A025618 -id:A025618 - cp's OEIS frontend

A025629 Numbers of form 6^i*10^j with i, j >= 0.

1, 6, 10, 36, 60, 100, 216, 360, 600, 1000, 1296, 2160, 3600, 6000, 7776, 10000, 12960, 21600, 36000, 46656, 60000, 77760, 100000, 129600, 216000, 279936, 360000, 466560, 600000, 777600, 1000000, 1296000, 1679616, 2160000, 2799360, 3600000, 4665600

Offset: 1

David W. Wilson

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

Cf. A025610, A025618, A025622, A025626, A025627, A025628.

Cf. A025612, A025616, A025621, A025625, A025632, A025634.

n = 10^6; Flatten[Table[6^i*10^j, {i, 0, Log[6, n]}, {j, 0, Log10[n/6^i]}]] // Sort (* Amiram Eldar, Sep 26 2020 *)

list(lim)=my(v=List(), N); for(n=0, logint(lim\=1, 10), N=10^n; while(N<=lim, listput(v, N); N*=6)); Set(v) \\ Charles R Greathouse IV, Jan 10 2018

Sum_{n>=1} 1/a(n) = (6*10)/((6-1)*(10-1)) = 4/3. - Amiram Eldar, Sep 26 2020
a(n) ~ exp(sqrt(2*log(6)*log(10)*n)) / sqrt(60). - Vaclav Kotesovec, Sep 26 2020
a(n) = 6^A025663(n) * 10^A025688(n). - R. J. Mathar, Jul 06 2025

A107988 Numbers of the form (4^i)*(11^j), with i, j >= 0.

Original entry on oeis.org

1, 4, 11, 16, 44, 64, 121, 176, 256, 484, 704, 1024, 1331, 1936, 2816, 4096, 5324, 7744, 11264, 14641, 16384, 21296, 30976, 45056, 58564, 65536, 85184, 123904, 161051, 180224, 234256, 262144, 340736, 495616, 644204, 720896, 937024, 1048576

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jun 12 2005

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Subsequence of A003596.

Cf. A025617, A025618, A025619, A025620, A025621, A107462, A003597, A003598, A003599, A108090.

import Data.Set (singleton, deleteFindMin, insert)
a107988 n = a107988_list !! (n-1)
a107988_list = f $ singleton (1,0,0) where
   f s = y : f (insert (4 * y, i + 1, j) $ insert (11 * y, i, j + 1) s')
         where ((y, i, j), s') = deleteFindMin s
-- Reinhard Zumkeller, May 15 2015

n = 10^6; Flatten[Table[4^i*11^j, {i, 0, Log[4, n]}, {j, 0, Log[11, n/4^i]}]] // Sort (* Amiram Eldar, Sep 24 2020 *)

Sum_{n>=1} 1/a(n) = (4*11)/((4-1)*(11-1)) = 22/15. - Amiram Eldar, Sep 24 2020

a(n) ~ exp(sqrt(2*log(4)*log(11)*n)) / sqrt(44). - Vaclav Kotesovec, Sep 24 2020

A107462 Numbers of the form (4^i)*(13^j), with i, j >= 0.

Original entry on oeis.org

1, 4, 13, 16, 52, 64, 169, 208, 256, 676, 832, 1024, 2197, 2704, 3328, 4096, 8788, 10816, 13312, 16384, 28561, 35152, 43264, 53248, 65536, 114244, 140608, 173056, 212992, 262144, 371293, 456976, 562432, 692224, 851968, 1048576, 1485172

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jun 09 2005

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

Subsequence of A107326.

Cf. A025617, A025618, A025619, A025620, A025621, A107364, A107466, A108056, A108090.

n = 10^6; Flatten[Table[4^i*13^j, {i, 0, Log[4, n]}, {j, 0, Log[13, n/4^i]}]] // Sort (* Amiram Eldar, Sep 24 2020 *)

Sum_{n>=1} 1/a(n) = (4*13)/((4-1)*(13-1)) = 13/9. - Amiram Eldar, Sep 24 2020

a(n) ~ exp(sqrt(2*log(4)*log(13)*n)) / sqrt(52). - Vaclav Kotesovec, Sep 24 2020

A025647 Exponent of 4 (value of i) in n-th number of form 4^i*6^j.

