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%I #55 Jul 02 2018 23:38:51
%S 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,
%T 36,43,46,53,61,58,49,50,58,66,56,52,61,70,73,83,83,94,82,93,105,110,
%U 122,117,116,128,141,143,149,142,125,137,150,163,146,160,174
%N a(n) is the sum of remainders n mod p, over primes p for which n falls between p and p+p^2.
%C "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).
%F a(n) = Sum_{primes p, sqrt(n) - 1/2 < p <= n} (n mod p).
%e 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.
%o (PARI) a(n) = sum(k=1, n, (n % k)*isprime(k)*(n <= (k^2+k))); \\ _Michel Marcus_, May 14 2018
%Y Similar to A024934, but waves generated by primes are wavelets.
%K nonn
%O 1,5
%A _Meir-Simchah Panzer_, May 06 2018
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