A029707 -id:A029707 - cp's OEIS frontend

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

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 }.

A334142 Indices of twin primes.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 17, 18, 20, 21, 26, 27, 28, 29, 33, 34, 35, 36, 41, 42, 43, 44, 45, 46, 49, 50, 52, 53, 57, 58, 60, 61, 64, 65, 69, 70, 81, 82, 83, 84, 89, 90, 98, 99, 104, 105, 109, 110, 113, 114, 116, 117, 120, 121, 140, 141, 142, 143, 144, 145, 148, 149

Offset: 1

Ilya Gutkovskiy, Apr 15 2020

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes.

Cf. A000720, A001097, A029707, A107770.

q:= n-> (p-> isprime(p-2) or isprime(p+2))(ithprime(n)):
select(q, [$1..150])[];  # Alois P. Heinz, Apr 15 2020

With[{p = PositionIndex[(#[[2]] - #[[1]] - 2) & /@ Partition[Prime[Range[150]], 2, 1]][0]}, Union[p, p + 1]] (* Amiram Eldar, Sep 06 2024 *)

a(n) = A000720(A001097(n)).

Set union of A029707 and A107770.

A340468 a(n) is the least prime of the form 2 + Product_{i=n..m} prime(i).

Original entry on oeis.org

5, 7, 79, 13, 223, 19, 439, 130753887906569681111538991218568790437537693430279000532630035672131604633987039552816424896353327834998483765849409837393409377729040653460715050958787058270805333463, 31, 34826927179023475480751694965449235272424989980919

Offset: 2

Robert Israel, Jan 08 2021

nonn

If n is in A029707, a(n) = 2+prime(n).
If n is not in A029707 but prime(n) is in A051507, a(n) = 2+prime(n)*prime(n+1).
a(15) > 10^1000 if it exists.

			a(2) = 2+3 = 5.
a(3) = 2+5 = 7.
a(4) = 2+7*11 = 79.
a(5) = 2+11 = 13.
a(6) = 2+13*17 = 223.
a(7) = 2+17 = 19.
a(8) = 2+19*23 = 439.
a(9) = 2+23*29*...*431.

Robert Israel, Table of n, a(n) for n = 2..14

Cf. A029707, A051507.

f:= proc(n) local i,t;
  t:= 1;
  for i from n do
    t:= t*ithprime(i);
    if isprime(t+2) then return t+2 fi;
  od
end proc:
seq(f(n),n=2..14);

from sympy import isprime, nextprime, prime
def a(n):
  prodpnpm = pm = prime(n)
  while not isprime(2+prodpnpm): pm = nextprime(pm); prodpnpm *= pm
  return 2+prodpnpm
print([a(n) for n in range(2, 12)]) # Michael S. Branicky, Jan 08 2021

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).

A367805 a(1) = 0; for n > 1, a(n) is the least positive integer k for which k*prime(n) + 2 is prime.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 1, 3, 3, 1, 5, 3, 1, 3, 7, 7, 1, 5, 5, 1, 5, 3, 3, 3, 3, 1, 3, 1, 5, 9, 3, 7, 1, 3, 1, 5, 5, 3, 3, 3, 1, 5, 1, 5, 1, 3, 9, 5, 1, 9, 3, 1, 15, 7, 3, 15, 1, 9, 11, 1, 9, 3, 21, 1, 3, 3, 5, 3, 1, 3, 3, 15, 3, 5, 9, 3, 13, 3, 19, 3, 1, 15, 1, 3, 3, 9, 13, 3, 1, 15

Offset: 1

Frank Hollstein, Dec 01 2023

nonn

			For n = 4: a(4) = 3, because prime(4) = 7, 3*7 + 2 = 23 which is prime.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000040, A029707, A035096, A117673, A279756, A001359.

a:= proc(n) local p, q, r; p:= ithprime(n); q:= p;
      while irem(q-2, p, 'r')<>0 do q:= nextprime(q) od; r
    end:
seq(a(n), n=1..99);  # Alois P. Heinz, Dec 04 2023

nmax=90; a[1]=0; For[n=2, n<=nmax, n++, For[k=1, k>0, k++, If[PrimeQ[k*Prime[n]+2], a[n]=k; k=-1]]]; Array[a,nmax] (* Stefano Spezia, Dec 04 2023 *)

a(n) = if (n==1, 0, my(k=1, p=prime(n)); while (!isprime(k*p+2), k++); k); \\ Michel Marcus, Dec 02 2023

from itertools import count, dropwhile
from sympy import prime, isprime
def A367805(n):
    if n==1:
        return 0
    else:
        p = prime(n)
        return next(dropwhile(lambda x:not isprime(x*p+2),count(1))) # Chai Wah Wu, Jan 04 2024

a(n) = (A279756(n) - 2)/A000040(n).

a(n) = 1 <=> n in A029707.