Original entry on oeis.org

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

Offset: 1

David W. Wilson

nonn

Cf. A025618, A025658, A235127.

Differs from A025653 at a(2805).

With[{max = 10^10}, SortBy[Flatten[Table[{i, j}, {i, 0, Log[4, max]}, {j, 0, Log[6, max/4^i]}], 1], 4^#[[1]] * 6^#[[2]] &][[;; , 1]]] (* Amiram Eldar, Jul 09 2025 *)

a(n) = A235127(A025618(n)/6^A025658(n)). - Amiram Eldar, Jul 09 2025

A025658 Exponent of 6 (value of j) in n-th number of form 4^i*6^j.

Original entry on oeis.org

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

Offset: 1

David W. Wilson

nonn

Cf. A025618, A025647, A122841.

Differs from A025673 at a(2805).

With[{max = 10^10}, IntegerExponent[Sort[Flatten[Table[4^i*6^j, {i, 0, Log[4, max]}, {j, 0, Log[6, max/4^i]}]]], 6]] (* Amiram Eldar, Jul 09 2025 *)

a(n) = A122841(A025618(n)). - Amiram Eldar, Jul 09 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

A025703 Index of 4^n within sequence of numbers of form 4^i*6^j.

Original entry on oeis.org

1, 2, 4, 7, 11, 15, 20, 26, 33, 40, 48, 57, 67, 78, 89, 101, 114, 128, 142, 157, 173, 190, 208, 226, 245, 265, 286, 307, 329, 352, 376, 400, 425, 451, 478, 506, 534, 563, 593, 624, 655, 687, 720, 754, 789, 824, 860, 897, 935, 973, 1012, 1052, 1093, 1135, 1177, 1220

Offset: 0

David W. Wilson

First differs from A025709 at a(84).

Positions of zeros in A025658. - R. J. Mathar, Jul 06 2025

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A025618, A025709.

N:= 200: # to get a(0) to a(N)
[$1..N+1] + ListTools:-PartialSums([seq(floor(j*log[6](4)),j=0..N)]); # Robert Israel, May 10 2015

a(n)=my(N=1); n+1+sum(i=1, n, logint(N*=4, 6)); \\ Charles R Greathouse IV, Jan 11 2018

first(n)=my(s, N=1/4); vector(n+1, i, s+=logint(N*=4, 6)+1) \\ Charles R Greathouse IV, Jan 11 2018

a(n) = n + 1 + Sum(j= 1 .. n, floor(j*log[6](4))). - Robert Israel, May 10 2015

Edited and offset corrected by Robert Israel, May 10 2015

A025714 Index of 6^n within sequence of numbers of form 4^i*6^j.

Original entry on oeis.org

1, 3, 6, 10, 16, 23, 31, 41, 52, 64, 77, 92, 108, 125, 144, 164, 185, 207, 231, 256, 282, 310, 339, 369, 401, 434, 468, 503, 540, 578, 617, 658, 700, 743, 787, 833, 880, 928, 978, 1029, 1081, 1134, 1189, 1245, 1302, 1361, 1421, 1482, 1545, 1609, 1674, 1740, 1808

Offset: 1

David W. Wilson

nonn

Positions of zeros in A025647. - R. J. Mathar, Jul 06 2025

Cf. A025618. Differs from A025729 at a(65).

a(n)=my(N=1); n+1+sum(i=1, n, logint(N*=6, 4)); \\ Charles R Greathouse IV, Jan 16 2018

first(n)=my(s, N=1/6); vector(n+1, i, s+=logint(N*=6, 4)+1) \\ Charles R Greathouse IV, Jan 16 2018

cp's OEIS Frontend

A025629 Numbers of form 6^i*10^j with i, j >= 0.

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A107988 Numbers of the form (4^i)*(11^j), with i, j >= 0.

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A107462 Numbers of the form (4^i)*(13^j), with i, j >= 0.

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A025647 Exponent of 4 (value of i) in n-th number of form 4^i*6^j.

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A025658 Exponent of 6 (value of j) in n-th number of form 4^i*6^j.

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

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Mathematica

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A025703 Index of 4^n within sequence of numbers of form 4^i*6^j.

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Maple

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A025714 Index of 6^n within sequence of numbers of form 4^i*6^j.

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