cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

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

Views

Author

Gus Wiseman, Aug 04 2022

Keywords

Comments

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.

Examples

			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).

Crossrefs

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.

Programs

  • Mathematica
    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

Views

Author

M. F. Hasler, Oct 19 2018

Keywords

Comments

Indices of the primes listed in A061779.

Links

Crossrefs

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).

Programs

  • Magma
    [n: n in [1..3000] | NthPrime(n+1) - NthPrime(n) eq 22]; // Vincenzo Librandi, Mar 22 2019
  • Mathematica
    Select[Range[3000], Prime[#] + 22 == Prime[# + 1] &] (* Vincenzo Librandi, Mar 22 2019 *)
  • PARI
    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

Formula

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

Views

Author

M. F. Hasler, Oct 19 2018

Keywords

Comments

Indices of the primes listed in A098974.

Links

Crossrefs

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).

Programs

  • Maple
    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
  • PARI
    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

Formula

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

Views

Author

M. F. Hasler, Oct 19 2018

Keywords

Comments

Indices of the primes listed in A124594.

Links

Crossrefs

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).

Programs

  • PARI
    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

Formula

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

Views

Author

M. F. Hasler, Oct 19 2018

Keywords

Comments

Indices of the primes listed in A124595.

Links

Crossrefs

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).

Programs

  • PARI
    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

Formula

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

Views

Author

M. F. Hasler, Oct 19 2018

Keywords

Comments

Indices of the primes listed in A126784.

Links

Crossrefs

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).

Programs

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

Formula

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

Views

Author

M. F. Hasler, Oct 19 2018

Keywords

Comments

Indices of the primes listed in A134116.

Links

Crossrefs

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.

Programs

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

Formula

a(n) = A000720(A134116(n)).
A320715 = { i>0 | prime(i+1) = prime(i) + 34 }.

A320716 Indices of primes followed by a gap (distance to next larger prime) of 36.

Original entry on oeis.org

1183, 1532, 1663, 1847, 2146, 2489, 2500, 2550, 2700, 2976, 3087, 3238, 3461, 4236, 4483, 4681, 4692, 4834, 4849, 4946, 5178, 5836, 6062, 6098, 6269, 6591, 6613, 6787, 6862, 6904, 7091, 7178, 7200, 7285, 7577, 7743, 8057, 8097, 8215, 8355, 8572, 8637, 8767, 8832, 8877, 9023, 9129, 9161
Offset: 1

Views

Author

M. F. Hasler, Oct 19 2018

Keywords

Comments

Indices of the primes listed in A134117.

Links

Crossrefs

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 A134117.
Indices of 36's in A001223.
Row 18 of A174349.

Programs

  • PARI
    A(N=100,g=36,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

Formula

a(n) = A000720(A134117(n)).
A320716 = { i>0 | prime(i+1) = prime(i) + 36 }.

A343496 First point of the straight lines in A340649.

Original entry on oeis.org

5, 31, 194, 1061, 6456, 40080, 251721, 1617206, 10553419, 69709769, 465769825
Offset: 1

Views

Author

Simon Strandgaard and Jamie Morken, Apr 17 2021

Keywords

Comments

prime(a(n)+1) - prime(a(n)) = n*2. E.g., for n=4: prime(a(4)+1) - prime(a(4)) = 4*2, prime(1062) - prime(1061) = 4*2, 8521 - 8513 = 8.

Examples

			For n=1, consider k's with prime gap 1*2 = 2, i.e., k's such that A001223(k)=2. k=5 is the first place where A001223(k)=2 and A141042(k)=A340649(k), so a(1)=5.
For n=2, consider k's with prime gap 2*2 = 4, i.e., k's such that A001223(k)=4. k=31 is the first place where A001223(k)=4 and A141042(k)=A340649(k), so a(2)=31.
For n=3, consider k's with prime gap 3*2 = 6, i.e., k's such that A001223(k)=6. k=194 is the first place where A001223(k)=6 and A141042(k)=A340649(k), so a(3)=194.

Links

Crossrefs

Cf. A000040, A001223, A141042, A340649, A029707, A029709, A320701, A320702, A320703.

Programs

  • Ruby
    n = 1
last_prime = 2
find_gap = 2
result = []
Prime.each(10_000) do |prime|
    next if prime == 2
    gap = prime - last_prime
    if gap == find_gap
        value = (n * prime) % last_prime
        if value == n * gap
            result << n
            find_gap += 2
        end
    end
    n += 1
    last_prime = prime
end
p result

Formula

a(n) = smallest k that satisfies A001223(k) = 2*n and A340649(k) = A141042(k).

A320717 Indices of primes followed by a gap (distance to next larger prime) of 38.

Original entry on oeis.org

3302, 4052, 4154, 4743, 5093, 5229, 5782, 5902, 6131, 6406, 6802, 7145, 7164, 7399, 7718, 7789, 8303, 8782, 9237, 9957, 10073, 10431, 10465, 10541, 10549, 10580, 10981, 11244, 11818, 11853, 12147, 12574, 13094, 13237, 13286, 13337, 13435, 13669, 13906, 14186, 14270, 14301, 14380, 14397
Offset: 1

Views

Author

M. F. Hasler, Oct 19 2018

Keywords

Comments

Indices of the primes listed in A134118.

Links

Crossrefs

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 A134118.
Indices of 38's in A001223.
Row 19 of A174349.

Programs

  • PARI
    A(N=100,g=38,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

Formula

a(n) = A000720(A134118(n)).
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