A003593 -id:A003593 - cp's OEIS frontend

A025614 Numbers of form 3^i*6^j, with i, j >= 0.

1, 3, 6, 9, 18, 27, 36, 54, 81, 108, 162, 216, 243, 324, 486, 648, 729, 972, 1296, 1458, 1944, 2187, 2916, 3888, 4374, 5832, 6561, 7776, 8748, 11664, 13122, 17496, 19683, 23328, 26244, 34992, 39366, 46656, 52488, 59049, 69984, 78732, 104976, 118098

Offset: 1

David W. Wilson

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

A025628 is a subsequence.
Cf. A003586, A003593, A003594, A003597, A107364.
Cf. A025610, A025618, A025622, A025626, A025627.

n = 10^6; Flatten[Table[3^i*6^j, {i, 0, Log[3, n]}, {j, 0, Log[6, n/3^i]}]] // Sort (* Amiram Eldar, Sep 26 2020 *)

Sum_{n>=1} 1/a(n) = (3*6)/((3-1)*(6-1)) = 9/5. - Amiram Eldar, Sep 26 2020
a(n) ~ exp(sqrt(2*log(3)*log(6)*n)) / sqrt(18). - Vaclav Kotesovec, Sep 26 2020
a(n) = 3^A025641(n) *6^A025657(n). - R. J. Mathar, Jul 06 2025

A025624 Numbers of form 5^i*9^j, with i, j >= 0.

Original entry on oeis.org

1, 5, 9, 25, 45, 81, 125, 225, 405, 625, 729, 1125, 2025, 3125, 3645, 5625, 6561, 10125, 15625, 18225, 28125, 32805, 50625, 59049, 78125, 91125, 140625, 164025, 253125, 295245, 390625, 455625, 531441, 703125, 820125, 1265625, 1476225, 1953125

Offset: 1

David W. Wilson

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

Subsequence of A003593.

f[upto_]:=Module[{maxi=Ceiling[Log[5,upto]],maxj=Ceiling[Log[9,upto]],s}, s=Union[Flatten[Outer[Times,5^Range[0,maxi], 9^Range[0,maxj]]]]; Select[s,#<=upto&]]; f[2000000] (* Harvey P. Dale, Mar 27 2011 *)

Sum_{n>=1} 1/a(n) = (5*9)/((5-1)*(9-1)) = 45/32. - Amiram Eldar, Sep 24 2020
a(n) ~ exp(sqrt(2*log(5)*log(9)*n)) / sqrt(45). - Vaclav Kotesovec, Sep 24 2020
a(n) = 5^A025654(n) * 9^A025679(n). - R. J. Mathar, Jul 06 2025

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.

A112756 Greatest prime factor of n-th number of the form 3^i*5^j.

Original entry on oeis.org

1, 3, 5, 3, 5, 5, 3, 5, 5, 3, 5, 5, 5, 3, 5, 5, 5, 5, 3, 5, 5, 5, 5, 3, 5, 5, 5, 5, 5, 3, 5, 5, 5, 5, 5, 5, 3, 5, 5, 5, 5, 5, 5, 3, 5, 5, 5, 5, 5, 5, 5, 3, 5, 5, 5, 5, 5, 5, 5, 5, 3, 5, 5, 5, 5, 5, 5, 5, 5, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 3, 5, 5, 5

Offset: 1

Reinhard Zumkeller, Sep 18 2005

nonn

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

Cf. A003593, A006530, A086411, A112764, A112755.

s = {}; m = 12; Do[n = 5^k; While[n <= 5^m, AppendTo[s, n]; n *= 3], {k, 0, m}]; FactorInteger[#][[-1, 1]] & /@ Union[s]  (* Amiram Eldar, Feb 07 2020 *)

a(n) = A006530(A003593(n));

A112755(n) <= a(n) <= 5.

A258023 Numbers of form (2^i)(3^j) or (3^i)(5^j).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 12, 15, 16, 18, 24, 25, 27, 32, 36, 45, 48, 54, 64, 72, 75, 81, 96, 108, 125, 128, 135, 144, 162, 192, 216, 225, 243, 256, 288, 324, 375, 384, 405, 432, 486, 512, 576, 625, 648, 675, 729, 768, 864, 972, 1024, 1125, 1152, 1215, 1296

Offset: 1

Reinhard Zumkeller, May 16 2015

Union of A003586 and A003593;

A006530(a(n)) <= 5; A001221(a(n)) <= 2; a(n) mod 10 != 0.

