A109895 -id:A109895 - cp's OEIS frontend

A079759 Let b(0)=0. For n >= 1, b(n) is the least k > b(n-1)+1 such that k divides (k-1)!/b(n-1)!, and a(n) = (b(n)-1)!/(b(n-1)!*b(n)).

1, 20, 4620, 12697776, 159845400, 941432800, 158800433792, 1895312483064000, 3438271897004237230080, 933561026438040, 2562849175892544, 640904462719404383808000, 1528364130975, 2352733350786, 959393282698730880000, 6142080926952

Offset: 1

Amarnath Murthy, Jan 10 2003

nonn

Group the natural numbers so that every 2n-th group product is divisible by the single number in the next group. (1), (2,3,4,5), (6), (7,8,9,10,11), (12), (13,14,15,16,17,18,19),(20), (21,22,23,24,25,26,27),(28),...Sequence contains the ratio of the product of terms in 2n-th group and the (2n+1)-th group.

			a(1) = 1*2*3*4*5/6 = 20, a(2) = 7*8*9*10*11/12 = 4620, a(3) = 13*14*15*16*17*18*19/20 = 12697776, a(4) = 159845400 = 21*22*...*27/28.

Robert Israel, Table of n, a(n) for n = 1..10001 (corrected by Robert Israel, Jan 20 2019)

Cf. A079759, A079760, A109895, A109896, A109897.

t:= 0:
for n from 1 to 30 do
  p:= t+1;
  for j from t+2 while not (p/j)::integer do p:= p*j od;
  A[n]:= p/j;
  t:= j;
od:
seq(A[i],i=1..30); # Robert Israel, Jul 16 2018

a[1] = 1; t = 0; nmax = 16; For[n = 1, n <= nmax, n++, p = t+1; For[j = t+2, Not[IntegerQ[p/j]], j++, p = p*j]; a[n+1] = p/j; t = j];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Mar 25 2019, after Robert Israel *)

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com) and Sascha Kurz, Jan 12 2003

Edited by N. J. A. Sloane, Nov 04 2018 at the suggestion of Georg Fischer. This entry now contains the merger of two identical sequences submitted by the same author.

A337486 Numbers k such that b(k+1) divides b(k), where b() is Recamán's multiplicative sequence A008336.

Original entry on oeis.org

1, 6, 10, 12, 14, 18, 20, 22, 26, 28, 30, 34, 36, 38, 42, 44, 45, 46, 50, 52, 54, 58, 60, 66, 68, 70, 72, 78, 82, 84, 86, 90, 92, 93, 94, 95, 98, 99, 100, 104, 106, 110, 111, 114, 116, 118, 119, 122, 124, 126, 130, 132, 134, 135, 136, 142, 146, 147, 148, 150, 154, 156, 158, 161, 162, 164, 165, 166

Offset: 1

Scott R. Shannon, Aug 29 2020

nonn

The old definition was: 1, together with the numbers formed by removing the required prime factors to form the number from a set which is initially empty and that has primes added via the addition of the prime factors of numbers which cannot be created from those currently in the set. Start by trying to create the number 2.
Consider an initially empty set of primes whose numbers are used to create a given number where each time a number is created those prime factors are removed from the set. If a number cannot be created as all its required prime factors are not currently in the set then all the prime factors of that number are instead added to the set. Start by trying to create the number 2 followed by all other integers. This sequence list the numbers that are created.
For the first 1 million terms the largest gap between terms is 15, between a(88018) = 189648 and a(88019) = 189663. The 1 millionth term is created after the addition of the number 2123404. The prime set at that point has 114323 primes, with the maximum number of entries for a single prime being eleven, for 887.
Note that if after the creation of a number the entire set of primes is cleared then the numbers created are those in A109895.

			a(2) = 6. As there are no primes initially in the set 2,3,4,5 cannot be created and instead these numbers add three 2's, one 3 and one 5 to the set. As there is now one 2 and one 3 the number 6 = 2*3 can be created. After 6 is created the set of primes now contains 2,2,5.
a(3) = 10. After 7,8,9, none of which can be created from the prime set, the prime set contains 2,2,2,2,2,3,3,5,7. As 2 and 5 are present 10 = 2*5 can be created, after which the set contains 2,2,2,2,3,3,7.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A005132, A008336, A331276, A330430, A109895.

First differences are A370969.

Block[{nn = 166, k = 1}, Reap[Do[If[Mod[k, i] == 0, k /= i; Sow[i], k *= i], {i, nn}]][[-1, 1]]] (* Michael De Vlieger, Sep 12 2020 *)

from itertools import count, islice
def A337486_gen(): # generator of terms
    c = 1
    for n in count(1):
        a, b = divmod(c,n)
        if not b:
            c = a
            yield n
        else:
            c *= n
A337486_list = list(islice(A337486_gen(),30)) # Chai Wah Wu, Apr 11 2024

A109896 Group the natural numbers so that every 2n-th group product is divisible by the single number in the next group. (1), (2,3,4,5), (6), (7,8,9,10,11), (12), (13,14,15,16,17,18,19),(20), (21,22,23,24,25,26,27),(28),... Sequence contains the number of terms in the 2n-th group.

