A332544 -id:A332544 - cp's OEIS frontend

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

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

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

A332543 a(n) = n + A332542(n) + 1.

Original entry on oeis.org

6, 12, 20, 10, 14, 56, 24, 15, 22, 33, 39, 26, 21, 48, 272, 34, 38, 76, 28, 33, 46, 69, 40, 50, 39, 36, 116, 58, 62, 992, 96, 51, 70, 45, 111, 74, 57, 65, 205, 82, 86, 172, 55, 69, 94, 141, 112, 70, 75, 68, 212, 106, 66, 77, 133, 87, 118, 177, 183, 122

Offset: 3

Scott R. Shannon, Feb 18 2020

nonn

Note that A332542 is well-defined for all n. - N. J. A. Sloane, Feb 20 2020

			See A332542.

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, April 2020

Cf. A332542, A332544, A082183.

Maple
```
See A332542.
```

cp's OEIS Frontend

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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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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A332543 a(n) = n + A332542(n) + 1.

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Maple

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Formula

A332542 a(n) is the smallest k such that n+(n+1)+(n+2)+...+(n+k) is divisible by n+k+1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Ruby

A332543 a(n) = n + A332542(n) + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

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