More terms from Michel Marcus, Dec 02 2023

A385660 Numbers k such that prime(k+1)-prime(k) divides k.

Original entry on oeis.org

1, 2, 4, 8, 10, 12, 18, 20, 24, 26, 28, 36, 44, 48, 52, 54, 60, 64, 72, 80, 84, 88, 96, 98, 102, 104, 108, 112, 116, 120, 128, 136, 140, 142, 144, 148, 152, 168, 174, 176, 178, 180, 182, 190, 192, 206, 210, 212, 216, 224, 230, 234, 236, 240, 244, 248, 252, 256, 262, 264, 268, 276, 286, 288, 294

Offset: 1

Vighnesh Patil, Jul 06 2025

			k = 18 is a term since prime(19) - prime(18) = 6 divides 18.

Vighnesh Patil, Table of n, a(n) for n = 1..1000

Cf. A000040, A001223, A029707.

q[k_] := Divisible[k, Prime[k + 1] - Prime[k]]; Select[Range[300], q] (* Amiram Eldar, Jul 09 2025 *)

from sympy import prime
def ok(k): return k % (prime(k+1) - prime(k)) == 0
print([k for k in range(1, 300) if ok(k)])

A114961 Numbers n such that PrimePi(prime(n + 1)^2) - PrimePi(prime(n)^2) < c*n with c=9/5.

Original entry on oeis.org

7, 10, 13, 20, 26, 28, 33, 35, 43, 45, 49, 52, 57, 60, 64, 89, 98, 109, 113, 116, 120, 140, 142, 144, 148, 152, 171, 173, 176, 178, 182, 190, 201, 209, 212, 215, 225, 230, 234, 236, 253, 256, 262, 265, 268, 277, 286, 288, 294, 296, 302, 307, 313, 315, 318, 320

Offset: 1

Robert G. Wilson v, Feb 21 2006

nonn

If c=2 instead of 1.8 then the sequence is A029707.
This sequence is probably finite with 699 terms with 14020 being the last.
If c=1.7 the sequence is just {7, 10, 13, 20, 26, 28, 33, 35, 45, 49, 57, 60, 64, 89, 98, 109, 113, 116, 171, 190, 201, 215, 225, 234, 236, 256, 288, 332, 384, 405, 430, 486, 495, 498, 524, 530, 601, 613, 625, 872}.
If c=1.6 the sequence is just {7, 13, 20, 28, 33, 57, 109}.
If c=3/2 the sequence has but one term, 33.

P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag, NY, 1995, page 248.

Cf. A029707.

t = {}; Do[ If[ PrimePi[ Prime[n + 1]^2] - PrimePi[ Prime[n]^2] < 9n/5, AppendTo[t, n]], {n, 10^5}]; t

A227203 Prime(n)^2 mod (prime(n) + prime(n+1)).

Original entry on oeis.org

4, 1, 1, 13, 1, 19, 1, 25, 9, 1, 9, 43, 1, 49, 9, 9, 1, 9, 73, 1, 9, 85, 9, 109, 103, 1, 109, 1, 115, 49, 133, 9, 1, 25, 1, 9, 9, 169, 9, 9, 1, 25, 1, 199, 1, 241, 253, 229, 1, 235, 9, 1, 25, 9, 9, 9, 1, 9, 283, 1, 25, 49, 313, 1, 319, 49, 9, 25, 1, 355, 9, 379

Offset: 1

Zak Seidov, Sep 18 2013

nonn

a(n) = 1 if and only if prime(n) and prime(n+1) are twin prime pair (that is, prime(n+1) = 2 + prime(n)).

Cf. A029707, A167770.

Table[Mod[Prime[n]^2, Prime[n] + Prime[n + 1]], {n, 200}]
PowerMod[#[[1]],2,Total[#]]&/@Partition[Prime[Range[80]],2,1] (* Harvey P. Dale, Dec 26 2017 *)

for(n=1,200,print1(prime(n)^2%(prime(n)+prime(n+1))","))

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A334142 Indices of twin primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A340468 a(n) is the least prime of the form 2 + Product_{i=n..m} prime(i).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Python

A343496 First point of the straight lines in A340649.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Ruby

Formula

A367805 a(1) = 0; for n > 1, a(n) is the least positive integer k for which k*prime(n) + 2 is prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions