A090396 Remainder when the sum of the first n primes is divided by n.

0, 1, 1, 1, 3, 5, 2, 5, 1, 9, 6, 5, 4, 1, 13, 13, 15, 15, 17, 19, 19, 21, 0, 3, 10, 17, 22, 27, 1, 3, 15, 27, 8, 19, 1, 15, 31, 11, 28, 7, 27, 3, 26, 3, 23, 41, 20, 5, 37, 17, 46, 25, 0, 33, 13, 49, 30, 7, 43, 19, 52, 29, 14, 61, 41, 19, 5, 59, 50, 37, 22, 7, 67, 55, 43, 29, 15, 3, 68, 57

Offset: 1

Author

Joseph L. Pe, Jan 31 2004

Keywords

• nonn
• look

Comments

a(n) = 0 if and only if n is a term of A045345. - Nicholas Drozd, Nov 18 2018

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Karl-Heinz Hofmann, Listening to the terms of A090396, YouTube video.
• Karl-Heinz Hofmann, Plot of 3 selected ranges, n = 1..8200, 59000..113000, 105000..154000.
• Hugo Pfoertner, Visualization of a(n)/n, covering time range of audio track in video (160000 terms).
• Hugo Pfoertner, Filtered spectrum of a(n)/n waveform, shifted to audible frequency range.

Crossrefs

Cf. A007504 (sum of first n primes), A045345 (indices of 0's).

Cf. A060620 (corresponding floor quotients).

Programs

Maple

N:= 1000; # to get the first N terms
pN:= ithprime(N):
C:= map(round,Statistics:-CumulativeSum(select(isprime,[$1..pN])));
seq(C[n] mod n, n = 1 .. N); # Robert Israel, May 29 2014

Mathematica

t = Table[Mod[ Sum[Prime[i], {i, 1, n}], n], {n, 1, 100}]
Module[{nn=80,pr},pr=Accumulate[Prime[Range[nn]]];Table[Mod[pr[[n]],n],{n,nn}]] (* Harvey P. Dale, Jul 03 2019 *)

PARI

a(n) = sum(k=1, n, prime(k)) % n;
for(n=1, 80, print1(a(n),", ")); \\ Indranil Ghosh, Mar 06 2017

Formula

a(n) = A007504(n) mod n. - Karl-Heinz Hofmann, May 05 2021