Original entry on oeis.org

4, 5, 7, 7, 7, 8, 10, 13, 9, 9, 13, 7, 7, 11, 7, 13, 13, 11, 11, 15, 11, 13, 15, 13, 15, 16, 14, 11, 11, 13, 9, 17, 17, 19, 15, 15, 13, 17, 14, 8, 15, 18, 20, 15, 17, 17, 17, 9, 19, 11, 20, 20, 13, 19, 19, 15, 15, 9, 14, 14, 17, 16, 15, 18, 13, 20, 14, 14, 14, 17, 15, 15, 15, 21, 13

Offset: 1

Amarnath Murthy, Jul 13 2005

nonn

a(n) = A109895(x+1)-A109895(x)-1 - Simon Nickerson (simonn(AT)maths.bham.ac.uk), Jul 15 2005

Cf. A079759, A109895, A109897.

More terms from Simon Nickerson (simonn(AT)maths.bham.ac.uk), Jul 15 2005

A109897 Group the natural numbers so that every 2n-th group product is divisible by the single number in the next group. (1), (2,3,4,5), (6), (7,8,9,10,11), (12), (13,14,15,16,17,18,19),(20), (21,22,23,24,25,26,27),(28),... Sequence contains the product of terms in the 2n-th group.

Original entry on oeis.org

120, 55440, 253955520, 4475671200, 33891580800, 7146019520640, 106137499051584000, 240679032790296606105600, 74684882115043200, 230656425830328960, 66654064122818055916032000, 171176782669200, 282328002094320, 126639913316232476160000, 859891329773280

Offset: 1

Amarnath Murthy, Jul 13 2005

nonn

			a(2) = 7*8*9*10*11 = 55440.

Cf. A079759, A109895, A109896.

m=1;for(k=2,150,if(m%k==0,print1(m", ");m=1,m*=k))

Terms from a(5) from Franklin T. Adams-Watters, Jun 26 2011

A352101 The numbers formed by removing the required prime factors to form the number from a set which is initially empty and that has primes added via the addition of the missing prime factors of numbers which cannot be created from those currently in the set. Start by trying to create the number 2.

Original entry on oeis.org

6, 10, 12, 20, 21, 22, 26, 30, 34, 35, 36, 38, 44, 46, 52, 58, 60, 63, 68, 70, 74, 76, 82, 84, 86, 90, 92, 93, 102, 104, 105, 106, 110, 114, 116, 118, 122, 126, 130, 132, 134, 136, 141, 142, 146, 148, 150, 154, 156, 161, 164, 166, 170, 171, 172, 178, 180, 182, 186, 194, 195, 198, 202, 203

Offset: 1

Scott R. Shannon, May 06 2022

nonn

This sequence is a variation of A337486; instead of adding to the set all the prime factors of the numbers that cannot be created from the current primes in the set, only add those primes that are missing from the set, i.e., only add enough primes so that the number could then be created from the set. See A337486 for further details.

			a(1) = 6. As there are no primes initially in the set, the numbers 2, 3, 4, 5 cannot be created. These numbers add a 2, a 3, another 2 (only one extra 2 is added as 4 = 2*2 and one 2 is already in the set), and a 5. As there is now one 2 and one 3 the number 6 = 2*3 can be created. After 6 is created the set of primes now contains 2 and 5.
a(2) = 10. After 7, 8, 9, none of which can be created from the prime set, the prime set contains 2, 2, 2, 3, 3, 5, 7. Note 8 only added two extra 2's as one 2 was already in the set. As 2 and 5 are present 10 = 2*5 can be created, after which the set contains 2, 2, 3, 3, 7.

Cf. A337486, A331276, A330430, A109895.

cp's OEIS Frontend

A079759 Let b(0)=0. For n >= 1, b(n) is the least k > b(n-1)+1 such that k divides (k-1)!/b(n-1)!, and a(n) = (b(n)-1)!/(b(n-1)!*b(n)).

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A337486 Numbers k such that b(k+1) divides b(k), where b() is Recamán's multiplicative sequence A008336.

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A109896 Group the natural numbers so that every 2n-th group product is divisible by the single number in the next group. (1), (2,3,4,5), (6), (7,8,9,10,11), (12), (13,14,15,16,17,18,19),(20), (21,22,23,24,25,26,27),(28),... Sequence contains the number of terms in the 2n-th group.

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A109897 Group the natural numbers so that every 2n-th group product is divisible by the single number in the next group. (1), (2,3,4,5), (6), (7,8,9,10,11), (12), (13,14,15,16,17,18,19),(20), (21,22,23,24,25,26,27),(28),... Sequence contains the product of terms in the 2n-th group.

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