A359262 a(n) is the largest number m such that prime(n)^m is in A359260.

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, 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, 1, 3, 1, 1, 3, 3, 3, 1, 1, 3, 1, 3, 1, 9, 1, 3, 3, 1, 1

Offset: 1

Author

Amiram Eldar, Dec 23 2022

Keywords

• nonn
• easy

Comments

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.

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).

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... .

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).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000720, A002110, A040976, A073917, A005867, A359260, A359261.

Cf. A002476, A073102, A132230.

Programs

Mathematica

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]

PARI

a(n) = {my(p = prime(n), k = 1, r = s = 1); while(!(s%k), k++; r *= p; s += r); k - 2; }

Formula

a(n) >= 1 for n >= 2.

a(n) >= 3 iff prime(n) == 1 (mod 6) (prime(n) is in A002476).

Conjectures:

a(n) >= 5 iff prime(n) == 1 (mod 30) (prime(n) is in A132230).

a(n) >= 9 iff prime(n) == 1 (mod 210) (prime(n) is in A073102).

a(n) >= prime(k) - 2 iff prime(n) == 1 (mod A002110(k-1)).