A332580 -id:A332580 - cp's OEIS frontend

A360406 a(n) = minimal positive k such that prime(n) * prime(n+1) * ... * prime(n+k) - 1 is divisible by prime(n+k+1), or -1 if no such k exists.

1, 1, 9, 14, 31, 826, 1, 34

Offset: 1

Scott R. Shannon, Feb 06 2023

Assuming a(9) exists it is greater than 1.75 million.

a(11) = 692, a(12) = 8, a(13) = 792. - Robert Israel, Feb 22 2023

			a(1) = 1 as prime(1) * prime(2) - 1 = 2 * 3 - 1 = 5, which is divisible by prime(3) = 5.
a(2) = 1 as prime(2) * prime(3) - 1 = 3 * 5 - 1 = 14, which is divisible by prime(4) = 7.
a(3) = 9 as prime(3) * ... * prime(12) - 1 = 1236789689134, which is divisible by prime(13) = 41.

Cf. A360376, A360297, A000040, A007504, A332542, A332580.

f:= proc(n) local P,k,p;
P:= ithprime(n); p:= nextprime(P);
for k from 0 to 10^6 do
  if P-1 mod p = 0 then return k fi;
  p:= nextprime(p);
 od;
FAIL
end proc:
map(f, [$1..8]); # Robert Israel, Feb 22 2023

from sympy import prime, nextprime
def A360406(n):
    p = prime(n)
    q = nextprime(p)
    s, k = p*q, 1
    while (s-1)%(q:=nextprime(q)):
        k += 1
        s *= q
    return k # Chai Wah Wu, Feb 06 2023

A263623 a(1)=1; thereafter, a(n) = smallest k such that the decimal concatenation [a(n-2)+1 a(n-2)+2, ... a(n-1)] divides the decimal concatenation [a(n-1)+1 a(n-1)+2 ... k].

Original entry on oeis.org

1, 2, 4, 8, 36

Offset: 1

N. J. A. Sloane, Oct 23 2015

a(6), if it exists, is > 10^6. - Lars Blomberg, Dec 01 2016

a(6) <= 86794654347484748866500883685475458354620023089553379437308257589024531796179370608623026912768. - Max Alekseyev, Dec 25 2024

			n=3: a(3) = 4 because k=4 is the smallest number such that 2 divides the concatenation 345...k.
n=4: a(4) = 8 because k=8 is the smallest number such that 34 divides the concatenation 567...k. See A002782 for the relevant concatenations.

Cf. A002782, A029455, A332580.

A337137 Variant of A332563 - binary version of Recamán concatenation sequence.

Original entry on oeis.org

2, 1, 3, 3, 2, 1, 8, 4, 6, 3, 2, 3, 2, 1, 3, 15, 10, 13, 4, 3, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 8, 7, 27, 29, 28, 27, 26, 10, 24, 23, 22, 21, 20, 19, 3, 15, 14, 15, 14, 13, 12, 3, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 62, 13, 2, 27, 58, 16, 15, 55, 22, 2, 52, 51, 2, 36, 16, 3, 46, 33, 7, 43, 2, 5, 3, 23, 38, 33, 4, 3, 34, 33, 13, 7, 22, 29, 16, 3, 26, 22, 16, 7, 22, 17, 2, 3, 2, 17, 16, 9, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 128

Offset: 1

Olivier Gérard, Sep 14 2020

Inspired by Neil Sloane's presentation at Rutgers' Experimental Mathematics Seminar (see the Links section).
In the original version (A332563), for a given n, one concatenate the binary representation of n||n+1||n+2||...||n+i until the corresponding number is divisible by n+i+1.
In this variant, one skips n+1 as an ingredient of the concatenation.
A337137(n) records the least i such that n||n+2||n+3||...||n+i is divisible by n+i+1.
This version is tamer than the one in A332563.
The scatterplot graph shows some interesting structures.

Olivier Gérard, Table of n, a(n) for n = 1..2048
N. J. A. Sloane, Conant's Gasket, Recamán Variations, the Enots Wolley Sequence, and Stained Glass Windows, Experimental Math Seminar, Rutgers University, Sep 10 2020 (video of Zoom talk).

Cf. A332580, A332563.

Module[{s, i, imax = 128},
Table[ s = IntegerDigits[n, 2]; i = 0;
  While[Mod[FromDigits[s, 2], n + i + 1] > 0 && i <= imax, i = i + 1;
   s = Join[s, IntegerDigits[n + i + 1, 2]]];
  i /. {imax + 1 -> Infinity} , {n, 1, 127}]]

cp's OEIS Frontend

A360406 a(n) = minimal positive k such that prime(n) * prime(n+1) * ... * prime(n+k) - 1 is divisible by prime(n+k+1), or -1 if no such k exists.

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A263623 a(1)=1; thereafter, a(n) = smallest k such that the decimal concatenation [a(n-2)+1 a(n-2)+2, ... a(n-1)] divides the decimal concatenation [a(n-1)+1 a(n-1)+2 ... k].

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A337137 Variant of A332563 - binary version of Recamán concatenation sequence.

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Author

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Crossrefs

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Mathematica