A036566 -id:A036566 - cp's OEIS frontend

A025630 Duplicate of A036566.

1, 7, 8, 49, 56, 64, 343, 392, 448, 512, 2401, 2744, 3136, 3584, 4096, 16807, 19208

Offset: 1

dead

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

A025669 Exponent of 7 (value of i) in n-th number of form 7^i*8^j.

Original entry on oeis.org

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

Offset: 1

David W. Wilson

nonn

Cf. A036566. Differs from A025581 at a(136).

A025675 Exponent of 8 (value of j) in n-th number of form 7^i*8^j.

Original entry on oeis.org

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

Offset: 1

David W. Wilson

nonn

Cf. A036566. Differs from A002262 at a(136).

A036567 Basic numbers used in Sedgewick-Incerpi upper bound for shell sort.

Original entry on oeis.org

1, 3, 7, 16, 41, 101, 247, 613, 1529, 3821, 9539, 23843, 59611, 149015, 372539, 931327, 2328307, 5820767, 14551919, 36379789, 90949471, 227373677, 568434193, 1421085473, 3552713687, 8881784201, 22204460497, 55511151233, 138777878081, 346944695197, 867361737989

Offset: 0

N. J. A. Sloane

			2.5^4 = 39.0625, and 41 is the next integer that is relatively prime to 1, 3, 7 and 16.

D. E. Knuth, The Art of Computer Programming, Vol. 3, Sorting and Searching, 2nd ed, section 5.2.1, p. 91.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Janet Incerpi and Robert Sedgewick, Improved upper bounds on shellsort, Journal of Computer and System Sciences, Volume 31, Issue 2, October 1985, Pages 210-224
Robert Sedgewick, Analysis of shellsort and related algorithms, Fourth European Symposium on Algorithms, Barcelona, September, 1996.
Index entries for sequences related to sorting

Cf. A003462, A033622, A036561, A036562, A036564, A036566, A036567, A036569, A055875, A055876, A102549, A108870, A112262, A112263, A154393, A204772, A205669, A205670, A361506, A361507.

a:= proc(n) option remember; local l, m;
      l:= [seq(a(i), i=1..n-1)];
      for m from ceil((5/2)^n) while ormap(x-> igcd(m, x)>1, l) do od; m
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 06 2022

A036567[1] = 3;
A036567[q_] :=
With[{prev = A036567 /@ Range[q - 1]},
  Block[{n = Ceiling[(5/2)^q]},
   While[Nand @@ ((# == 1 &) /@ GCD[prev, n]), n++];
   n]]; (* Morgan Owens, Oct 08 2020 *)
Array[A036567, 10]

a036567(m)={my(v=vector(m)); for(n=1,m,my(b=ceil((5/2)^n));for(j=b,oo,my(g=1); for(k=1,n-1,if(gcd(j,v[k])>1,g=0;break));if(g,v[n]=j;break)));v};
a036567(28) \\ Hugo Pfoertner, Oct 15 2020

a(n) is the smallest number >= 2.5^n that is relatively prime to all previous terms in the sequence.

Better description and more terms from Jud McCranie, Jan 05 2001

a(0)=1 prepended by Alois P. Heinz, Dec 04 2023

A108238 Numbers of the form (7^i)*(12^j), with i, j >= 0.

Original entry on oeis.org

1, 7, 12, 49, 84, 144, 343, 588, 1008, 1728, 2401, 4116, 7056, 12096, 16807, 20736, 28812, 49392, 84672, 117649, 145152, 201684, 248832, 345744, 592704, 823543, 1016064, 1411788, 1741824, 2420208, 2985984, 4148928, 5764801, 7112448

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jun 17 2005

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

Cf. A036566, A025631, A025632, A003599, A108056, A108201, A064476.

