A003594 -id:A003594 - 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

A025631 Numbers of form 7^i*9^j, with i, j >= 0.

Original entry on oeis.org

1, 7, 9, 49, 63, 81, 343, 441, 567, 729, 2401, 3087, 3969, 5103, 6561, 16807, 21609, 27783, 35721, 45927, 59049, 117649, 151263, 194481, 250047, 321489, 413343, 531441, 823543, 1058841, 1361367, 1750329, 2250423, 2893401, 3720087, 4782969

Offset: 1

David W. Wilson

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

Subsequence of A003594.

With[{nn=40},Take[Union[7^First[#] 9^Last[#]&/@Tuples[Range[0,nn/4],2]],nn]] (* Harvey P. Dale, Mar 06 2014 *)

Sum_{n>=1} 1/a(n) = (7*9)/((7-1)*(9-1)) = 21/16. - Amiram Eldar, Sep 24 2020
a(n) ~ exp(sqrt(2*log(7)*log(9)*n)) / sqrt(63). - Vaclav Kotesovec, Sep 24 2020
a(n) = 7^A025670(n) * 9^A025681(n). - R. J. Mathar, Jul 06 2025

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

Original entry on oeis.org

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

A306999 Numbers m such that 1 < gcd(m, 21) < m and m does not divide 21^e for e >= 0.

Original entry on oeis.org

6, 12, 14, 15, 18, 24, 28, 30, 33, 35, 36, 39, 42, 45, 48, 51, 54, 56, 57, 60, 66, 69, 70, 72, 75, 77, 78, 84, 87, 90, 91, 93, 96, 98, 99, 102, 105, 108, 111, 112, 114, 117, 119, 120, 123, 126, 129, 132, 133, 135, 138, 140, 141, 144, 150, 153, 154, 156, 159, 161

Offset: 1

Michael De Vlieger, Aug 22 2019

Complement of the union of A003594 and A160545.

Analogous to A081062 and A105115 regarding terms 1 and 2 of A120944, respectively. This sequence applies to A120944(5) = 21.

			6 is in the sequence since gcd(6, 21) = 3 and 6 does not divide 21^e with integer e >= 0.
5 is not in the sequence since it is coprime to 21.
3 is not in the sequence since 3 | 21.
9 is not in the sequence since 9 | 21^2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003594, A081062, A105115, A120944, A160545.

filter:= proc(n) local g;
  g:= igcd(n,21);
  if g = 1 or g = n then return false fi;
  3^padic:-ordp(n,3)*7^padic:-ordp(n,7) < n
end proc:
select(filter, [$1..200]); # Robert Israel, Aug 28 2019

With[{nn = 161, k = 21}, Select[Range@ nn, And[1 < GCD[#, k] < #, PowerMod[k, Floor@ Log2@ nn, #] != 0] &]]

A025642 Exponent of 3 (value of i) in n-th number of form 3^i*7^j.

Original entry on oeis.org

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

Offset: 1

David W. Wilson

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003594.

N:= 10^9: # for 3^i * 7^j <= N
R:= sort([seq(seq(3^i*7^j, j=0 .. ilog[7](N/3^i)),i=0..ilog[3](N))]):
map(padic:-ordp,R,3); # Robert Israel, Jun 04 2025

A025665 Exponent of 7 (value of j) in n-th number of form 3^i*7^j.

Original entry on oeis.org

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

Offset: 1

David W. Wilson

nonn

Cf. A003594.

A036316 Composite numbers whose prime factors contain no digits other than 3 and 7.

Original entry on oeis.org

9, 21, 27, 49, 63, 81, 111, 147, 189, 219, 243, 259, 333, 343, 441, 511, 567, 657, 729, 777, 999, 1011, 1029, 1119, 1323, 1369, 1533, 1701, 1813, 1971, 2187, 2199, 2319, 2331, 2359, 2401, 2611, 2701, 2997, 3033, 3087, 3357, 3577, 3969, 4107, 4599, 5103

Offset: 1

Patrick De Geest, Dec 15 1998

All terms are a product of at least two terms of A020463. - David A. Corneth, Oct 09 2020

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 800 terms from Vincenzo Librandi)
Index entries for sequences related to prime factors.

Cf. A003594, A020463, A036302-A036325.

