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

%I A364633 #39 Sep 05 2023 12:21:50
%S A364633 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 A364633 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 A364633 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 A364633 a(n) is the smallest nonnegative number k such that prime(n) + k is divisible by n + 1.
%C A364633 The sequence presents a pattern with large discontinuities at regular intervals in the logarithmic plot (See plots in Links).
%H A364633 Andres Cicuttin, <a href="/A364633/a364633_2.png">Log-log plot</a>
%H A364633 Andres Cicuttin, <a href="/A364633/a364633_3.png">Linear plot</a>
%F A364633 a(n) = Min_{k | (n+1) divides (prime(n)+k)}.
%F A364633 a(n) = (n+1)*ceiling(prime(n)/(n+1)) - prime(n)
%e A364633 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 A364633    n   n+1 prime(n) a(n)
%e A364633    1    2     2       0
%e A364633    2    3     3       0
%e A364633    3    4     5       3
%e A364633    4    5     7       3
%e A364633    5    6    11       1
%e A364633    6    7    13       1
%e A364633    7    8    17       7
%e A364633    8    9    19       8
%e A364633    9   10    23       7
%e A364633   10   11    29       4
%t A364633 a[n_]:=Module[{k},k=0;
%t A364633 While[Mod[Prime[n]+k,n+1]!=0,k++];k];
%t A364633 Table[a[n],{n,1,70}]
%o A364633 (Python)
%o A364633 from sympy import prime
%o A364633 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 A364633 (PARI) a(n) = my(k=0, p=prime(n)); while ((p+k) % (n+1), k++); k; \\ _Michel Marcus_, Sep 05 2023
%Y A364633 Cf. A068901.
%K A364633 nonn,look
%O A364633 1,3
%A A364633 _Andres Cicuttin_, Jul 30 2023