This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A359262 #14 Dec 26 2022 09:45:41
%S A359262 0,1,1,3,1,3,1,3,1,1,5,3,1,3,1,1,1,5,3,1,3,3,1,1,3,1,3,1,3,1,3,1,1,3,
%T A359262 1,5,3,3,1,1,1,5,1,3,1,3,9,3,1,3,1,1,5,1,1,1,1,5,3,1,3,1,3,1,3,1,5,3,
%U A359262 1,3,1,1,3,3,3,1,1,3,1,3,1,9,1,3,3,1,1
%N A359262 a(n) is the largest number m such that prime(n)^m is in A359260.
%C A359262 a(n) is the largest number m such that the arithmetic mean of {1, p, p^2, ..., p^k} is an integer for all k in 1..m.
%C A359262 Apparently, all the terms are of the form prime(k)-2 (A040976). Conjecture: The asymptotic density of the occurrences of prime(k)-2 is (1/s(k-1)-1/s(k)), where s(k) = A005867(k) = phi(prime(k)#), and prime(k)# is the k-th primorial number (A002110).
%C A359262 The sums of the first 10^k terms, for k = 1, 2, ..., are 15, 221, 2291, 23287, 233641, 2337007, 23379901, 233814475, 2338211029, 23382168187, ... . If the mentioned above conjecture is correct, then the asymptotic mean of this sequence is Sum_{k>=1} (prime(k)-2)*(1/s(k-1)-1/s(k)) = 2.33821872365981424748... .
%C A359262 Apparently, the indices of records after n = 1 occur at A000720(A073917(n)) (verified for the first 12 terms of A073917) with record values a(A000720(A073917(n))) = prime(n+1) - 2 (verified for the first 150 terms of A073917).
%H A359262 Amiram Eldar, <a href="/A359262/b359262.txt">Table of n, a(n) for n = 1..10000</a>
%F A359262 a(n) >= 1 for n >= 2.
%F A359262 a(n) >= 3 iff prime(n) == 1 (mod 6) (prime(n) is in A002476).
%F A359262 Conjectures:
%F A359262 a(n) >= 5 iff prime(n) == 1 (mod 30) (prime(n) is in A132230).
%F A359262 a(n) >= 9 iff prime(n) == 1 (mod 210) (prime(n) is in A073102).
%F A359262 a(n) >= prime(k) - 2 iff prime(n) == 1 (mod A002110(k-1)).
%t A359262 a[n_] := Module[{p = Prime[n], k = 1, r = s = 1}, While[Divisible[s, k], k++; r *= p; s += r]; k - 2]; Array[a, 100]
%o A359262 (PARI) a(n) = {my(p = prime(n), k = 1, r = s = 1); while(!(s%k), k++; r *= p; s += r); k - 2; }
%Y A359262 Cf. A000720, A002110, A040976, A073917, A005867, A359260, A359261.
%Y A359262 Cf. A002476, A073102, A132230.
%K A359262 nonn,easy
%O A359262 1,4
%A A359262 _Amiram Eldar_, Dec 23 2022