = (A022328(n)+1)*(A022329(n)+1).

A352072 a(n) = least k such that A003586(n) | 12^k.

Original entry on oeis.org

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

Offset: 1

Michael De Vlieger, Mar 08 2022

Also, number of digits in the duodecimal expansion of terminating unit fractions 1/A003586.

			a(1) = 0 since A003586(1) = 1 | 12^0.
a(2) = 1 since A003586(2) = 2 | 12^1; 1/2 expanded in base 12 = .6.
a(6) = 2 since A003586(6) = 8 | 12^2; 1/8 in base 12 = .16.
a(12) = 3 since A003586(12) = 27 | 12^3; 1/27 in base 12 = .054, etc.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Chapter IX: The Representation of Numbers by Decimals, Theorem 136. 8th ed., Oxford Univ. Press, 2008, 144-145.

Michael De Vlieger, Table of n, a(n) for n = 1..10283 (A003586(10283) = 12^50)
Eric Weisstein's World of Mathematics, Duodecimal.
Wikipedia, Duodecimal.

Cf. A003586, A086415, A117920.

With[{nn = 40000}, Sort[Join @@ Table[{2^a*3^b, Max[Ceiling[a/2], b]}, {a, 0, Log2[nn]}, {b, 0, Log[3, nn/(2^a)]}] ][[All, -1]] ]

A352218 a(n) = least k such that A003592(n) | 20^k.

Original entry on oeis.org

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

Offset: 1

Michael De Vlieger, Mar 08 2022

Also, number of digits in the vigesimal (base 20) expansion of terminating unit fractions 1/A003592.

			a(1) = 0 since A003592(1) = 1 | 20^0.
a(4) = 1 since A003592(4) = 5 | 20^1; 1/5 in base 20 = 0.4.
a(5) = 2 since A003592(5) = 8 | 20^2; 1/8 in base 20 = 0.2a, where "a" is digit 10, etc.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Chapter IX: The Representation of Numbers by Decimals, Theorem 136. 8th ed., Oxford Univ. Press, 2008, 144-145.

Michael De Vlieger, Table of n, a(n) for n = 1..10212 (A003592(10212) = 20^50)
Eric Weisstein's World of Mathematics, Vigesimal.
Wikipedia, Vigesimal.

Cf. A003592, A086415, A117920.

With[{nn = 360000}, Sort[Join @@ Table[{2^a*5^b, Max[Ceiling[a/2], b]}, {a, 0, Log2[nn]}, {b, 0, Log[5, nn/(2^a)]}]][[All, -1]] ]

A352219 a(n) is the least k such that A051037(n) | 60^k.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 3, 1, 3, 2, 2, 2, 2, 2, 3, 1, 3, 2, 2, 2, 4, 2, 3, 2, 3, 2, 3, 4, 3, 2, 2, 3, 4, 2, 3, 2, 3, 2, 2, 5, 3, 4, 3, 3, 2, 3, 4, 2, 3, 4, 2, 4, 3, 2, 3, 5, 3, 5, 3, 3, 2, 4, 4, 4, 3, 2, 6, 3, 4, 3, 4, 3, 2, 3, 5, 3, 5

Offset: 1

Michael De Vlieger, Mar 08 2022

Also, number of digits in the sexagesimal expansion of terminating unit fractions 1/A051037.

			a(1) = 0 since A051037(1) = 1 | 60^0.
a(2) = 1 since A051037(2) = 2 | 60^1; 1/2 in base 60 is represented by digit 30 after the radix point ":", i.e., :30.
a(7) = 2 since A051037(7) = 8 | 60^2; 1/8 in base 60 is :7,30, etc.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Chapter IX: The Representation of Numbers by Decimals, Theorem 136. 8th ed., Oxford Univ. Press, 2008, 144-145.

Michael De Vlieger, Table of n, a(n) for n = 1..10540 (A051037(10540) = 60^10)
Eric Weisstein's World of Mathematics, Sexagesimal
Wikipedia, Regular number.

Cf. A051037, A086415, A117920.

With[{nn = 1024}, Sort[Flatten[Table[{2^a * 3^b * 5^c, Max[Ceiling[a/2], b, c]}, {a, 0, Log2[nn]}, {b, 0, Log[3, nn/(2^a)]}, {c, 0, Log[5, nn/(2^a*3^b)]}], 2]][[All, -1]] ]

a(n) ≍ n^(1/3), with lim sup a(n)/n^(1/3) being (6*log(2)*log(3)*log(5))^(1/3)/log(3) = 1.770... where A051037(n) is a power of 3 and the lim inf being (6*log(2)*log(3)*log(5))^(1/3)/log(60) = 0.4749... where A051037(n) is a power of 60. - Charles R Greathouse IV, Mar 08 2022

cp's OEIS Frontend

A069352 Total number of prime factors of 3-smooth numbers.

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A086414 Minimal exponent in prime factorization of 3-smooth numbers.

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A086416 Number of divisors of 3-smooth numbers.

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A352072 a(n) = least k such that A003586(n) | 12^k.

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A352218 a(n) = least k such that A003592(n) | 20^k.

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A352219 a(n) is the least k such that A051037(n) | 60^k.

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