A107926 -id:A107926 - cp's OEIS frontend

A231156 Trisection of A107926: The least number k such that there are primes p and q with p - q = 6*n, p + q = k, and p the least such prime >= k/2.

4, 16, 50, 44, 98, 92, 242, 256, 406, 272, 442, 292, 1054, 476, 530, 572, 682, 1604, 1106, 1408, 1846, 968, 2230, 724, 3478, 688, 1862, 2396, 3586, 1052, 3514, 1808, 2666, 2764, 3182, 3412, 4498, 4964, 5146, 2228, 3382, 6268, 3526, 5036, 6070, 3472, 3502

Offset: 0

Robert G. Wilson v, Nov 04 2013

nonn

All terms found to date are not congruent to 0 (mod 6).

Robert G. Wilson v, Table of n, a(n) for n = 0..1794

a(n) = A107926(3n).

A234955 Trisection of A107926: The least number k such that there are primes p and q with p - q = 6*n+2, p + q = k, and p the least such prime >= k/2.

Original entry on oeis.org

8, 54, 108, 234, 228, 414, 516, 1182, 612, 1038, 1776, 1074, 3312, 1398, 1728, 2706, 2844, 4902, 1152, 3870, 2724, 4974, 2328, 6222, 5040, 13194, 10236, 5838, 8952, 9642, 9816, 12906, 21900, 11958, 14712, 6294, 15984, 9498, 31752, 31602, 6096, 37854, 41208, 6114

Offset: 1

Robert G. Wilson v, Jan 01 2014

nonn

All terms found to date are congruent to 0 (mod 6), except for a(1).

Record values: 8, 54, 108, 228, 414, 516, 612, 1038, 1074, 1152, 2328, 5040, 5838, 6096, 6114, 22194, 37764, 37902, 99432, 136116, 176856, 318144, 410712, 1079952, 1436448, 2549346, 3278118, 7012944, 8268534, 11283126, 11284134, 22614234, 37510062, 41607234, 94089894, 139419954, 144049014, 305966316, 378180246, 490373322, 998189838, 1326486408, 1373334486, 1445744268, 2016602694, 2247482688, 3239350182, 3884888976, 5147119596, 7172019282, …, .

Robert G. Wilson v, Table of n, a(n) for n = 1..867

Cf. A107926, A231156, A234956.

f[n_] := Block[{p = n/2}, While[ !PrimeQ[n - p], p = NextPrime@ p]; p - n/2]; t = Table[0, {10000}]; k = 4;  While[k < 12475000001, If[ t[[f@ k]] == 0, t[[f@ k]] = k; Print[{f@ k, k}]]; k += 2]; Table[ t[[n]], {n, 2, 5000, 3}]

a(n) = A107926(3n-2).

A234956 Trisection of A107926: The least number k such that there are primes p and q with p - q = 6*n+4, p + q = k, and p the least such prime >= k/2.

Original entry on oeis.org

18, 48, 102, 444, 174, 432, 582, 672, 846, 984, 1902, 636, 1122, 1464, 2730, 3348, 3342, 1752, 5154, 8424, 1842, 5244, 5802, 5076, 9714, 10392, 11898, 11928, 12966, 14796, 7662, 21516, 23202, 39216, 18234, 10572, 8742, 21732, 16770, 38076, 30102, 19884, 54822, 44604

Offset: 1

Robert G. Wilson v, Jan 01 2014

nonn

All terms found to date are congruent to 0 (mod 6).

Record values: 18, 48, 102, 174, 432, 582, 636, 1122, 1464, 1752, 1842, 5076, 7662, 8742, 16770, 16938, 27072, 37416, 49086, 50736, 63552, 80568, 93654, 126582, 136362, 255672, 500208, 1070574, 2549718, 3328608, 4436316, 4743834, 7906854, 8303664, 8818122, 11747676, 21461364, 26582496, 30738636, 36170334, 42304728, 45413748, 100573404, 101901222, 142408062, 215780022, 222856404, 276403416, 397812606, 578042658, 695661546, 1217194032, 1540728846, 1752132852, 1760999466, 1896604482, 3024520584, 8602478358, 12860956476, 12987816186, 13162543146, 13319210952, …, .

Robert G. Wilson v, Table of n, a(n) for n = 1..855

Cf. A107926, A231156, A234955.

f[n_] := Block[{p = n/2}, While[ !PrimeQ[n - p], p = NextPrime@ p]; p - n/2]; t = Table[0, {10000}]; k = 4;  While[k < 12475000001, If[ t[[f@ k]] == 0, t[[f@ k]] = k; Print[{f@ k, k}]]; k += 2]; Table[ t[[n]], {n, 2, 5000, 3}]

a(n) = A107926(3n-1).

