A003591 -id:A003591 - cp's OEIS frontend

A108056 Numbers of the form (7^i)*(13^j).

1, 7, 13, 49, 91, 169, 343, 637, 1183, 2197, 2401, 4459, 8281, 15379, 16807, 28561, 31213, 57967, 107653, 117649, 199927, 218491, 371293, 405769, 753571, 823543, 1399489, 1529437, 2599051, 2840383, 4826809, 5274997, 5764801, 9796423, 10706059, 18193357, 19882681

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jun 02 2005

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

Cf. A003586, A003592, A003593, A003591, A003594, A003595, A003596, A003597, A003598, A003599, A107326, A107364, A107466.

n = 10^7; Flatten[Table[7^i*13^j, {i, 0, Log[7, n]}, {j, 0, Log[13, n/7^i]}]] // Sort (* Amiram Eldar, Sep 23 2020 *)

list(lim)=my(v=List(),N);for(n=0,log(lim)\log(13),N=13^n;while(N<=lim,listput(v,N);N*=7));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

from sympy import integer_log
def A108056(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(integer_log(x//13**i,7)[0]+1 for i in range(integer_log(x,13)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Oct 22 2024

Sum_{n>=1} 1/a(n) = (7*13)/((7-1)*(13-1)) = 91/72. - Amiram Eldar, Sep 23 2020

a(n) ~ exp(sqrt(2*log(7)*log(13)*n)) / sqrt(91). - Vaclav Kotesovec, Sep 23 2020

More terms from Amiram Eldar, Sep 23 2020

A108347 Numbers of the form (3^i)(5^j)(7^k), with i, j, k >= 0.

Original entry on oeis.org

1, 3, 5, 7, 9, 15, 21, 25, 27, 35, 45, 49, 63, 75, 81, 105, 125, 135, 147, 175, 189, 225, 243, 245, 315, 343, 375, 405, 441, 525, 567, 625, 675, 729, 735, 875, 945, 1029, 1125, 1215, 1225, 1323, 1575, 1701, 1715, 1875, 2025, 2187, 2205, 2401, 2625, 2835, 3087

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jul 01 2005

nonn

The Heinz numbers of the partitions into parts 2,3, and 4 (including the number 1, the Heinz number of the empty partition). We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [2,3,3,4] the Heinz number is 3*5*5*7 = 525; it is in the sequence. - Emeric Deutsch , May 21 2015

Numbers m | 105^e with integer e >= 0. - Michael De Vlieger, Aug 22 2019

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A003586, A003591-A003595, A051037, A108319, A108513, A215366.

The odd terms of A002473.

[n: n in [1..4000] | PrimeDivisors(n) subset [3,5,7]]; // Bruno Berselli, Sep 24 2012

with(numtheory): S := {}: for j to 3100 do if `subset`(factorset(j), {3, 5, 7}) then S := `union`(S, {j}) else end if end do: S; # Emeric Deutsch, May 21 2015
# alternative
isA108347 := proc(n)
      if n = 1 then
        true;
    else
        return (numtheory[factorset](n) minus {3, 5, 7} = {} );
    end if;
end proc:
A108347 := proc(n)
     option remember;
     if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA108347(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A108347(n),n=1..80); # R. J. Mathar, Jun 06 2024

With[{n = 3087}, Sort@ Flatten@ Table[3^i * 5^j * 7^k, {i, 0, Log[3, n]}, {j, 0, Log[5, n/2^i]}, {k, 0, Log[7, n/(3^i*5^j)]}]] (* Michael De Vlieger, Aug 22 2019 *)

from sympy import integer_log
def A108347(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n+x
        for i in range(integer_log(x,7)[0]+1):
            for j in range(integer_log(m:=x//7**i,5)[0]+1):
                c -= integer_log(m//5**j,3)[0]+1
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

Sum_{n>=1} 1/a(n) = (3*5*7)/((3-1)*(5-1)*(7-1)) = 35/16. - Amiram Eldar, Sep 22 2020

a(n) ~ exp((6*log(3)*log(5)*log(7)*n)^(1/3)) / sqrt(105). - Vaclav Kotesovec, Sep 23 2020

A108319 Numbers of the form (2^i)(3^j)(7^k), with i, j, k >= 0.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 27, 28, 32, 36, 42, 48, 49, 54, 56, 63, 64, 72, 81, 84, 96, 98, 108, 112, 126, 128, 144, 147, 162, 168, 189, 192, 196, 216, 224, 243, 252, 256, 288, 294, 324, 336, 343, 378, 384, 392, 432, 441, 448, 486, 504, 512, 567

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jun 30 2005

Numbers m | 42^e with integer e >= 0. - Michael De Vlieger, Aug 22 2019

Sum_{n>=1} 1/a(n) = (2*3*7)/((2-1)*(3-1)*(7-1)) = 7/2. - Amiram Eldar, Sep 24 2020

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

Cf. A051037, A003586, A003591, A003594.

