A329158
Let P1>=3, P2, P3 be consecutive primes, with P3-P2=2. a(n)=(P2+P3)/12 when P2-P1 sets a record.
Original entry on oeis.org
1, 2, 5, 25, 87, 192, 500, 1158, 1668, 4217, 4713, 5955, 17127, 28905, 61838, 76967, 96147, 139725, 260342, 1061923, 1205080, 4663498, 8871842, 11732765, 32534740, 42313103, 77638122, 92523718, 282054523, 728833340, 2940948542, 3344803093, 11810906035
Offset: 1
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p1=3;p2=5;r=0;forprime(p3=7,1e9,if(p3-p2==2,d=p2-p1;if(d>r,r=d;print1((p2+p3)/12,", ")));p1=p2;p2=p3)
A329159
2*a(n) are the lengths of record prime gaps immediately preceding twin primes.
Original entry on oeis.org
1, 2, 3, 5, 6, 11, 14, 15, 17, 20, 24, 26, 30, 35, 39, 42, 45, 50, 66, 68, 74, 75, 89, 98, 104, 107, 117, 123, 144, 159, 165, 180, 183, 192, 195
Offset: 1
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p1=3;p2=5;r=0;forprime(p3=7,1e9,if(p3-p2==2,d=p2-p1;if(d>r,r=d;print1(d/2,", ")));p1=p2;p2=p3)
A329160
Let P1>=5, P2, P3 be consecutive primes, with P2-P1=2. a(n)=(P1+P2)/12 when P3-P2 sets a record.
Original entry on oeis.org
1, 5, 23, 33, 87, 278, 495, 1293, 2027, 2690, 4245, 6773, 13283, 24938, 28893, 44270, 67475, 139708, 224922, 315893, 971000, 1723960, 3319792, 6228255, 7013717, 13194622, 25321985, 31864375, 32163975, 65155398, 86090027, 381175405, 452803425
Offset: 1
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p1=5; p2=7; r=0; forprime(p3=11, 1e9, if(p2-p1==2, d=p3-p2; if(d>r, r=d; print1((p1+p2)/12, ", "))); p1=p2; p2=p3)
A329161
2*a(n) are the lengths of record prime gaps immediately following twin primes.
Original entry on oeis.org
2, 3, 5, 6, 9, 12, 14, 15, 17, 21, 26, 27, 29, 30, 35, 41, 45, 50, 59, 60, 63, 71, 74, 75, 84, 92, 96, 98, 104, 110, 125, 129, 144, 155, 156, 165, 180, 201, 204, 207
Offset: 1
a(1) = 2 because the prime gap after 5, 7 has length 11 - 7 = 2*a(1) = 4,
a(2) = 3 because the prime gap after 29,31 has length 37 - 31 = 2*a(2) = 6, with location given by 6*A329160(2) = 30 +- 1. The gaps before all have length 4 (11,13 -> 17), (17,19 -> 23).
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p1=5; p2=7; r=0; forprime(p3=11, 1e9, if(p2-p1==2, d=p3-p2; if(d>r, r=d; print1(d/2, ", "))); p1=p2; p2=p3)
A329164
Let P1, P2, P3, P4 be consecutive primes, with P2-P1=P4-P3=2. a(n)=(P1+P2)/12 when P3-P2 sets a new record.
Original entry on oeis.org
1, 23, 322, 495, 3407, 8113, 28893, 139708, 716182, 2497092, 5130198, 5761777, 7315173, 13194622, 145995245, 201544467, 417649822, 566513637, 833667068, 2266818768, 4710228962, 5186737183, 5192311957, 7454170028, 9853412390, 11817808908
Offset: 1
Values of P1, P2, P3, P4 corresponding to record gaps:
P3-P1 P1 P2 P3 P4 a(n)
6 5 7 11 13 (5+7)/12 = 1
12 137 139 149 151 (137+139)/12 = 23
18 1931 1933 1949 1951 (1931+1933)/12 = 322
30 2969 2971 2999 3001 (2969+2971)/12 = 495
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p1=3;p2=5;p3=7;r=0;forprime(p4=11,1e9,if(p2-p1==2&&p4-p3==2,d=p3-p2;if(d>r,r=d;print1((p1+p2)/12,", ")));p1=p2;p2=p3;p3=p4)
A329250
Let P1, P2, P3, P4 be consecutive primes, with P2 - P1 = P4 - P3 = 2. a(n) = (P1 + P2)/12 for the first occurrence of (P3 - P1)/6 = n.
Original entry on oeis.org
1, 23, 322, 1573, 495, 3407, 10498, 85067, 8113, 112912, 166302, 28893, 189052, 510548, 598532, 812752, 139708, 716182, 2582073, 4576458, 2497092, 5130198, 5761777, 25381573, 7315173, 20200532, 40629683, 33185292, 69948743, 38771927, 13194622
Offset: 1
a(4) = 1573, because the 4 primes P1 = 6*1573 - 1 = 9437, P2 = 6*1573 + 1 = 9439, P3 = P1 + 6*4 = 9461, P4 = 9463 produce the first occurrence of the gap P3 - P2 = 9461 - 9439 = 6*4 - 2 = 22. See also example in A329164.
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my(v=vector(31),p1=3,p2=5,p3=7,r=0,d);forprime(p4=11,5e8,if(p2-p1==2&&p4-p3==2,d=(p3-p1)/6;if(v[d]==0,v[d]=(p1+p2)/12));p1=p2;p2=p3;p3=p4);v
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