A364633 - OEIS

%I #39 Sep 05 2023 12:21:50

%S 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,

%T 11,1,1,33,1,31,34,33,32,33,32,31,34,29,32,33,36,29,22,23,26,27,26,29,

%U 24,23,22,21,24,23,24,27,22,13,14,17,18,9,8,3,6,7,6,3,2,1,2,1

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

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

%H Andres Cicuttin, <a href="/A364633/a364633_2.png">Log-log plot</a>

%H Andres Cicuttin, <a href="/A364633/a364633_3.png">Linear plot</a>

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

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

%e 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:

%e    n   n+1 prime(n) a(n)

%e    1    2     2       0

%e    2    3     3       0

%e    3    4     5       3

%e    4    5     7       3

%e    5    6    11       1

%e    6    7    13       1

%e    7    8    17       7

%e    8    9    19       8

%e    9   10    23       7

%e   10   11    29       4

%t a[n_]:=Module[{k},k=0;

%t While[Mod[Prime[n]+k,n+1]!=0,k++];k];

%t Table[a[n],{n,1,70}]

%o (Python)

%o from sympy import prime

%o def A364633(n): return (n+1)*(prime(n)//(n+1)+1)-prime(n) if n>2 else 0 # _Chai Wah Wu_, Sep 04 2023

%o (PARI) a(n) = my(k=0, p=prime(n)); while ((p+k) % (n+1), k++); k; \\ _Michel Marcus_, Sep 05 2023

%Y Cf. A068901.

%K nonn,look

%O 1,3

%A _Andres Cicuttin_, Jul 30 2023