With[{n = 567}, Sort@ Flatten@ Table[2^i * 3^j * 7^k, {i, 0, Log2@ n}, {j, 0, Log[3, n/2^i]}, {k, 0, Log[7, n/(2^i*3^j)]}]] (* Michael De Vlieger, Aug 22 2019 *)

list(lim)=my(v=List(), s, t); for(i=0, logint(lim\=1, 7), t=7^i; for(j=0, logint(lim\t, 3), s=t*3^j; while(s<=lim, listput(v, s); s<<=1))); Set(v) \\ Charles R Greathouse IV, Nov 20 2024

a(n) ~ exp((6*log(2)*log(3)*log(7)*n)^(1/3)) / sqrt(42). - Vaclav Kotesovec, Sep 23 2020

A025626 Numbers of form 6^i*7^j, with i, j >= 0.

Original entry on oeis.org

1, 6, 7, 36, 42, 49, 216, 252, 294, 343, 1296, 1512, 1764, 2058, 2401, 7776, 9072, 10584, 12348, 14406, 16807, 46656, 54432, 63504, 74088, 86436, 100842, 117649, 279936, 326592, 381024, 444528, 518616, 605052, 705894, 823543, 1679616, 1959552

Offset: 1

David W. Wilson

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

Cf. A025610, A025622, A025627, A025628, A025629.

Cf. A025619, A025631, A025632, A003591, A003594, A003595, A036566.

n = 10^6; Flatten[Table[6^i*7^j, {i, 0, Log[6, n]}, {j, 0, Log[7, n/6^i]}]] // Sort (* Amiram Eldar, Sep 25 2020 *)

Sum_{n>=1} 1/a(n) = (6*7)/((6-1)*(7-1)) = 7/5. - Amiram Eldar, Sep 25 2020
a(n) ~ exp(sqrt(2*log(6)*log(7)*n)) / sqrt(42). - Vaclav Kotesovec, Sep 25 2020
a(n) = 6^A025660(n) * 7^A025668(n). - R. J. Mathar, Jul 06 2025

A177805 Numbers k such that k divides 15^k - 1.

Original entry on oeis.org

1, 2, 4, 7, 8, 14, 16, 28, 32, 49, 56, 64, 98, 112, 128, 136, 196, 224, 256, 272, 343, 392, 448, 452, 512, 544, 686, 784, 812, 896, 904, 952, 1024, 1088, 1372, 1568, 1624, 1792, 1808, 1904, 2048, 2176, 2312, 2401, 2744, 3136, 3164, 3248, 3584, 3616, 3808, 4096

Offset: 1

Alexander Adamchuk, May 17 2010

nonn

A000420 are the only odd terms of the sequence. - Robert Israel, Feb 25 2020

Robert Israel, Table of n, a(n) for n = 1..2500

Cf. A000420, A014960, A014956, A014951, A014949, A014946, A014945, A067945, A128358, A128360.

Contains A003591.

filter:= n -> 15 &^ n - 1 mod n = 0:
select(filter, [$1..10000]); # Robert Israel, Feb 25 2020

is(n)=Mod(15,n)^n==1 \\ Charles R Greathouse IV, Nov 04 2016

A025619 Numbers of form 4^i*7^j, with i, j >= 0.

Original entry on oeis.org

1, 4, 7, 16, 28, 49, 64, 112, 196, 256, 343, 448, 784, 1024, 1372, 1792, 2401, 3136, 4096, 5488, 7168, 9604, 12544, 16384, 16807, 21952, 28672, 38416, 50176, 65536, 67228, 87808, 114688, 117649, 153664, 200704, 262144, 268912, 351232, 458752, 470596

Offset: 1

David W. Wilson

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

Subsequence of A003591.

n=10^6; Flatten[Table[4^i*7^j, {i, 0, Log[4, n]}, {j, 0, Log[7, n/4^i]}]]//Sort (* Amiram Eldar, Sep 24 2020 *)

Sum_{n>=1} 1/a(n) = (4*7)/((4-1)*(7-1)) = 14/9. - Amiram Eldar, Sep 24 2020
a(n) ~ exp(sqrt(2*log(4)*log(7)*n)) / sqrt(28). - Vaclav Kotesovec, Sep 24 2020
a(n) = 4^A025648(n) * 7^A025666(n). - R. J. Mathar, Jul 06 2025

A316991 Numbers m such that 1 < gcd(m, 14) < m and m does not divide 14^e for e >= 0.

