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A339944
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Let N(p,i) denote the result of applying "nextprime" i times to p; a(n) = smallest prime p such that N(p,4) - p = 2*n, or -1 if no such prime exists.
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2
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-1, -1, -1, -1, 3, 5, 17, 13, 19, 47, 79, 73, 113, 109, 193, 317, 313, 521, 503, 523, 887, 1499, 1231, 1319, 1373, 1321, 1307, 3947, 2473, 2143, 2477, 7369, 5573, 5939, 9967, 16111, 18587, 20773, 18593, 31883, 17209, 19597, 24251, 19609, 25471, 31397, 44389, 18803, 38459, 38461, 66191, 69557, 103183
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OFFSET
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1,5
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COMMENTS
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This sequence is the fourth row of A337767.
From Robert G. Wilson v, Dec 30 2020: (Start)
a(n) > -1 for all n >= 5.
It seems likely that for almost all values of n there is more than one integer k such that prime(k+4) - prime(k) = 2*n; a(n) = prime(k) for the smallest such k.
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n | list of numbers k such that prime(k+4) - prime(k) = 2*n
---+-----------------------------------------------------------------
5 | 3.
6 | 5, 7, 11, 97, 101, 1481, 1867, 3457, 5647, 15727, 16057, ...
7 | 17, 29, 59, 227, 269, 1277, 1289, 1607, 2129, 2789, 3527, ...
8 | 13, 31, 37, 67, 223, 1087, 1291, 1423, 1483, 1597, 1861, ...
9 | 19, 23, 41, 43, 53, 61, 71, 89, 149, 163, 179, ...
10 | 47, 83, 131, 137, 173, 191, 251, 257, 347, 419, 443, ...
etc.
(End)
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LINKS
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Martin Raab, Table of n, a(n) for n = 1..493 (Terms 1..342 from Robert G. Wilson)
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EXAMPLE
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a(1) = 3. This represents the beginning of the run of primes {3, 5, 7, 11, 13}. (13 - 3)/2 = 5 and it is the only prime to do so.
a(2) = 5. This represents the beginning of the run of primes {5, 7, 11, 13, 17}. (17 - 5)/2 = 6 and it is the first prime to do so. Others are 7, 11, 97, 101, etc.
a(3) = 17. This represents the beginning of the run of primes {17, 19, 23, 29, 31}. (31 - 17)/2 = 7 and it is the first prime to do so. Others are 29, 59, 227, 269, etc.
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MATHEMATICA
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p = 3; q = 5; r = 7; s = 11; t = 13; tt[_] := 0; While[p < 450000, d = (t - p)/2; If[ tt[d] == 0, tt[d] = p]; {p, q, r, s, t} = {q, r, s, t, NextPrime@ t}]; tt@# & /@ Range@ 75 (* Robert G. Wilson v, Dec 30 2020 *)
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CROSSREFS
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Cf. A337767, A000230, A144103, A339943.
Sequence in context: A242104 A158895 A085418 * A359395 A292008 A139427
Adjacent sequences: A339941 A339942 A339943 * A339945 A339946 A339947
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KEYWORD
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sign
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AUTHOR
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Robert G. Wilson v, Dec 23 2020
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STATUS
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approved
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