A320701 -id:A320701 - cp's OEIS frontend

A320713 Indices of primes followed by a gap (distance to next larger prime) of 30.

590, 650, 708, 757, 842, 890, 928, 985, 1006, 1051, 1108, 1556, 1570, 1648, 1650, 1675, 1754, 1900, 1919, 2027, 2125, 2149, 2321, 2391, 2397, 2429, 2631, 2637, 2699, 2781, 2866, 2918, 2989, 2993, 3010, 3085, 3153, 3207, 3315, 3340, 3350, 3373, 3420, 3511, 3551, 3580, 3637, 3751, 3777, 3948

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A124596.

Index entries for primes, gaps between.

Equals A000720 o A124596.
Indices of 30's in A001223.
Row 15 of A174349.
Subsequence of A107730 (prime(n+1) ends in same digit as prime(n)).
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

A(N=100,g=30,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A124596(n)).

A320713 = { i>0 | prime(i+1) = prime(i) + 30 } = A001223^(-1)({30}).

A320718 Indices of primes followed by a gap (distance to next larger prime) of 40.

Original entry on oeis.org

2191, 2344, 2524, 2788, 3562, 4058, 4677, 5030, 5349, 6076, 6145, 6256, 6320, 6442, 6454, 6902, 7232, 7488, 8119, 8152, 8245, 8366, 8553, 8567, 8591, 8746, 9260, 9361, 10536, 10735, 11095, 11407, 11534, 11781, 12227, 12312, 12663, 12815, 12940, 13015, 13333, 13676, 13873, 14065, 14123

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A126721.

Index entries for primes, gaps between.

Equals A000720 o A126721.
Row 20 of A174349.
Subsequence of A107730 (prime(n+1) ends in same digit as prime(n)).
Indices of 40's in A001223.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

A(N=100,g=40,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A126721(n)).

A320718 = { i > 0 | prime(i+1) = prime(i) + 40 } = A001223^(-1)({40}).

A320719 Indices of primes followed by a gap (distance to next larger prime) of 42.

Original entry on oeis.org

1879, 3732, 4059, 4135, 4714, 5355, 5948, 6160, 6841, 7434, 7724, 7746, 7952, 7980, 8081, 8269, 8580, 9303, 9395, 9971, 10045, 10305, 10968, 11023, 11135, 11251, 11338, 11399, 11515, 11807, 11888, 11901, 12089, 12374, 12488, 13277, 13447, 14497, 14802, 15086, 15089, 15350, 15612, 15785

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A134120.

Harvey P. Dale, Table of n, a(n) for n = 1..3000
Index entries for primes, gaps between.

Cf. A029707, A029709 (analog for gaps 2 & 4), A320701, A320702, ... A320720 (analog for gaps 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).
Equals A000720 o A134120.
Indices of 42's in A001223.
Row 21 of A174349.

Position[Differences[Prime[Range[16000]]],42]//Flatten (* Harvey P. Dale, Feb 22 2020 *)

A(N=100,g=42,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A134120(n)).

A356222 Array read by antidiagonals upwards where A(n,k) is the position of the k-th appearance of 2n in the sequence of prime gaps A001223. If A001223 does not contain 2n at least k times, set A(n,k) = -1.

Original entry on oeis.org

2, 4, 3, 9, 6, 5, 24, 11, 8, 7, 34, 72, 15, 12, 10, 46, 42, 77, 16, 14, 13, 30, 47, 53, 79, 18, 19, 17, 282, 62, 91, 61, 87, 21, 22, 20, 99, 295, 66, 97, 68, 92, 23, 25, 26, 154, 180, 319, 137, 114, 80, 94, 32, 27, 28, 189, 259, 205, 331, 146, 121, 82, 124, 36, 29, 33

Offset: 1

Gus Wiseman, Aug 04 2022

Prime gaps (A001223) are the differences between consecutive prime numbers. They begin: 1, 2, 2, 4, 2, 4, 2, 4, 6, ...

This is a permutation of the positive integers > 1.

