A364633 a(n) is the smallest nonnegative number k such that prime(n) + k is divisible by n + 1.

0, 0, 3, 3, 1, 1, 7, 8, 7, 4, 5, 2, 1, 2, 1, 15, 13, 15, 13, 13, 15, 13, 13, 11, 7, 7, 9, 9, 11, 11, 1, 1, 33, 1, 31, 34, 33, 32, 33, 32, 31, 34, 29, 32, 33, 36, 29, 22, 23, 26, 27, 26, 29, 24, 23, 22, 21, 24, 23, 24, 27, 22, 13, 14, 17, 18, 9, 8, 3, 6, 7, 6, 3, 2, 1, 2, 1

Offset: 1

Author

Andres Cicuttin, Jul 30 2023

Keywords

• nonn
• look

Comments

The sequence presents a pattern with large discontinuities at regular intervals in the logarithmic plot (See plots in Links).

Examples

			The following table shows the first 10 terms where the fourth column, a(n), plus the third column, prime(n), is divisible by the second column n+1:
   n   n+1 prime(n) a(n)
   1    2     2       0
   2    3     3       0
   3    4     5       3
   4    5     7       3
   5    6    11       1
   6    7    13       1
   7    8    17       7
   8    9    19       8
   9   10    23       7
  10   11    29       4
		

Links

• Andres Cicuttin, Log-log plot
• Andres Cicuttin, Linear plot

Crossrefs

Cf. A068901.

Programs

Mathematica

a[n_]:=Module[{k},k=0;
While[Mod[Prime[n]+k,n+1]!=0,k++];k];
Table[a[n],{n,1,70}]

PARI

a(n) = my(k=0, p=prime(n)); while ((p+k) % (n+1), k++); k; \\ Michel Marcus, Sep 05 2023

Python

from sympy import prime
def A364633(n): return (n+1)*(prime(n)//(n+1)+1)-prime(n) if n>2 else 0 # Chai Wah Wu, Sep 04 2023

Formula

a(n) = Min_{k | (n+1) divides (prime(n)+k)}.

a(n) = (n+1)*ceiling(prime(n)/(n+1)) - prime(n)