Original entry on oeis.org

6, 10, 12, 18, 20, 21, 22, 24, 26, 30, 34, 35, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 58, 60, 62, 63, 66, 68, 70, 72, 74, 76, 77, 78, 80, 82, 84, 86, 88, 90, 91, 92, 94, 96, 100, 102, 104, 105, 106, 108, 110, 114, 116, 118, 119, 120, 122, 124, 126, 130, 132

Offset: 1

Michael De Vlieger, Aug 02 2018

Complement of A000027 and union of A003591 and A162699.
Analogous to A081062 and A105115 that apply to A120944(1) and A120944(2), respectively.
This sequence applies to A120944(3).

			6 is in the sequence since gcd(6, 14) = 2 and 6 does not divide 14^e with integer e >= 0.
2 is not in the sequence since 2 | 14.
4 is not in the sequence since 4 | 14^2.
3 and 5 are not in the sequence since they are coprime to 14.

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

Cf. A003591, A081062, A105115, A120944, A162699.

With[{nn = 132, k = 14}, Select[Range@ nn, And[1 < GCD[#, k] < #, PowerMod[k, Floor@ Log2@ nn, #] != 0] &]]

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.

A108513 Numbers of the form (2^i)(5^j)(7^k), with i, j, k >= 0.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 14, 16, 20, 25, 28, 32, 35, 40, 49, 50, 56, 64, 70, 80, 98, 100, 112, 125, 128, 140, 160, 175, 196, 200, 224, 245, 250, 256, 280, 320, 343, 350, 392, 400, 448, 490, 500, 512, 560, 625, 640, 686, 700, 784, 800, 875, 896, 980, 1000, 1024, 1120

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jul 05 2005

Numbers m | 70^e with integer e >= 0. - Michael De Vlieger, Aug 22 2019

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

Cf. A051037, A108319, A108347, A003592, A003591, A003595.

With[{n = 1120}, Sort@ Flatten@ Table[2^i * 5^j * 7^k, {i, 0, Log2@ n}, {j, 0, Log[5, n/2^i]}, {k, 0, Log[7, n/(2^i*5^j)]}]] (* Michael De Vlieger, Aug 22 2019 *)

isok(n) = (n/(2^valuation(n,2)*5^valuation(n,5)*7^valuation(n,7)) == 1); \\ Michel Marcus, Oct 01 2013

Sum_{n>=1} 1/a(n) = (2*5*7)/((2-1)*(5-1)*(7-1)) = 35/12. - Amiram Eldar, Sep 23 2020

a(n) ~ exp((6*log(2)*log(5)*log(7)*n)^(1/3)) / sqrt(70). - Vaclav Kotesovec, Sep 23 2020

A025637 Exponent of 2 (value of i) in n-th number of form 2^i*7^j.

Original entry on oeis.org

0, 1, 2, 0, 3, 1, 4, 2, 5, 0, 3, 6, 1, 4, 7, 2, 5, 8, 0, 3, 6, 9, 1, 4, 7, 10, 2, 5, 8, 11, 0, 3, 6, 9, 12, 1, 4, 7, 10, 13, 2, 5, 8, 11, 14, 0, 3, 6, 9, 12, 15, 1, 4, 7, 10, 13, 16, 2, 5, 8, 11, 14, 0, 17, 3, 6, 9, 12, 15, 1, 18, 4, 7, 10, 13, 16, 2, 19, 5, 8, 11, 14, 0, 17, 3, 20, 6, 9, 12, 15, 1

Offset: 1

David W. Wilson

nonn

Cf. A003591.

cp's OEIS Frontend

A108056 Numbers of the form (7^i)*(13^j).

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A108347 Numbers of the form (3^i)*(5^j)*(7^k), with i, j, k >= 0.

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A108319 Numbers of the form (2^i)*(3^j)*(7^k), with i, j, k >= 0.

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A025626 Numbers of form 6^i*7^j, with i, j >= 0.

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A177805 Numbers k such that k divides 15^k - 1.

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Maple

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A025619 Numbers of form 4^i*7^j, with i, j >= 0.

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A316991 Numbers m such that 1 < gcd(m, 14) < m and m does not divide 14^e for e >= 0.

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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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A108347 Numbers of the form (3^i)(5^j)(7^k), with i, j, k >= 0.

A108319 Numbers of the form (2^i)(3^j)(7^k), with i, j, k >= 0.

A108513 Numbers of the form (2^i)(5^j)(7^k), with i, j, k >= 0.