			Array begins:
        k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9
  n=1:   2   3   5   7  10  13  17  20  26
  n=2:   4   6   8  12  14  19  22  25  27
  n=3:   9  11  15  16  18  21  23  32  36
  n=4:  24  72  77  79  87  92  94 124 126
  n=5:  34  42  53  61  68  80  82 101 106
  n=6:  46  47  91  97 114 121 139 168 197
  n=7:  30  62  66 137 146 150 162 223 250
  n=8: 282 295 319 331 335 378 409 445 476
  n=9:  99 180 205 221 274 293 326 368 416
For example, the positions in A001223 of appearances of 2*3 begin: 9, 11, 15, 16, 18, 21, 23, ..., which is row n = 3 (A320701).

The row containing n is A028334(n).
Row n = 1 is A029707.
Row n = 2 is A029709.
Column k = 1 is A038664.
The column containing n is A274121(n).
Column k = 2 is A356221.
The diagonal A(n,n) is A356223.
A001223 lists the prime gaps.
A073491 lists numbers with gapless prime indices.
A356224 counts even divisors with gapless prime indices, complement A356225.
Cf. A066205, A119313, A193829, A287170, A328457, A356226, A356232.

gapa=Differences[Array[Prime,10000]];
Table[Position[gapa,2*(k-n+1)][[n,1]],{k,6},{n,k}]

A320709 Indices of primes followed by a gap (distance to next larger prime) of 22.

Original entry on oeis.org

189, 297, 344, 375, 457, 487, 522, 549, 557, 721, 836, 914, 1010, 1158, 1170, 1197, 1233, 1242, 1272, 1290, 1370, 1390, 1444, 1471, 1625, 1633, 1672, 1683, 1757, 1858, 1975, 1983, 2039, 2074, 2158, 2243, 2248, 2250, 2327, 2370, 2388, 2614, 2638, 2703, 2725, 2838, 2842, 2872

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A061779.

Vincenzo Librandi, Table of n, a(n) for n = 1..10040
Index entries for primes, gaps between.

Equals A000720 o A061779.
Row 11 of A174349.
Indices of 22's in A001223.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

[n: n in [1..3000] | NthPrime(n+1) - NthPrime(n) eq 22]; // Vincenzo Librandi, Mar 22 2019

Select[Range[3000], Prime[#] + 22 == Prime[# + 1] &] (* Vincenzo Librandi, Mar 22 2019 *)

A(N=100,g=22,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A061779(n)).

A320709 = { i > 0 | prime(i+1) = prime(i) + 22 }.

A320710 Indices of primes followed by a gap (distance to next larger prime) of 24.

Original entry on oeis.org

263, 327, 574, 615, 641, 697, 804, 834, 869, 909, 938, 987, 1022, 1045, 1127, 1336, 1399, 1421, 1446, 1452, 1551, 1577, 1865, 1883, 1908, 1938, 1939, 1968, 2032, 2064, 2128, 2130, 2176, 2214, 2313, 2508, 2555, 2592, 2612, 2652, 2736, 2762, 2991, 3162, 3243, 3285, 3483, 3489

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A098974.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between.

Equals A000720 o A098974.
Row 12 of A174349.
Indices of 24's in A001223.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

P:= [seq(ithprime(i),i=1..4001)]:
B:= P[2..-1]-P[1..-2]:
select(t -> B[t]=24, [$1..4000]); # Robert Israel, May 03 2019

A(N=100,g=24,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A098974(n)).

A320710 = { i > 0 | prime(i+1) = prime(i) + 24 }.

A320711 Indices of primes followed by a gap (distance to next larger prime) of 26.

Original entry on oeis.org

367, 446, 732, 1357, 1440, 1475, 1746, 1864, 1912, 1933, 2293, 2714, 2888, 2912, 3159, 3204, 3362, 3523, 3715, 3786, 3801, 3840, 3870, 3920, 3931, 4107, 4164, 4235, 4240, 4502, 4643, 4809, 4957, 4990, 5110, 5371, 5440, 5451, 5581, 5712, 5736, 5743, 5870, 5882, 5906, 5923, 5933, 6018, 6277

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A124594.

Index entries for primes, gaps between.

