A025631 -id:A025631 - cp's OEIS frontend

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

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

A108238 Numbers of the form (7^i)*(12^j), with i, j >= 0.

Original entry on oeis.org

1, 7, 12, 49, 84, 144, 343, 588, 1008, 1728, 2401, 4116, 7056, 12096, 16807, 20736, 28812, 49392, 84672, 117649, 145152, 201684, 248832, 345744, 592704, 823543, 1016064, 1411788, 1741824, 2420208, 2985984, 4148928, 5764801, 7112448

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jun 17 2005

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

Cf. A036566, A025631, A025632, A003599, A108056, A108201, A064476.

Take[7^#[[1]]*12^#[[2]]&/@Tuples[Range[0,10],2]//Union,40] (* Harvey P. Dale, Mar 05 2017 *)
n = 10^6; Flatten[Table[7^i*12^j, {i, 0, Log[7, n]}, {j, 0, Log[12, n/7^i]}]] // Sort (* Amiram Eldar, Sep 26 2020 *)

from sympy import integer_log
def A108238(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = 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//12**i,7)[0]+1 for i in range(integer_log(x,12)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Mar 26 2025

Sum_{n>=1} 1/a(n) = (7*12)/((7-1)*(12-1)) = 14/11. - Amiram Eldar, Sep 26 2020

a(n) ~ exp(sqrt(2*log(7)*log(12)*n)) / sqrt(84). - Vaclav Kotesovec, Sep 26 2020

A025670 Exponent of 7 (value of i) in n-th number of form 7^i*9^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, 0, 6, 5, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 9, 0, 8, 7, 6, 5, 4, 3, 2, 10, 1, 9, 0, 8, 7, 6, 5, 4, 3, 11, 2, 10, 1, 9, 0, 8, 7, 6, 5, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 7, 6, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0, 8, 7, 6, 14, 5

Offset: 1

David W. Wilson

nonn

Cf. A025631.

A025681 Exponent of 9 (value of j) in n-th number of form 7^i*9^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, 5, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0, 8, 1, 2, 3, 4, 5, 6, 7, 0, 8, 1, 9, 2, 3, 4, 5, 6, 7, 0, 8, 1, 9, 2, 10, 3, 4, 5, 6, 7, 0, 8, 1, 9, 2, 10, 3, 11, 4, 5, 6, 7, 0, 8, 1, 9, 2, 10, 3, 11, 4, 12, 5, 6, 7, 0, 8, 1, 9, 2

Offset: 1

David W. Wilson

nonn

Cf. A025631.

cp's OEIS Frontend

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

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A108238 Numbers of the form (7^i)*(12^j), with i, j >= 0.

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A025670 Exponent of 7 (value of i) in n-th number of form 7^i*9^j.

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A025681 Exponent of 9 (value of j) in n-th number of form 7^i*9^j.

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