A003595 -id:A003595 - 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

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

A108201 Numbers of the form (5^i)*(12^j), with i, j >= 0.

Original entry on oeis.org

1, 5, 12, 25, 60, 125, 144, 300, 625, 720, 1500, 1728, 3125, 3600, 7500, 8640, 15625, 18000, 20736, 37500, 43200, 78125, 90000, 103680, 187500, 216000, 248832, 390625, 450000, 518400, 937500, 1080000, 1244160, 1953125, 2250000, 2592000

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jun 15 2005

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

Cf. A025622, A003595, A025623, A025624, A025625, A003598, A107466, A064476.

Take[Union[5^First[#] 12^Last[#]&/@Tuples[Range[0,20],2]],50] (* Harvey P. Dale, Mar 23 2012 *)

Sum_{n>=1} 1/a(n) = 15/11. - Amiram Eldar, Mar 29 2025

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

A307589 Numbers m such that 1 < gcd(m, 35) < m and m does not divide 35^e for e >= 0.

Original entry on oeis.org

10, 14, 15, 20, 21, 28, 30, 40, 42, 45, 50, 55, 56, 60, 63, 65, 70, 75, 77, 80, 84, 85, 90, 91, 95, 98, 100, 105, 110, 112, 115, 119, 120, 126, 130, 133, 135, 140, 145, 147, 150, 154, 155, 160, 161, 165, 168, 170, 180, 182, 185, 189, 190, 195, 196, 200, 203, 205

Offset: 1

Michael De Vlieger, Aug 22 2019

Complement of the union of A003595 and A235933.

Analogous to A081062 and A105115 for terms 1 and 2 of A120944. This sequence applies to A120944(6) = 35.

			10 is in the sequence since gcd(10, 35) = 5 and 10 does not divide 35^e with integer e >= 0.
2 is not in the sequence since 2 is coprime to 35.
7 is not in the sequence since 7 | 35.
25 is not in the sequence since 25 | 35^2.

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

Cf. A003595, A081062, A105115, A120944, A235933, A306999.

With[{nn = 205, k = 35}, Select[Range@ nn, And[1 < GCD[#, k] < #, PowerMod[k, Floor@ Log2@ nn, #] != 0] &]]

is(n)=gcd(n,35)>1 && n/5^valuation(n,5)/7^valuation(n,7)>1 \\ Charles R Greathouse IV, Sep 07 2022

a(n) = 35n/11 + O(log^2 n). - Charles R Greathouse IV, Sep 07 2022

A025652 Exponent of 5 (value of i) in n-th number of form 5^i*7^j.

Original entry on oeis.org

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

Offset: 1

David W. Wilson

nonn

Cf. A003595.

A025667 Exponent of 7 (value of j) in n-th number of form 5^i*7^j.

Original entry on oeis.org

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

Offset: 1

David W. Wilson

nonn

Cf. A003595.

A036320 Composite numbers whose prime factors contain no digits other than 5 and 7.

Original entry on oeis.org

25, 35, 49, 125, 175, 245, 343, 625, 875, 1225, 1715, 2401, 2785, 2885, 3125, 3785, 3899, 4039, 4375, 5299, 6125, 8575, 12005, 13925, 14425, 15625, 16807, 18925, 19495, 20195, 21875, 26495, 27293, 27785, 28273, 30625, 37093, 37885, 38785

Offset: 1

Patrick De Geest, Dec 15 1998

All terms are a product of at least two terms of A020467. - David A. Corneth, Oct 09 2020

David A. Corneth, Table of n, a(n) for n = 1..10000
Index entries for sequences related to prime factors.

Cf. A003595, A020467, A036302-A036325.

Sum_{n>=1} 1/a(n) = Product_{p in A020467} (p/(p - 1)) - Sum_{p in A020467} 1/p - 1 = 0.1179595738... . - Amiram Eldar, May 22 2022

A057490 Numbers k that divide 8^k + 7^k + 6^k + 5^k + 4^k + 3^k + 2^k.

Original entry on oeis.org

1, 5, 7, 25, 35, 49, 125, 175, 245, 301, 343, 455, 625, 875, 1225, 1295, 1435, 1505, 1715, 1765, 2107, 2191, 2401, 3125, 4375, 6125, 7525, 8575, 10535, 11375, 12005, 12943, 14063, 14749, 15625, 16807, 21875, 22295, 30625, 35875, 37625, 42875, 52675, 60025, 64715, 70315, 73745, 78125, 80375, 84035, 90601, 93275

Offset: 1

Robert G. Wilson v, Sep 22 2000

nonn

Contains A003595. The first term not in A003595 is 301. Is 1 the only term not divisible by 5 or 7? - Robert Israel, Feb 22 2017

No term is divisible by 3. 5 and 7 are the only primes in this sequence. - Altug Alkan, Feb 23 2017

Cf. A003595.

select(t -> add(i &^ t, i=2..8) mod t = 0, [$1..10^6]); # Robert Israel, Feb 22 2017

Select[ Range[ 10^5 ], Mod[ PowerMod[ 8, #, # ] + PowerMod[ 7, #, # ] + PowerMod[ 6, #, # ] + PowerMod[ 5, #, # ] + PowerMod[ 4, #, # ] + PowerMod[ 3, #, # ] + PowerMod[ 2, #, # ], # ] == 0 & ]
Select[Range[100000],Mod[Total[PowerMod[Range[2,8],#,#]],#]==0&] (* Harvey P. Dale, Jul 28 2021 *)

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

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A108201 Numbers of the form (5^i)*(12^j), with i, j >= 0.

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

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A307589 Numbers m such that 1 < gcd(m, 35) < m and m does not divide 35^e for e >= 0.

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A025652 Exponent of 5 (value of i) in n-th number of form 5^i*7^j.

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A025667 Exponent of 7 (value of j) in n-th number of form 5^i*7^j.

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A036320 Composite numbers whose prime factors contain no digits other than 5 and 7.

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

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