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A354370
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Successive pairs of terms (i, j) such that (i + j) is a prime number and at least i is a prime number. This is the lexicographically earliest infinite sequence of distinct terms > 1 with this property.
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4
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2, 3, 5, 6, 7, 4, 11, 8, 13, 10, 17, 12, 19, 18, 23, 14, 29, 24, 31, 16, 37, 22, 41, 20, 43, 28, 47, 26, 53, 30, 59, 38, 61, 36, 67, 34, 71, 32, 73, 40, 79, 48, 83, 44, 89, 42, 97, 52, 101, 50, 103, 46, 107, 56, 109, 54, 113, 60, 127, 64, 131, 62, 137, 74, 139, 58, 149, 78, 151, 72, 157, 66, 163, 70
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OFFSET
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1,1
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COMMENTS
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The terms 1, 9, 15, 21, 25, 27, 33, 35, 39, 45, ... will never appear in the sequence; they form A014076, the "Odd nonprimes". Two prime terms can form a pair (2 and 3 for instance) but the first term must always be a prime [the pair (5, 6) is ok].
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LINKS
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Michael De Vlieger, Annotated log-log scatterplot of a(n), n = 1..10^4, showing records in red and local minima in blue, highlighting fixed points in gold and composite powers of 2 in magenta.
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EXAMPLE
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The earliest pairs with their prime sum: (2, 3) = 5, (5, 6) = 11, (7, 4) = 11, (11, 8) = 19, (13, 10) = 23, (17, 12) = 29, (19, 18) = 37, (23, 14) = 37, etc.
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MATHEMATICA
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nn = 120; c[_] = 0; a[1] = 2; c[2] = 1; u = 3; Do[If[EvenQ[i], k = u; While[Nand[c[k] == 0, PrimeQ[# + k]] &[a[i - 1]], k++], k = u; While[Nand[c[k] == 0, PrimeQ[k]], k++]]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[Or[c[u] > 0, And[OddQ[u], CompositeQ[u]]], u++]], {i, 2, nn}]; Array[a, nn] (* Michael De Vlieger, May 24 2022 *)
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PROG
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(Python)
from sympy import isprime
from itertools import islice
def agen(): # generator of terms
i, j, v, aset = 2, 3, 4, set()
while True:
aset.update((i, j)); yield from (i, j)
i = j = v
while i in aset or not isprime(i): i += 1
while j == i or j in aset or not isprime(i+j): j += 1
while v in aset: v += 1
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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