A332559 -id:A332559 - 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

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 *)

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

A332560 a(n) = (n + A332558(n))!/(n-1)!.

Original entry on oeis.org

120, 120, 60, 840, 15120, 332640, 55440, 7920, 2162160, 240240, 98017920, 8910720, 253955520, 19535040, 1395360, 19769460480, 1235591280, 72681840, 4037880, 212520, 4475671200, 173059286400, 7866331200, 342014400, 14250600, 19033511777280, 732058145280

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, April 2020

Cf. A061836, A332558, A332559, A332561.

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

A333538 Indices of records in A333537.

Original entry on oeis.org

1, 5, 21, 91, 355, 456, 666, 2927, 4946, 6064, 6188, 6192, 13858, 14884, 39592, 54429, 77603, 87566, 210905, 245770, 422097, 585876, 908602, 976209, 1240768, 1340675, 1573890, 2589172, 4740893, 5168099, 8525972, 8646462, 10478354, 12636785, 17943798, 19524935

Offset: 1

N. J. A. Sloane, Apr 12 2020

nonn

The first few primes that are not record values of A333537 are 2, 11, 53, 59, 71, 73, 89, 97, 103, 107 (see A333541, A333542). - Robert Israel, Apr 12 2020

a(72) > 5*10^9. - David A. Corneth, Apr 14 2020

David A. Corneth, Table of n, a(n) for n = 1..71 (first 36 terms from Robert Israel)
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, A333537, A333541, A333542.

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:
R:= 1: g:= 3: count:= 1:
for n from 2 while count < 20 do
  x:= max(numtheory:-factorset(f(n)));
  if x > g then count:= count+1; g:= x; R:= R, n;  fi
od:
R; # Robert Israel, Apr 12 2020

f[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]]]]]];
R = {1}; g = 3; count = 1;
For[n = 2, count < 20, n++, x = f[n]; If[x > g, count++; g = x; AppendTo[R, n]]];
R (* Jean-François Alcover, Aug 17 2020, after Robert Israel *)

a(13)-a(20) from Robert Israel, Apr 12 2020

More terms from Jinyuan Wang, Apr 12 2020

A333541 Records in A333537.

Original entry on oeis.org

3, 5, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 61, 67, 79, 83, 101, 109, 113, 137, 139, 149, 151, 157, 167, 199, 211, 227, 239, 257, 269, 277, 283, 307, 313, 317, 353, 373, 379, 389, 397, 409, 433, 439, 499, 503, 569, 571, 593, 607, 617, 631, 701, 709, 727, 743, 757, 769, 773

Offset: 1

N. J. A. Sloane, Apr 20 2020, using data from Robert Israel's comment in A333538

nonn

For the primes that are not records, see A333542.

			For n = 91 as A332558(91) = 12 we have (91 + A332558(91) + 1) = (91 + 12 + 1) | (91 * 92 * ... * (91 + 12)) = (91 * 92 * ... * (91 + A332558(91))). The largest prime factor of 91 + 12 + 1 = 104 is 13. For no m < 91 the largest prime factor of m + A332558(m) + 1 = A332559(m) is at least 13 so 13 is a new record in A333537. - _David A. Corneth_, Apr 21 2020

David A. Corneth, Table of n, a(n) for n = 1..71
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. A332558, A333537, A333538, A333542.

More terms from David A. Corneth, Apr 21 2020

cp's OEIS Frontend

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

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A333537 Greatest prime factor of A332559.

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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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A332560 a(n) = (n + A332558(n))!/(n-1)!.

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A333538 Indices of records in A333537.

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A333541 Records in A333537.

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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.