Equals A000720 o A124594.
Row 13 of A174349.
Indices of 26's in A001223.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

A(N=100,g=26,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A124594(n)).

A320711 = { i > 0 | prime(i+1) = prime(i) + 26 }.

A320712 Indices of primes followed by a gap (distance to next larger prime) of 28.

Original entry on oeis.org

429, 462, 685, 781, 1116, 1231, 1274, 1288, 1327, 1392, 1585, 1708, 1710, 1891, 1944, 2065, 2154, 2367, 2417, 2606, 2663, 2729, 2980, 3012, 3069, 3227, 3519, 3653, 3990, 4018, 4168, 4196, 4595, 4603, 4618, 4797, 4856, 4867, 5123, 5191, 5294, 5375, 5432, 5476, 5498, 5593, 5627, 5688, 5703

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A124595.

Index entries for primes, gaps between.

Equals A000720 o A124595.
Indices of 28's in A001223.
Row 14 of A174349.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

A(N=100,g=28,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A124595(n)).

A320712 = { i > 0 | prime(i+1) = prime(i) + 28 }.

A320714 Indices of primes followed by a gap (distance to next larger prime) of 32.

Original entry on oeis.org

738, 1315, 3022, 3042, 3607, 4112, 4300, 4362, 4555, 4619, 4761, 4893, 5204, 5358, 5615, 5637, 6215, 6265, 6479, 6610, 6706, 6933, 7295, 7829, 7884, 8049, 8198, 8548, 9085, 9155, 9524, 9588, 9641, 9826, 9924, 10463, 10824, 11367, 11590, 11701, 11729, 11869, 12159, 12258, 12275, 12327

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A126784.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between.

Equals A000720 o A126784.
Indices of 32's in A001223.
Row 16 of A174349.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

p:= 2: Res:= NULL: count:= 0:
for k from 1 while count < 100 do
  q:= nextprime(p);
  if q - p = 32 then
    count:= count+1;
    Res:= Res, k;
  fi;
  p:= q;
od:
Res; # Robert Israel, Oct 25 2018

PrimePi/@Select[Partition[Prime[Range[15000]],2,1],#[[2]]-#[[1]]==32&][[;;,1]] (* Harvey P. Dale, Jun 19 2024 *)

A(N=100,g=32,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A126784(n)).

A320714 = { i>0 | prime(i+1) = prime(i) + 32 }.

A320715 Indices of primes followed by a gap (distance to next larger prime) of 34.

Original entry on oeis.org

217, 1059, 1229, 1409, 1457, 1986, 2169, 2310, 2406, 3221, 3505, 3692, 3995, 4324, 4923, 5130, 5518, 6050, 6152, 6168, 6434, 7257, 7362, 7604, 7694, 7915, 8293, 8555, 8584, 8651, 8859, 9017, 9341, 9598, 9796, 9869, 10028, 10092, 10116, 10150, 10211, 10234, 10317, 10657, 10744, 10762

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A134116.

Index entries for primes, gaps between.

Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).
Equals A000720 o A134116.
Indices of 34's in A001223.
Row 17 of A174349.

Position[Differences[Prime[Range[11000]]],34]//Flatten (* Harvey P. Dale, Jan 19 2021 *)

A(N=100,g=34,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A134116(n)).

A320715 = { i>0 | prime(i+1) = prime(i) + 34 }.

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A320713 Indices of primes followed by a gap (distance to next larger prime) of 30.

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A320718 Indices of primes followed by a gap (distance to next larger prime) of 40.

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A320719 Indices of primes followed by a gap (distance to next larger prime) of 42.

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A356222 Array read by antidiagonals upwards where A(n,k) is the position of the k-th appearance of 2n in the sequence of prime gaps A001223. If A001223 does not contain 2n at least k times, set A(n,k) = -1.

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A320709 Indices of primes followed by a gap (distance to next larger prime) of 22.

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Magma

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A320710 Indices of primes followed by a gap (distance to next larger prime) of 24.

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A320711 Indices of primes followed by a gap (distance to next larger prime) of 26.

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A320712 Indices of primes followed by a gap (distance to next larger prime) of 28.

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