[n: n in [9..6000] | not IsPrime(n) and forall{f: f in PrimeDivisors(n) | Intseq(f) subset [3,7]}]; // Bruno Berselli, Aug 26 2013

dpfQ[n_]:=Module[{d=Union[Flatten[IntegerDigits/@Transpose[FactorInteger[n]][[1]]]]}, !PrimeQ[n]&&(d == {3}||d == {7}||d == {3, 7})]; Select[Range[6000], dpfQ] (* Vincenzo Librandi, Aug 25 2013 *)

Sum_{n>=1} 1/a(n) = Product_{p in A020463} (p/(p - 1)) - Sum_{p in A020463} 1/p - 1 = 0.3143000293... . - Amiram Eldar, May 22 2022

A057233 Numbers n such that n | 8^n + 7^n + 6^n.

Original entry on oeis.org

1, 3, 7, 9, 21, 27, 49, 51, 63, 81, 147, 189, 243, 343, 441, 567, 729, 931, 1029, 1323, 1593, 1701, 2187, 2349, 2401, 2493, 2499, 3087, 3969, 4077, 4131, 5103, 6561, 7203, 9261, 11907, 15309, 16807, 19683, 21609, 27783, 31743, 35721, 45927, 50421

Offset: 1

Robert G. Wilson v, Sep 20 2000

nonn

Robert Israel, Table of n, a(n) for n = 1..238

Contains A003594.

select(n -> 6 &^ n + 7 &^ n + 8 &^ n) mod n = 0, [$1..10^5]); # Robert Israel, Jan 30 2020

Select[ Range[ 10^6 ], Mod[ PowerMod[ 8, #, # ] + PowerMod[ 7, #, # ] + PowerMod[ 6, #, # ], # ] == 0 & ]

A057493 Numbers n such that n | 11^n + 10^n.

Original entry on oeis.org

1, 3, 7, 9, 21, 27, 49, 63, 81, 111, 147, 171, 189, 203, 243, 333, 343, 441, 513, 567, 609, 729, 777, 999, 1029, 1143, 1197, 1323, 1421, 1539, 1701, 1791, 1827, 2187, 2331, 2401, 2943, 2997, 3087, 3249, 3429, 3591, 3969, 4107, 4263, 4617, 5103, 5373

Offset: 1

Robert G. Wilson v, Sep 20 2000

nonn

From Robert Israel, Feb 23 2017: (Start)
Contains A003594.
Every term other than 1 is divisible by 3 or 7.
If m and n are in the sequence, then so is m*n. (End)

Robert Israel, Table of n, a(n) for n = 1..5305 (first 580 terms from Vincenzo Librandi)

Cf. A003594.

select(t -> (10 &^ t + 11 &^ t mod t = 0), [seq(i,i=1..10000,2)]); # Robert Israel, Feb 23 2017

Select[ Range[ 10000 ], Mod[ PowerMod[ 11, #, # ] + PowerMod[ 10, #, # ], # ] == 0 & ]

is(n)=Mod(11,n)^n+Mod(10,n)^n==0 \\ Charles R Greathouse IV, Feb 23 2017

A025721 Index of 7^n within sequence of numbers of form 3^i*7^j for i >= 0, j >= 1.

Original entry on oeis.org

1, 3, 7, 13, 21, 30, 41, 54, 69, 85, 103, 123, 145, 169, 194, 221, 250, 281, 313, 347, 383, 421, 460, 501, 544, 589, 636, 684, 734, 786, 840, 895, 952, 1011, 1072, 1134, 1198, 1264, 1332, 1402, 1473, 1546, 1621, 1698, 1776, 1856, 1938, 2022, 2108, 2195, 2284

Offset: 1

David W. Wilson

nonn

Positions of zeros in A025642. - R. J. Mathar, Jul 06 2025

			a(1) = 1, a(2) = 3 and a(3) = 7 because the first 7 numbers of the form 3^i * 7^j with i >= 0 and j >= 1 are 7, 21, 49, 63, 147, 189 and 343. - _Robert Israel_, Aug 20 2024

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003594.

dA:= map(t -> 1+ilog[3](7^t), [$0..100]):
ListTools:-PartialSums(dA); # Robert Israel, Aug 20 2024

Name clarified by Robert Israel, Aug 20 2024

cp's OEIS Frontend

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

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A025631 Numbers of form 7^i*9^j, with i, j >= 0.

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

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A306999 Numbers m such that 1 < gcd(m, 21) < m and m does not divide 21^e for e >= 0.

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Maple

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A025642 Exponent of 3 (value of i) in n-th number of form 3^i*7^j.

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Maple

A025665 Exponent of 7 (value of j) in n-th number of form 3^i*7^j.

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A036316 Composite numbers whose prime factors contain no digits other than 3 and 7.

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Magma

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A057233 Numbers n such that n | 8^n + 7^n + 6^n.

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Maple

Mathematica

A057493 Numbers n such that n | 11^n + 10^n.

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