			.   n |  a(n) |                 n |  a(n) |
. ----+-------+----------     ----+-------+------------
.   1 |    1  |  1             16 |   32  |  2^5
.   2 |    2  |  2             17 |   36  |  2^2 * 3^2
.   3 |    3  |  3             18 |   45  |  3^2 * 5
.   4 |    4  |  2^2           19 |   48  |  2^4 * 3
.   5 |    5  |  5             20 |   54  |  2 * 3^3
.   6 |    6  |  2 * 3         21 |   64  |  2^6
.   7 |    8  |  2^3           22 |   72  |  2^3 * 3^2
.   8 |    9  |  3^2           23 |   75  |  3 * 5^2
.   9 |   12  |  2^2 * 3       24 |   81  |  3^4
.  10 |   15  |  3 * 5         25 |   96  |  2^5 * 3
.  11 |   16  |  2^4           26 |  108  |  2^2 * 3^3
.  12 |   18  |  2 * 3^2       27 |  125  |  5^3
.  13 |   24  |  2^3 * 3       28 |  128  |  2^7
.  14 |   25  |  5^2           29 |  135  |  3^3 * 5
.  15 |   27  |  3^3           30 |  144  |  2^4 * 3^2

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (65000000 terms)

Cf. A003586, A003593, A051037, A006530, A001221, A010879, subsequence of: A051037, A257997, A337800, A337801.

import Data.List.Ordered (union)
a258023 n = a258023_list !! (n-1)
a258023_list = union a003586_list a003593_list

n = 10^4; Join[Table[2^i*3^j, {i, 0, Log[2, n]}, {j, 0, Log[3, n/2^i]}], Table[3^i*5^j, {i, 0, Log[3, n]}, {j, 0, Log[5, n/3^i]}]] // Flatten // Union (* Amiram Eldar, Sep 23 2020 *)

a(n) ~ exp(sqrt(2*log(2)*log(3)*log(5)*n / log(10))) / sqrt(3). - Vaclav Kotesovec, Sep 22 2020

Sum_{n>=1} 1/a(n) = 27/8. - Amiram Eldar, Sep 23 2020

A337801 Numbers of the form (2^i)(5^j) or (3^i)(5^j).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 10, 15, 16, 20, 25, 27, 32, 40, 45, 50, 64, 75, 80, 81, 100, 125, 128, 135, 160, 200, 225, 243, 250, 256, 320, 375, 400, 405, 500, 512, 625, 640, 675, 729, 800, 1000, 1024, 1125, 1215, 1250, 1280, 1600, 1875, 2000, 2025, 2048, 2187, 2500

Offset: 1

Vaclav Kotesovec, Sep 22 2020

Union of A003592 and A003593.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (42000000 terms)

Cf. A003592, A003593, A257997, A258023, A337800.

n = 10^4; Join[Table[2^i*5^j, {i, 0, Log[2, n]}, {j, 0, Log[5, n/2^i]}], Table[3^i*5^j, {i, 0, Log[3, n]}, {j, 0, Log[5, n/3^i]}]] // Flatten // Union (* Amiram Eldar, Sep 23 2020 *)

a(n) ~ exp(sqrt(2*log(2)*log(3)*log(5)*n / log(6))) / sqrt(5).

Sum_{n>=1} 1/a(n) = 25/8. - Amiram Eldar, Sep 23 2020

A369417 Powerful numbers k with multiple distinct prime factors such that rad(k) is not a primorial, where rad(k) = A007947(k).

Original entry on oeis.org

100, 196, 200, 225, 392, 400, 441, 484, 500, 675, 676, 784, 800, 968, 1000, 1089, 1125, 1156, 1225, 1323, 1352, 1372, 1444, 1521, 1568, 1600, 1764, 1936, 2000, 2025, 2116, 2312, 2500, 2601, 2704, 2744, 2888, 3025, 3087, 3136, 3200, 3249, 3267, 3364, 3375, 3528

Offset: 1

Michael De Vlieger, Jan 22 2024

nonn

Numbers k such that Omega(k) > omega(k) > 1, where all prime power factors p^m have exponents m > 1, such that squarefree kernel rad(k) not in A002110, where Omega = A001222 and omega = A001221.

			Let S = A366413 = {A120944 \ A002110}.
This sequence is the union of the following infinite sets:
S(1)^2 * A003592 = 10^2 * A003592 = {100, 200, 400, 500, 800, 1000, ...}
                 = { m*S(1)^2 : rad(m) | S(1) }.
S(2)^2 * A003591 = 14^2 * A003591 = {196, 392, 784, 1372, 1568, ...}
                 = { m*S(2)^2 : rad(m) | S(2) }.
S(3)^2 * A003593 = 15^2 * A003593 = {225, 675, 1125, 2025, 3375, ...}
                 = { m*S(3)^2 : rad(m) | S(3) }, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A002110, A055932, A120944, A126706, A286708, A366413, A369374.

