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A330775
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Irregular triangle read by rows: row n gives the primes of the form m*prime(n)+1 where m is an even number <= prime(n) and prime(n) is the n-th prime, or 0 if no such prime exists for any n.
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1
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5, 7, 11, 29, 43, 23, 67, 89, 53, 79, 131, 157, 103, 137, 239, 191, 229, 47, 139, 277, 461, 59, 233, 349, 523, 311, 373, 683, 149, 223, 593, 1259, 83, 739, 821, 1231, 1559, 173, 431, 947, 1033, 1291, 1549, 1721, 283, 659, 941, 1129, 1223, 1693, 1787, 2069, 107, 743, 1061, 1697, 2333
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OFFSET
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1,1
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COMMENTS
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All safe primes are in this sequence.
Conjecture: For every prime p, there is at least one even m <= p such that m*p+1 is prime; this implies that no row is empty and there is no "0" in the sequence.
Conjecture: For every prime p, there is always a positive integer k <= p such that k*p+m is prime for any odd integer m, 0 < m < p. For example, for p = 11, k*11+m is prime for pairs {k,m}: {2,1}, {4,3}, {6,5}, {2,7}, {2,9}. - Metin Sariyar, Jan 26 2021
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LINKS
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FORMULA
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EXAMPLE
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For n = 4, m = {4, 6}, prime(4) = 7, and 4*7+1 = 29, 6*7+1 = 43 are primes.
Rows of the triangle:
n=1 => {5}
n=2 => {7}
n=3 => {11}
n=4 => {29, 43}
n=5 => {23, 67, 89}
n=6 => {53, 79, 131, 157}
n=7 => {103, 137, 239}
n=8 => {191, 229}
n=9 => {47, 139, 277, 461}
...
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MATHEMATICA
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row[n_] := Select[2 * Range[Floor[(p = Prime[n])/2]] * p + 1, PrimeQ]; row /@ Range[16] //Flatten (* Amiram Eldar, Jan 02 2020 *)
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PROG
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(PARI) row(n) = select(x->isprime(x), vector(prime(n)\2, k, 2*k*prime(n)+1)); \\ Michel Marcus, Feb 05 2020
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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