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User: Meir-Simchah Panzer

Meir-Simchah Panzer has authored 3 sequences.

A350011 Numbers which are the minimal "product-weight" of a simple connected graph over the sequence of primes. See comments for precise definition.

6, 16, 30, 31, 35, 45, 52, 57, 60, 66, 67, 74, 78, 101

Offset: 1

Meir-Simchah Panzer, Dec 08 2021

Label the nodes of a simple connected graph on N nodes with integers from a sequence s starting from s(1), then s(2), s(3), ..., s(N). Assign to each edge a value equal to the product of its nodes' labels. Sum the edge values of the graph. The possible values of a graph are its 'product-weight' over the sequence s. This sequence a(n) comprises the minimal product-weight of simple, connected graphs over the sequence of primes, listed in ascending order.
Among N-node graphs, the graph with the smallest minimal product-weight will be a star graph, and the graph with the largest minimal product-weight will be the complete graph. Minimal product-weights are partially ordered by number of edges (e.g., 5-node graphs with 4 edges have smaller product-weights than the complete graph over 4-nodes which has 6 edges); so there are values missing from the sequence above, which includes simple connected graphs for 2, 3, and 4 nodes, as well as 4 5-node graphs and the 6-node star graph; e.g., if the terms listed so far included minimal product-weights for simple connected graphs of 2, 3, and 4 nodes, it would have a maximum value 101 and it would have skipped the value of 78 which is the minimal product-weight of the 6-node start graph.
Can different graphs have the same minimal product-weight?

			Consider a graph of 4 nodes A, B, C, D, with edges AB, AC, AD, BC. The labeling which corresponds to the minimal product-weight labels A as 2, B as 3, C as 5, and D as 7, and its minimal product-weight is 45. (Its maximal product-weight is 85.)

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

Original entry on oeis.org

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

Meir-Simchah Panzer, May 06 2018

nonn

"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).

			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.

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

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

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

A274069 a(n) is the concatenation of a(n-1) and the Hamming distance between a(n-1) and its reverse (i.e., the minimum number of bitflips needed to make them identical). Sequence written in binary.

Original entry on oeis.org

1, 10, 1010, 1010100, 101010010, 101010010110, 101010010110100, 101010010110100100, 1010100101101001001000, 10101001011010010010001110, 101010010110100100100011101100, 10101001011010010010001110110010100, 101010010110100100100011101100101001100, 10101001011010010010001110110010100110011000, 10101001011010010010001110110010100110011000100000

Offset: 1

Meir-Simchah Panzer, Jun 09 2016

			Let a'(n) be the reverse of a(n). E.g., if a(n) = 10100, then a'(n) = 00101. Let hamm(b,c) denote the Hamming distance between b and c. Let concat designate concatenation of arguments.
a(1):=1.
a(2) is the concatenation of a(1) and hamm(a(1),a'(1)). a'(1) = 1. So hamm(a(1),a'(1)) = hamm('1','1') = 0. So a(2) = concat('1','0') = 10.
a(3) is the concatenation of a(2) and hamm(a(2),a'(2)). hamm(a(2),a'(2)) = hamm('10','01') = 2 or 10 in base 2. So a(3) = concat('10','10') = 1010.
a(4) is the concatenation of a(3) and hamm(a(3),a'(3)). hamm(a(3),a'(3)) = hamm('1010','0101') = 4 or 100 in base 2. So a(3) = concat('1010','100') = 10100.

Cf. A144078.

A274069aux := proc(n)
    option remember;
    if n = 1 then
        [1];
    else
        d := procname(n-1) ;
        dreve := ListTools[Reverse](d) ;
        ham := 0 ;
        for i from 1 to nops(d) do
            if op(i,d) <> op(i,dreve) then
                ham := ham+1 ;
            end if;
        end do:
        if ham = 0 then
            [op(d),0] ;
        else
            ListTools[Reverse](convert(ham,base,2)) ;
            [op(d),op(%) ] ;
        end if ;
    end if;
end proc:
A274069 := proc(n)
    digcatL(A274069aux(n)) ;
end proc:
seq(A274069(n),n=1..30) ; # R. J. Mathar, May 08 2019

Edited by Meir-Simchah Panzer, Jun 12 2018

More terms from R. J. Mathar, May 08 2019.

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User: Meir-Simchah Panzer

A350011 Numbers which are the minimal "product-weight" of a simple connected graph over the sequence of primes. See comments for precise definition.

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

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A274069 a(n) is the concatenation of a(n-1) and the Hamming distance between a(n-1) and its reverse (i.e., the minimum number of bitflips needed to make them identical). Sequence written in binary.

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