A065978 For even k >= 4, let f(k) = A066285(k/2) be the minimal difference between primes p and q whose sum is k. Such a k is in the sequence if f(k) > f(m) for all even m with 4 <= m < k.

Original entry on oeis.org

4, 8, 16, 44, 92, 242, 256, 272, 292, 476, 530, 572, 682, 688, 1052, 1808, 2228, 3382, 3472, 3502, 3562, 4952, 6194, 7102, 10262, 17008, 20684, 37052, 45128, 49552, 80144, 137414, 251806, 349826, 362534, 742856, 1655152, 1872236, 2108282, 2319728, 2707118

Offset: 1

Jon Perry, Dec 09 2001

The values of f(a(n)) (given in A066286) appear to be divisible by 6, except the first two.

			4 = 2+2; the gap is 0. 6=3+3 (0). 8=3+5; the gap is 2, and this is the largest gap to date, so 8 is in the sequence.
10=5+5 (0), 12=5+7 (2), 14=7+7 (0), 16=5+11 (6), so 16 is in the sequence.

Gilmar Rodriguez Pierluissi, Table of n, a(n) for n = 1..64 (terms 1..50 from Jon Perry, Robert G. Wilson and Dean Hickerson, terms 51..55 from Gilmar Rodriguez Pierluissi, terms 56..63 from Robert G. Wilson v)

Cf. A066285, A066286, A107926.

f[n_] := For[p=n/2, True, p--, If[PrimeQ[p]&&PrimeQ[n-p], Return[n-2p]]]; For[n=4; max=-1, True, n+=2, If[f[n]>max, Print[n]; max=f[n]]]

More terms from Robert G. Wilson v and Dean Hickerson, Dec 10 2001

Changed offset to 1 (this is a list). - N. J. A. Sloane, Sep 07 2013

A103147 Least k such that k+n and k-n are both prime but k-m and k+m are not both prime for any 0 <= m < n.

Original entry on oeis.org

2, 4, 9, 8, 27, 24, 25, 54, 51, 22, 117, 222, 49, 114, 87, 46, 207, 216, 121, 258, 291, 128, 591, 336, 203, 306, 423, 136, 519, 492, 221, 888, 951, 146, 537, 318, 527, 1656, 561, 238, 699, 732, 265, 864, 1365, 286, 1353, 1674, 341, 1422, 1671, 802, 2451, 876, 553

Offset: 0

Lei Zhou, Jan 26 2005

nonn

First appearance of n in A047160.
It appears that a(3n) is less than a(3n-1) and a(3n+1) for all n except 2 and 12. The lower and upper primes are A155766(n) and A155767(n). - T. D. Noe, Jan 26 2009
No odd primes are in this sequence. - Lei Zhou, Mar 06 2012

			a(0)=2 because 2-0 and 2+0 are primes. 2 is the least such value.
a(1)=4 because 4-1 and 4+1 are prime, but 4-0 and 4-0 are not prime. 4 is the least such value.
a(2)=9 because 9-2 and 9+2 are prime, but (8,10) and (9,9) are not prime pairs. 9 is the least such value.
a(3)=8 because 8-3 and 8+3 are prime, but (6,10), (7,9) and (8,8) are not prime pairs. 8 is the least such value.
a(11)=222 because 211 and 233 are prime, but (222-m,222+m) is not a prime pair for any m<11. 222 is the least such value.

Cf. A047160, A104883, A107926.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a103147 = (+ 2) . fromJust . (`elemIndex` a047160_list)
-- Reinhard Zumkeller, Aug 10 2014

primePairQ[k_, n_] := PrimeQ[k+n]&&PrimeQ[k-n]; SetAttributes[primePairQ, Listable]; Table[k=n+2; While[ !primePairQ[k, n] || (Or@@primePairQ[k, Range[0, n-1]]), k++ ]; k, {n, 0, 55}]

Edited by Ray Chandler and T. D. Noe, Feb 01 2005

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A231156 Trisection of A107926: The least number k such that there are primes p and q with p - q = 6*n, p + q = k, and p the least such prime >= k/2.

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A234955 Trisection of A107926: The least number k such that there are primes p and q with p - q = 6*n+2, p + q = k, and p the least such prime >= k/2.

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A234956 Trisection of A107926: The least number k such that there are primes p and q with p - q = 6*n+4, p + q = k, and p the least such prime >= k/2.

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A065978 For even k >= 4, let f(k) = A066285(k/2) be the minimal difference between primes p and q whose sum is k. Such a k is in the sequence if f(k) > f(m) for all even m with 4 <= m < k.

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A103147 Least k such that k+n and k-n are both prime but k-m and k+m are not both prime for any 0 <= m < n.

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