A302867 a(n) is the sum of remainders n mod p, over primes p for which n falls between p and p+p^2.

0, 0, 1, 1, 3, 1, 3, 6, 6, 4, 7, 8, 11, 8, 7, 11, 15, 20, 25, 26, 25, 20, 26, 33, 35, 29, 36, 36, 43, 46, 53, 61, 58, 49, 50, 58, 66, 56, 52, 61, 70, 73, 83, 83, 94, 82, 93, 105, 110, 122, 117, 116, 128, 141, 143, 149, 142, 125, 137, 150, 163, 146, 160, 174

Offset: 1

Author

Meir-Simchah Panzer, May 06 2018

Keywords

• nonn

Comments

"Jubilees". Motivation: 7 years are counted 7 times and capped off with a 50th year, the Jubilee (Leviticus 25:8); similarly, 7 days are counted 7 times and capped off with "Chag ha-Atzeret" (The Festival of Stopping) in the Omer-counting cycle (ibid 23:15); and these iterative cycles overlay other iterative cycles, like the lunar cycle nested not-quite-evenly within the solar year. This sequence idealizes the overlaying of multiple cycles. Each prime p generates a "swell" of p waves each with max amplitude = p-1, a kind of wavelet that is added into the total signal that is the sequence (e.g., the swell generated by 3 is (3^2)+1 terms in length, running for n=3,...,12 and has values n mod 3 = 0,1,2,0,1,2,0,1,2,0).

Examples

			For n = 12, we sum over primes 3, 5, 7, 11: a(12) = 12 mod 3 + 12 mod 5 + 12 mod 7 + 12 mod 11 = 0 + 2 + 5 + 1 = 8. In contrast with A024934, the sum does not include 12 mod 2 since 12 > 2+2^2.
		

Crossrefs

Similar to A024934, but waves generated by primes are wavelets.

Programs

PARI

a(n) = sum(k=1, n, (n % k)*isprime(k)*(n <= (k^2+k))); \\ Michel Marcus, May 14 2018

Formula

a(n) = Sum_{primes p, sqrt(n) - 1/2 < p <= n} (n mod p).