A320702 -id:A320702 - cp's OEIS frontend

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

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)).

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 }.

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

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A134117.

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

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

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

Simon Strandgaard and Jamie Morken, Apr 17 2021

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.

			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.

Simon Strandgaard, Visualization of the first 5 terms.

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

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

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

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A134118.

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

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

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

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

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

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

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

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

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