Take[7^#[[1]]*12^#[[2]]&/@Tuples[Range[0,10],2]//Union,40] (* Harvey P. Dale, Mar 05 2017 *)
n = 10^6; Flatten[Table[7^i*12^j, {i, 0, Log[7, n]}, {j, 0, Log[12, n/7^i]}]] // Sort (* Amiram Eldar, Sep 26 2020 *)

from sympy import integer_log
def A108238(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(integer_log(x//12**i,7)[0]+1 for i in range(integer_log(x,12)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Mar 26 2025

Sum_{n>=1} 1/a(n) = (7*12)/((7-1)*(12-1)) = 14/11. - Amiram Eldar, Sep 26 2020

a(n) ~ exp(sqrt(2*log(7)*log(12)*n)) / sqrt(84). - Vaclav Kotesovec, Sep 26 2020

A120027 Triangle, generated from (3^(n-k) * 5^k) table.

Original entry on oeis.org

1, 3, 5, 9, 15, 25, 27, 45, 75, 125, 81, 135, 225, 375, 625, 243, 405, 675, 1125, 1875, 3125, 729, 1215, 2025, 3375, 5625, 9375, 15625, 2187, 3645, 6075, 10125, 16875, 28125, 46875, 78125, 6561, 10935, 18225, 30375, 50625, 84375, 140625, 234375

Offset: 0

Gary W. Adamson, Jun 04 2006

Row 1 of the array (3, 15, 75, 375, ...) = A005053, (3 * 5^n), deleting the "1".

			First few rows of the array:
  1,  5,  25,  125, ...
  3, 15,  75,  375, ...
  9, 45, 225, 1125, ...
First few rows of the triangle are:
   1;
   3,  5;
   9, 15, 25;
  27, 45, 75, 125;
  ...
Example: a(17) = 675 = (3,2) in the array, = 3^3 * 5^2.

Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732 [math.CO], 2012.

Cf. A005053, A036561, A036565, A036566.

Table[3^(n - k)*5^k, {n, 0, 8}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jun 06 2006 *)

Antidiagonals of the (3^i * 5^j) multiplication table, as an array.
From Boris Putievskiy, Jan 09 2013: (Start)
T(n,k) = 3^(k-1)*5^(n-1) n, k >0 read by antidiagonals.
a(n) = 3^(A004736(n)-1) * 5^(A002260(n)-1), n > 0, or
a(n) = 3^(j-1) * 5^(i-1), n > 0,
where i = n - t*(t+1)/2, j = (t*t + 3*t + 4)/2 - n, t = floor((-1+sqrt(8*n-7))/2). (End)
G.f.: 1/((1 - 3*x)(1 - 5*x*y)). - Ilya Gutkovskiy, Jun 03 2017

More terms from Robert G. Wilson v, Jun 06 2006

A025725 Index of 7^n within sequence of numbers of form 7^i*8^j.

Original entry on oeis.org

1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 136, 152, 169, 187, 206, 226, 247, 269, 292, 316, 341, 367, 394, 422, 451, 481, 511, 542, 574, 607, 641, 676, 712, 749, 787, 826, 866, 907, 949, 992, 1036, 1080, 1125, 1171, 1218, 1266, 1315, 1365, 1416

Offset: 1

David W. Wilson

nonn

Positions of zeros in A025675. - R. J. Mathar, Jul 06 2025

Cf. A025675, A036566.

A025731 Index of 8^n within sequence of numbers of form 7^i*8^j.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 137, 155, 174, 194, 215, 237, 260, 284, 309, 335, 362, 390, 419, 449, 480, 513, 547, 582, 618, 655, 693, 732, 772, 813, 855, 898, 942, 987, 1033, 1081, 1130, 1180, 1231, 1283, 1336, 1390, 1445, 1501, 1558, 1616

Offset: 1

David W. Wilson

nonn

Positions of zeros in A025669. - R. J. Mathar, Jul 06 2025

Cf. A001018, A036566.

cp's OEIS Frontend

A025630 Duplicate of A036566.

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

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A025669 Exponent of 7 (value of i) in n-th number of form 7^i*8^j.

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A025675 Exponent of 8 (value of j) in n-th number of form 7^i*8^j.

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A036567 Basic numbers used in Sedgewick-Incerpi upper bound for shell sort.

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A108238 Numbers of the form (7^i)*(12^j), with i, j >= 0.

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A120027 Triangle, generated from (3^(n-k) * 5^k) table.

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A025725 Index of 7^n within sequence of numbers of form 7^i*8^j.

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A025731 Index of 8^n within sequence of numbers of form 7^i*8^j.

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