A332560 -id:A332560 - cp's OEIS frontend

A332561 a(n) = A332560(n)/A332559(n).

20, 20, 10, 105, 1512, 27720, 4620, 660, 144144, 16016, 5445440, 495040, 12697776, 976752, 69768, 823727520, 51482970, 3028410, 168245, 8855, 159845400, 5768642880, 262211040, 11400480, 475020, 543814622208, 20915947008, 774664704, 941432800

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 24 2020

Jinyuan Wang, Table of n, a(n) for n = 1..10000
J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020.

Cf. A061836, A332558, A332559, A332560.

A332558(n) = {my(r=n*(n+1)); for(k=2, oo, r=r*(n+k); if(r%(n+k+1)==0, return(k))); }
a(n) = {my(s=A332558(n)+n); s!/(n-1)!/(s+1); } \\ Jinyuan Wang, Feb 25 2020

A332558 a(n) is the smallest k such that n(n+1)(n+2)...(n+k) is divisible by n+k+1.

Original entry on oeis.org

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

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 24 2020

This is a multiplicative analog of A332542.

a(n) always exists because one can take k to be 2^m - 1 for m large.

Robert Israel, Table of n, a(n) for n = 1..10000
David A. Corneth, PARI program
J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004.14000 [math.NT], April 2020.

Cf. A061836 (k+1), A332559 (n+k+1), A332560 (the final product), A332561 (the quotient).
For records, see A333532 and A333533 (and A333537), which give the records in the essentially identical sequence A061836.
Additive version: A332542, A332543, A332544, A081123.
"Concatenate in base 10" version: A332580, A332584, A332585.

f:= proc(n) local k,p;
  p:= n;
  for k from 1 do
    p:= p*(n+k);
    if (p/(n+k+1))::integer then return k fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Feb 25 2020

a[n_] := Module[{k, p = n}, For[k = 1, True, k++, p *= (n+k); If[Divisible[p, n+k+1], Return[k]]]];
Array[a, 100] (* Jean-François Alcover, Jun 04 2020, after Maple *)

a(n) = {my(r=n*(n+1)); for(k=2, oo, r=r*(n+k); if(r%(n+k+1)==0, return(k))); } \\ Jinyuan Wang, Feb 25 2020

PARI
```
\\ See Corneth link
```

def a(n):
    k, p = 1, n*(n+1)
    while p%(n+k+1): k += 1; p *= (n+k)
    return k
print([a(n) for n in range(1, 85)]) # Michael S. Branicky, Jun 06 2021

a(n) = A061836(n) - 1 for n >= 1.

a(n + 1) >= a(n) - 1. a(n + 1) = a(n) - 1 mostly. - David A. Corneth, Apr 14 2020

A332559 a(n) = n + A332558(n) + 1.

Original entry on oeis.org

6, 6, 6, 8, 10, 12, 12, 12, 15, 15, 18, 18, 20, 20, 20, 24, 24, 24, 24, 24, 28, 30, 30, 30, 30, 35, 35, 35, 36, 36, 40, 40, 40, 40, 40, 45, 45, 45, 45, 48, 48, 48, 54, 54, 54, 56, 56, 56, 56, 60, 60, 60, 60, 60, 60, 63, 70, 70, 70, 70, 70, 70, 70, 72, 72, 72

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 24 2020

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020.

Cf. A061836, A332558, A332560, A332561.

f:= proc(n) local k,p;
  p:= n;
  for k from 1 do
    p:= p*(n+k);
    if (p/(n+k+1))::integer then return n+k+1 fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Feb 25 2020

a[n_] := Module[{k, p = n}, For[k = 1, True, k++, p *= (n+k); If[Divisible[p, n+k+1], Return[n+k+1]]]];
Array[a, 100] (* Jean-François Alcover, Jul 18 2020, after Maple *)

def a(n):
    k, p = 1, n*(n+1)
    while p%(n+k+1): k += 1; p *= (n+k)
    return n + k + 1
print([a(n) for n in range(1, 67)]) # Michael S. Branicky, Jun 06 2021

A333537 Greatest prime factor of A332559.

Original entry on oeis.org

3, 3, 3, 2, 5, 3, 3, 3, 5, 5, 3, 3, 5, 5, 5, 3, 3, 3, 3, 3, 7, 5, 5, 5, 5, 7, 7, 7, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 3, 3, 3, 3, 3, 3, 7, 7, 7, 7, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 5, 5, 5, 5, 5, 5, 5, 3, 3, 3, 3, 5, 5, 13, 7, 7, 7, 7, 7, 7, 7, 3, 7, 7, 7, 7, 7, 7, 5

Offset: 1

N. J. A. Sloane, Apr 12 2020

nonn

For rate of growth, see the Myers et al. link. - N. J. A. Sloane, Apr 30 2020

N. J. A. Sloane, Table of n, a(n) for n = 1..10000.
J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020.

Cf. A006530, A061836, A332558, A332559, A332560, A332561.

For records see A333538, A333541, A333542.

a[n_] := Module[{k, p = n}, For[k = 1, True, k++, p *= (n+k); If[Divisible[ p, n+k+1], Return[FactorInteger[n+k+1][[-1, 1]]]]]];
Array[a, 1000] (* Jean-François Alcover, Aug 17 2020 *)

cp's OEIS Frontend

A332561 a(n) = A332560(n)/A332559(n).

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A332558 a(n) is the smallest k such that n*(n+1)*(n+2)*...*(n+k) is divisible by n+k+1.

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Maple

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Formula

A332559 a(n) = n + A332558(n) + 1.

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Maple

Mathematica

Python

A333537 Greatest prime factor of A332559.

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Programs

Mathematica

A332558 a(n) is the smallest k such that n(n+1)(n+2)...(n+k) is divisible by n+k+1.