A332558 -id:A332558 - cp's OEIS frontend

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

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

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

A061836 a(n) = smallest k>0 such that k+n divides k!.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Jun 22 2001

nonn

Comments from M. F. Hasler, Feb 20 2020 (Start)
The index at which any n > 2 appears for the last time is given by A005096(n) = n! - n.
For m>2, a(n) > m for n > A005096(m).
The integer 1 appears only once as a(0), the integer 2 is the only positive integer which never appears. (End)
It would be nice to have an estimate for the growth of the upper envelope of this sequence - what is lim sup a(n)? The answer seems to be controlled by A333537. - N. J. A. Sloane, Apr 12 2020
Paul Zimmermann suggests that perhaps a(n) is O(log(n)^2). My estimate was n^(1/3), although that seems a bit low. - N. J. A. Sloane, Apr 09 2020

Rémy Sigrist, Table of n, a(n) for n = 0..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.
N. J. A. Sloane, Table of n, a(n) for n = 0..100000

Cf. A332584 for a "concatenation in base 10" variant.
See also A005096, A332558 (essentially identical to this one).
For records, see A333532 and A333533 (and A333537).

f[n_] := (k = 1; While[ !IntegerQ[ k! / (k + n) ], k++ ]; k); Table[ f[n], {n, 0, 75} ]

a(n) = my (f=1); for (k=1, oo, if ((f*=k)%(n+k)==0, return (k))) \\ Rémy Sigrist, Feb 17 2020

"k>0" added to definition at the suggestion of Chai Wah Wu, Apr 09 2020. - N. J. A. Sloane, Apr 22 2020

A332542 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

2, 7, 14, 3, 6, 47, 14, 4, 10, 20, 25, 11, 5, 31, 254, 15, 18, 55, 6, 10, 22, 44, 14, 23, 11, 7, 86, 27, 30, 959, 62, 16, 34, 8, 73, 35, 17, 24, 163, 39, 42, 127, 9, 22, 46, 92, 62, 19, 23, 15, 158, 51, 10, 20, 75, 28, 58, 116, 121, 59, 29, 127, 254, 11

Offset: 3

Scott R. Shannon, Feb 18 2020

nonn

Note that (n+(n+1)+(n+2)+...+(n+k))/(n+k+1) = A332544(n)/(n+k+1) = A082183(n-1). See the Myers et al. link for proof. - N. J. A. Sloane, Apr 30 2020

We can always take k = n^2-2*n-1, for then the sum in the definition becomes (n+1)*n*(n-1)*(n-2)/2, which is an integral multiple of n+k+1 = n*(n-1). So a(n) always exists. - N. J. A. Sloane, Feb 20 2020

			n=4: we get 4 -> 4+5=9 -> 9+6=15 -> 15+7=22 -> 22+8=30 -> 30+9=39 -> 39+10=49 -> 49+11=60, which is divisible by 12, and took k=7 steps, so a(4) = 7. Also A332543(4) = 12, A332544(4) = 60, and A082183(3) = 60/12 = 5.

Seiichi Manyama, Table of n, a(n) for n = 3..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], 2020-2021.

Cf. A332543, A332544, A082183.

See A332558-A332561 for a multiplicative analog.

grow2 := proc(n,M) local p,q,k; # searches out to a limit of M
# returns n, k (A332542(n)), n+k+1 (A332543(n)), p (A332544(n)), and q (which appears to match A082183(n-1))
for k from 1 to M do
   if ((k+1)*n + k*(k+1)/2) mod (n+k+1) = 0 then
   p := (k+1)*n+k*(k+1)/2;
   q := p/(n+k+1); return([n,k,n+k+1,p,q]);
   fi;
od:
# if no success, return -1's
[n,-1,-1,-1,-1]; end; # N. J. A. Sloane, Feb 18 2020

a[n_] := NestWhile[#1+1&,0,!IntegerQ[Divide[(#+1)*n+#*(#+1)/2,n+#+1]]&]
a/@Range[3,100] (* Bradley Klee, Apr 30 2020 *)

a(n) = my(k=1); while (sum(i=0, k, n+i) % (n+k+1), k++); k; \\ Michel Marcus, Aug 26 2021

def a(n):
    k, s = 1, 2*n+1
    while s%(n+k+1) != 0: k += 1; s += n+k
    return k
print([a(n) for n in range(3, 67)]) # Michael S. Branicky, Aug 26 2021

def A(n)
  s = n
  t = n + 1
  while s % t > 0
    s += t
    t += 1
  end
  t - n - 1
end
def A332542(n)
  (3..n).map{|i| A(i)}
end
p A332542(100) # Seiichi Manyama, Feb 19 2020

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

Original entry on oeis.org

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 *)

A333532 Record high values in A061836.

Original entry on oeis.org

1, 5, 6, 7, 8, 9, 11, 13, 14, 16, 17, 18, 20, 21, 22, 24, 25, 27, 30, 32, 35, 40, 41, 45, 48, 49, 53, 54, 58, 62, 63, 64, 68, 72, 73, 76, 85, 86, 90, 102, 117, 136, 143, 144, 153, 154, 161, 165, 166, 182, 183, 187, 189, 200, 202, 217, 219, 233, 234, 235, 238, 249

Offset: 1

N. J. A. Sloane, Apr 08 2020

nonn

Paul Zimmermann suggests that perhaps A061836(n) is O(log(n)^2). - N. J. A. Sloane, Apr 09 2020

Chai Wah Wu, Table of n, a(n) for n = 1..104

Cf. A061836, A332558, A333533.

More terms from Jinyuan Wang, Apr 08 2020

A333533 Indices of record high values in A061836.

Original entry on oeis.org

0, 1, 6, 11, 16, 26, 43, 57, 106, 144, 191, 222, 330, 518, 650, 666, 668, 1013, 1090, 1312, 2205, 2300, 3455, 4946, 5867, 6069, 7192, 12000, 12152, 12174, 17481, 23801, 24772, 26790, 29490, 36860, 39767, 60502, 65668, 87566, 87568, 172232, 293452, 434972, 453591

Offset: 1

N. J. A. Sloane, Apr 08 2020

nonn

Paul Zimmermann suggests that perhaps A061836(n) is O(log(n)^2). - N. J. A. Sloane, Apr 09 2020

Chai Wah Wu, Table of n, a(n) for n = 1..104

Cf. A061836, A332558, A333532.

Name clarified and more terms from Jinyuan Wang, Apr 08 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

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

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

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A061836 a(n) = smallest k>0 such that k+n divides k!.

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A332542 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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A332561 a(n) = A332560(n)/A332559(n).

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A333537 Greatest prime factor of A332559.

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A333532 Record high values in A061836.

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A333533 Indices of record high values in A061836.

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A333538 Indices of records in A333537.

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