With[{nn = 2^14},
  Select[
    Select[
      Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
    Not@*PrimePowerQ],
  Nand[EvenQ[#],
    Union@ Differences@ PrimePi[FactorInteger[#][[All, 1]]] == {1}] &] ]

{a(n)} = { A286708 \ A055932 }.
{a(n)} = { m*s^2 : Omega(s) = omega(s) > 1, s not in A002110, rad(m) | s }.
A286708 is the union of A369374 and this sequence.

A112751 Number of numbers of the form 3^i*5^j that are less than or equal to n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Sep 18 2005

nonn

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

Cf. A003593, A008683, A071520.

[&+[MoebiusMu(15*k)*Floor(n/k):k in [1..n]]: n in [1..97]]; // Marius A. Burtea, Jul 30 2019

with(numtheory): seq(add(mobius(15*k)*floor(n/k), k=1..n), n=1..90); # Ridouane Oudra, Jul 29 2019

Accumulate[Table[Boole[n == Times @@ ({3, 5}^IntegerExponent[n, {3, 5}])], {n, 1, 100}]] (* Amiram Eldar, May 04 2025 *)

From Ridouane Oudra, Jul 29 2019: (Start)
a(n) = Card_{ k | A003593(k) <= n }.
a(n) = Sum_{k=1..n} mu(15*k)*floor(n/k), where mu is the Möbius function (A008683).
a(n) = Sum_{k=1..n} (floor(15^k/k)-floor((15^k-1)/k)). (End)
From Ridouane Oudra, Jul 17 2020: (Start)
a(n) = Sum_{i=0..floor(log_5(n))} (floor(log_3(n/5^i)) + 1).
a(n) = Sum_{i=0..floor(log_3(n))} (floor(log_5(n/3^i)) + 1). (End)

A112755 Smallest prime factor of n-th number of the form 3^i*5^j.

Original entry on oeis.org

1, 3, 5, 3, 3, 5, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3

Offset: 1

Reinhard Zumkeller, Sep 18 2005

nonn

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

Cf. A003593, A020639, A086410, A112756, A112763.

s = {}; m = 12; Do[n = 5^k; While[n <= 5^m, AppendTo[s, n]; n *= 3], {k, 0, m}]; FactorInteger[#][[1, 1]] & /@ Union[s] (* Amiram Eldar, Feb 07 2020 *)

a(n) = A020639(A003593(n));

a(n) <= A112756(n) <= 5.

A264997 Number of partitions of n into distinct parts of the form 3^a*5^b.

Original entry on oeis.org

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

Offset: 0

Joseph Myers, Nov 29 2015

			28 = 27 + 1 = 25 + 3 = 15 + 9 + 3 + 1, so a(28) = 3.

Alois P. Heinz, Table of n, a(n) for n = 0..20000 (first 1001 terms from Joseph Myers)
British Mathematical Olympiad 2015/16, Olympiad Round 1, Problem 6, Friday, 27 November 2015.

Cf. A003593, A264998.

import Data.MemoCombinators (memo2, list, integral)
a264997 n = a264997_list !! (n-1)
a264997_list = f 0 [] a003593_list where
   f u vs ws'@(w:ws) | u < w = (p' vs u) : f (u + 1) vs ws'
                     | otherwise = f u (vs ++ [w]) ws
   p' = memo2 (list integral) integral p
   p _  0 = 1
   p [] _ = 0
   p (k:ks) m = if m < k then 0 else p' ks (m - k) + p' ks m
-- Reinhard Zumkeller, Dec 18 2015

nmax = 100; A003593 = Select[Range[nmax], PowerMod[15, #, #] == 0 &]; CoefficientList[Series[Product[(1 + x^(A003593[[k]])), {k, 1, Length[A003593]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 01 2015 *)

G.f.: (1+x)(1+x^3)(1+x^5)(1+x^9)(1+x^15)....

cp's OEIS Frontend

A025614 Numbers of form 3^i*6^j, with i, j >= 0.

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A025624 Numbers of form 5^i*9^j, with i, j >= 0.

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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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A112756 Greatest prime factor of n-th number of the form 3^i*5^j.

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A258023 Numbers of form (2^i)*(3^j) or (3^i)*(5^j).

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Haskell

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A337801 Numbers of the form (2^i)*(5^j) or (3^i)*(5^j).

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A369417 Powerful numbers k with multiple distinct prime factors such that rad(k) is not a primorial, where rad(k) = A007947(k).

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A112751 Number of numbers of the form 3^i*5^j that are less than or equal to n.

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Magma

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A258023 Numbers of form (2^i)(3^j) or (3^i)(5^j).

A337801 Numbers of the form (2^i)(5^j) or (3^i)(5^j).