A025643 -id:A025643 - cp's OEIS frontend

A025615 Numbers of form 3^i*8^j, with i, j >= 0.

1, 3, 8, 9, 24, 27, 64, 72, 81, 192, 216, 243, 512, 576, 648, 729, 1536, 1728, 1944, 2187, 4096, 4608, 5184, 5832, 6561, 12288, 13824, 15552, 17496, 19683, 32768, 36864, 41472, 46656, 52488, 59049, 98304, 110592, 124416, 139968, 157464, 177147, 262144

Offset: 1

David W. Wilson

Subset of A003586 3-smooth numbers: numbers of the form 2^i*3^j with i, j >= 0. - Daniel Forgues, Feb 24 2011

Indices for which a term is a power of 3 are in A025699 and a power of 8 are in A025728- Bernard Schott, Dec 27 2021

Georg Fischer, Table of n, a(n) for n = 1..300

Cf. A003586, A025699, A025728.

lim = 262144; Select[Sort[Flatten[Table[3^i 8^j, {i, 0, Log[3, lim]}, {j, 0, Log[8, lim]}]]], # <=lim &] (* T. D. Noe, Mar 01 2012 *)

list(lim)=my(v=List(), N); for(n=0, logint(lim\=1, 3), N=3^n; while(N<=lim, listput(v, N); N<<=3)); Set(v) \\ Charles R Greathouse IV, Jan 10 2018

Sum_{n>=1} 1/a(n) = 12/7. - Amiram Eldar, Feb 18 2021
From Bernard Schott, Dec 27 2021: (Start)
a(A025699(n)) = 3^(n-1).
a(A025728(n)) = 8^(n-1). (End)
a(n) = 3^A025643(n) * 8^A025672(n). - R. J. Mathar, Jul 06 2025

A384688 Runs of t in the range 0 <= t <= k and the same parity as k, for successive k >= 0.

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

Offset: 0

Kevin Ryde, Jun 07 2025

The corresponding k is A055086(n), or k+1 = A000267(n).
A run is 0, 2, 4, ..., k when k even, or 1, 3, 5, ..., k when k odd, and has length floor(k/2) + 1.
Runs start at quarter squares n = A002620(k+1), with those beginning 0 at oblong numbers n = A002378(i) and those starting 1 at the squares n = (i+1)^2 (for i >= 0 in both cases).
Starts to differ from A025643 at n=109.

			Runs and their corresponding k = A055086(n) begin,
  n          = 0  1  2    4    6      9
  a(n)       = 0, 1, 0,2, 1,3, 0,2,4, 1,3,5, ...
  A055086(n) = 0, 1, 2,2, 3,3, 4,4,4, 5,5,5, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..5001

Cf. A000196, A000267, A053186, A055086, A055087, A079813, A216607.

Cf. A002620, A002378 (indices of 0's), A000290 (indices of 1's).

ClearAll[a] a[n_Integer]:=Module[{s,r},s=Floor[Sqrt[n]]; r=n-s^2; If[rVincenzo Librandi, Jul 06 2025 *)

PARI
```
a(n) = my(r,s=sqrtint(n,&r)); if(r
				
```

a(n) = 2*r+1 if r < s or a(n) = 2*(r-s) otherwise, where square root and remainder n = s^2 + r being  s=A000196(n), r=A053186(n).
a(n) = ceiling(A053186(4*n+1) / 2).
a(n) = A055086(n) - 2*A216607(n+1).
a(n) = 2*A055087(n) + A079813(n+1).

A025728 Index of 8^n within sequence of numbers of form 3^i*8^j (A025615).

Original entry on oeis.org

1, 3, 7, 13, 21, 31, 43, 57, 73, 91, 110, 131, 154, 179, 206, 235, 266, 299, 334, 370, 408, 448, 490, 534, 580, 628, 678, 730, 783, 838, 895, 954, 1015, 1078, 1143, 1210, 1279, 1350, 1422, 1496, 1572, 1650, 1730, 1812, 1896, 1982, 2070, 2159, 2250, 2343, 2438

Offset: 1

David W. Wilson

nonn

Positions of zeros in A025643. - R. J. Mathar, Jul 06 2025

Cf. A025615, A025699 (similar for 3^n).

A025615(a(n)) = 8^(n-1). - Bernard Schott, Dec 27 2021

cp's OEIS Frontend

A025615 Numbers of form 3^i*8^j, with i, j >= 0.

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A384688 Runs of t in the range 0 <= t <= k and the same parity as k, for successive k >= 0.

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A025728 Index of 8^n within sequence of numbers of form 3^i*8^j (A025615).

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