A025658 -id:A025658 - cp's OEIS frontend

A025618 Numbers of form 4^i*6^j, with i, j >= 0.

1, 4, 6, 16, 24, 36, 64, 96, 144, 216, 256, 384, 576, 864, 1024, 1296, 1536, 2304, 3456, 4096, 5184, 6144, 7776, 9216, 13824, 16384, 20736, 24576, 31104, 36864, 46656, 55296, 65536, 82944, 98304, 124416, 147456, 186624, 221184, 262144, 279936

Offset: 1

David W. Wilson

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

Subsequence of A003586.

f[upto_]:=Module[{max4=Floor[Log[upto]/Log[4]],max6 = Floor[ Log[ upto]/ Log[6]], seq},seq=Union[4^First[#] 6^Last[#]&/@Tuples[ {Range[ 0,max4],Range[ 0,max6]}]];Select[seq,#<=upto&]];f[300000]  (* Harvey P. Dale, Mar 05 2011 *)

Sum_{n>=1} 1/a(n) = (4*6)/((4-1)*(6-1)) = 8/5. - Amiram Eldar, Sep 24 2020
a(n) ~ exp(sqrt(2*log(4)*log(6)*n)) / sqrt(24). - Vaclav Kotesovec, Sep 24 2020
a(n) = 4^A025647(n) * 6^A025658(n). - R. J. Mathar, Jul 06 2025

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

A025673 Exponent of 8 (value of j) in n-th number of form 5^i*8^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. A025623, A025653, A244413.

Differs from A025658 at a(2805).

Take[Sort[Flatten[Table[{5^i 8^j,j},{i,0,20},{j,0,20}],1]][[;;,2]],120] (* Harvey P. Dale, Nov 10 2024 *)
With[{max = 10^11}, IntegerExponent[Sort[Flatten[Table[5^i*8^j, {i, 0, Log[5, max]}, {j, 0, Log[8, max/5^i]}]]], 8]] (* Amiram Eldar, Jul 09 2025 *)

a(n) = A244413(A025623(n)). - Amiram Eldar, Jul 09 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

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A025618 Numbers of form 4^i*6^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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A025673 Exponent of 8 (value of j) in n-th number of form 5^i*8^j.

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

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