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A098213 The values of some algorithm. 2
1, 1, 1, 8, 30, 97, 374, 2185, 7399, 60475, 303535, 2332720, 16630294 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
The algorithm: Take n consecutive primes starting with the a(n)-th prime: {p(a(n)), ..., p(a(n)+n-1)}. Calculate the absolute differences abs(p(i)-p(j)) for all relevant pairs i,j. The number of distinct entries of this n X n difference matrix equals binomial(n,2) = n(n-1)/2, the maximum possible. Also a(n) is the earliest index such that this diversity of differences reaches the maximum possible, binomial(n,2).
The diagonal [{p(i)-p(i)}] of difference matrices consists only of zeros and can be disregarded.
The complete diversity of k-1 consecutive prime differences [as in A079007] is a necessary but not sufficient condition for providing binomial(k,2) distinct entries in the corresponding k X k difference matrix of k consecutive primes. Consecutive prime differences are para-diagonal entries in the difference matrix. So the conditions here are stronger than in A079007.
LINKS
FORMULA
Subscripts at which sequences like A098726, A098206-A098212 reach first their possible maximum, i.e., binomial(k,2) with the corresponding k.
EXAMPLE
At n=1, 2, 3, the maxima are binomial(n,2) = 0, 1, 3 reached at a(n) = 1, 1, 1 respectively;
n=7: a(7)=374, primes = {p(374), ..., p(80)} = {2551, 2557, 2579, 2591, 2593, 2609, 2617}. Building the 7 X 7 matrix of values of abs(p(i)-p(j)), the number of its distinct positive entries equals binomial(7,2)=21, namely: {2, 6, 8, 12, 14, 16, 18, 22, 24, 26, 28, 30, 34, 36, 38, 40, 42, 52, 58, 60, 66};
n=12: a(12)=2332720, list of 12 primes = {p(n), ..., p(n+11)} = {38238461, ..., 38238737}; 12 X 12 matrix = {abs((p(i)-p(j))}, number of distinct entries = binomial(12,2)=66, that of {2, 6, 8, ..., 266, 274, 276}.
CROSSREFS
Sequence in context: A113751 A343520 A107233 * A372252 A163613 A279217
KEYWORD
nonn,more
AUTHOR
Labos Elemer, Oct 05 2004
EXTENSIONS
Edited by Jon E. Schoenfield, Oct 27 2019
STATUS
approved

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Last modified June 23 13:28 EDT 2024. Contains 373648 sequences. (Running on oeis4.)