A025610 -id:A025610 - cp's OEIS frontend

A025712 Index of 6^n within sequence of numbers of form 2^i*6^j.

1, 4, 10, 18, 29, 42, 58, 77, 98, 122, 148, 177, 209, 243, 280, 319, 361, 405, 452, 502, 554, 609, 666, 726, 789, 854, 922, 992, 1065, 1140, 1218, 1299, 1382, 1468, 1556, 1647, 1741, 1837, 1936, 2037, 2141, 2247, 2356, 2468, 2582, 2699, 2818, 2940, 3065, 3192

Offset: 1

David W. Wilson

nonn

Positions of zeros in A025636. - R. J. Mathar, Jul 06 2025

Michael De Vlieger, Table of n, a(n) for n = 1..1000
Beata Bajorska-Harapińska, Barbara Smoleń, Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.

Cf. A025610.

With[{n = 6^49}, Position[Union@ Flatten@ Table[2^a * 6^b, {a, 0, Log2@ n}, {b, 0, Log[6, n/2^a]}], ?(IntegerQ@ Log[6, #] &)][[All, 1]]] (* _Michael De Vlieger, Jul 16 2019 *)

A307252 Records in A319100.

Original entry on oeis.org

1, 2, 6, 12, 24, 36, 48, 72, 144, 216, 288, 432, 864, 1296, 1728, 2592, 5184, 7776, 10368, 15552, 31104, 46656, 62208, 93312, 186624, 373248, 559872, 1119744, 2239488, 3359232, 4478976, 6718464, 13436928, 20155392, 26873856, 40310784, 80621568, 120932352

Offset: 1

Jianing Song, Mar 31 2019

nonn

All terms are of the form 6^u*2^j. Other than the term 48, k = 6^i*2^j is a term if and only if for all i', j' such that F(i',j') < F(i,j) we have 6^i'*2^j' < 6^i*2^j, where F(i,j) = Product_{s=1..i} (p_s)*Product_{t=1..j} (q_t), where p_1 = 7, p_2 = 9, p_s = A002476(s-1) for s >= 3; q_1 = 4, q_2 = 2, q_t = A007528(t-2) for t >= 3. Or equivalently: (a) for any u, v such that u <= i and 6^u < 2^v, Product_{s=i-u+1..i} (p_s) < Product_{t=j+1..j+v} (q_t); (b) for any u, v such that v <= j and 6^u > 2^v, Product_{s=i+1..i+u} (p_s) > Product_{t=j-v+1..j} (q_t). For example, 746496 = 6^6*2^4 is not a term because (q_3)*(q_4) = 5*11 > p_7 = 43.

			A319100(168) = 48 which is larger than A319100(i) for i < 168, so 48 is a term.

Cf. A319100, A307251, A002476, A007528, A025610.

P(n) = if(!n, 1, if(n==1, 7, my(i=0,N=9); forprime(p=7, oo, if(p%3==1, i++; N*=p); if(i==n-1, return(N)))))
Q(n) = if(!n, 1, if(n==1, 4, my(i=0,N=4); forprime(p=2, oo, if(p%3==2, i++; N*=p); if(i==n-1, return(N)))))
v = []; for(i=0, 15, for(j=0, 15, if(P(i)*Q(j) < min(P(16), Q(16)), v=concat(v, [P(i)*Q(j)])))); v=vecsort(v);
u = []; for(i=1, #v, if(sum(j=1, i-1, A319100(v[j]) >= A319100(v[i]))==0, u=concat(u, [A319100(v[i])])));
vecsort(concat(u, [48])) \\ See A319100 for its program

A333964 Numbers of the form 2^i * 6^j * 30^k * 210^m where i, j, k, m >= 0.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 210, 216, 240, 256, 288, 360, 384, 420, 432, 480, 512, 576, 720, 768, 840, 864, 900, 960, 1024, 1080, 1152, 1260, 1296, 1440, 1536, 1680, 1728, 1800, 1920, 2048, 2160, 2304, 2520, 2592

Offset: 1

David A. Corneth, Apr 20 2020

nonn

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

Intersection of A025487 and A002473.

Cf. A025610, A279537.

mx = 2600; Select[Sort[Flatten[Table[2^i*6^j*30^k*210^m, {i, 0, Log[2, mx]}, {j, 0, Log[6, mx]}, {k, 0, Log[30, mx]}, {m, 0, Log[210, mx]}]]], # <= mx &] (* Amiram Eldar, Apr 24 2020 after Robert G. Wilson v at A279537 *)

Sum_{n>=1} 1/a(n) = 15120/6061. - Amiram Eldar, Feb 18 2021

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A025712 Index of 6^n within sequence of numbers of form 2^i*6^j.

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A307252 Records in A319100.

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A333964 Numbers of the form 2^i * 6^j * 30^k * 210^m where i, j, k, m >= 0.

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