A029707 -id:A029707 - cp's OEIS frontend

A280941 Least integer k such that prime(k+1) - prime(k) = 2 and prime(k+2) - prime(k+1) = 2n, or 0 if no such k exists.

2, 3, 10, 0, 33, 45, 0, 294, 98, 0, 296, 262, 0, 428, 984, 0, 1456, 3086, 0, 2343, 1878, 0, 14938, 8422, 0, 2809, 4259, 0, 7809, 13819, 0, 51036, 45506, 0, 15782, 30764, 0, 57764, 24553, 0, 23282, 51942, 0, 44902, 34214, 0, 1242641, 95929, 0, 66761

Offset: 1

Michel Lagneau, Jan 11 2017

nonn

Or least integer k such that prime(k+2) - prime(k+1) = 2n where prime(k) is in A001359 (lesser of twin primes).
The corresponding prime(k) are 3, 5, 29, 137, 197, 1931, 521, 1949, 1667, 2969, 7757, 12161, 28349, 20807, ...
a(n) is a subsequence of A029707(n) or subsequence of A107770(n) - 1.
a(n) = 0 for n == 1 mod 3 for n > 1.
Proof: prime(k+1) - prime(k) = 2 => prime(k+1) == 1 mod 6 and prime(k) == -1 mod 6. If prime(k+2) - prime(k+1) = 2n, then prime(k+2) = 2(n+1) + prime(k). Combining n == 1 mod 3 and prime(k) == -1 mod 6 we obtain prime(k+2) == 3 mod 6, a contradiction because prime(k+2) == +-1 mod 6. Hence, a(n) = 0.

			a(3) = 10 because prime(11) - prime(10) = 31 - 29 = 2 and prime(12) - prime(11) = 37 - 31 = 6 = 2*3.
a(11) = 296 because prime(297) - prime(296) = 1951 - 1949 = 2 and prime(298) - prime(297) = 1973 - 1951 = 22 = 2*11.

Cf. A000040, A001359, A006512, A029707, A107770.

nn:=50:m:=10^5:
for n from 1 to 50 do:
ii:=0:
  for k from 1 to m while(ii=0) do:
   p1:=ithprime(k):p2:=ithprime(k+1):p3:=ithprime(k+2):
    if p2-p1 = 2 and p3-p2 = 2*n
    then
    ii:=1:printf(`%d %d \n`,n,k):
    else
    fi:
   od:
    if ii=0 then printf(`%d %d \n`,n,0):
    else
    fi:
od:

Table[If[And[n > 1, Mod[n, 3] == 1], 0, k = 1; While[Nand[# - Prime@ k == 2, Prime[k + 2] - # == 2 n] &@ Prime[k + 1], k++]; k], {n, 40}] (* Michael De Vlieger, Jan 14 2017 *)

A305558 If (p1,p2) is the n-th twin prime pair and p the prime before p1 and q the prime after p2 then a(n) = p + q - (p1 + p2).

Original entry on oeis.org

1, 2, 0, 0, 0, 0, 0, 2, 0, 0, 4, -4, 4, -6, 8, 0, 4, 0, 6, 0, -6, 0, -4, 0, 6, 0, 0, 8, -6, 6, -2, -6, 6, 0, 0, 4, -4, 0, -4, 0, -12, 0, -14, 0, 0, -6, 0, 2, -6, 0, -2, 0, 20, 6, -2, 8, 0, 6, -2, 6, 0, 0, -8, 6, 4, -10, 6, -12, -12, 10, 0, 2, 0, 4, -6, 0, 2, 0, -6, 12, 22, -18, 6, 8, -18, 8, -22, 6, -2, 6, 0, 0, 18, -6

Offset: 1

Dimitris Valianatos, Jun 21 2018

			For n = 8, the 8th prime pair is (71, 73), the prime before 71 is 67 and prime after 73 is 79. So a(8) = 67 + 79 - 71 - 73 = 2.

Cf. A001097, A054735, A036263.

Map[#1 + #4 - (#2 + #3) & @@ # &, Select[Partition[Prime@ Range[500], 4, 1], And[#3 - #2 == 2] & @@ # &]] (* Michael De Vlieger, Jun 30 2018 *)

{
print1(2+7-(5+3)", ");
forstep(n=6,100,6,
        if(isprime(n-1)&&isprime(n+1),
           a=precprime(n-2);b=nextprime(n+2);
           print1(a+b-2*n", ")
          )
       )
}

a(n) = A000040(A029707(n)-1) + A000040(A107770(n)+1) - (A001359(n) + A006512(n)). - Jianing Song, Jun 22 2018

Definition clarified by Jianing Song, Jun 22 2018

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)).

A345366 a(n) = (p*q+1) mod (p+q) where p=prime(n) and q=prime(n+1).

Original entry on oeis.org

2, 0, 0, 6, 0, 12, 0, 18, 44, 0, 60, 36, 0, 42, 92, 104, 0, 120, 66, 0, 144, 78, 164, 78, 96, 0, 102, 0, 108, 192, 126, 260, 0, 264, 0, 300, 312, 162, 332, 344, 0, 348, 0, 192, 0, 170, 182, 222, 0, 228, 464, 0, 468, 500, 512, 524, 0, 540, 276, 0, 552, 552, 306

Offset: 1

Simon Strandgaard, Jun 16 2021

nonn

The graph of this function consists of three branches: the upper one corresponds to cases where q-p == 2 (mod 4) except the twin primes, the middle one to cases where q-p == 0 (mod 4), and the lower one (where a(n)=0) to cases where q-p = 2, the twin primes.

All terms are even.

			a(1) = ( 2* 3+1) mod ( 2+ 3) =   7 mod  5 = 2,
a(2) = ( 3* 5+1) mod ( 3+ 5) =  16 mod  8 = 0,
a(3) = ( 5* 7+1) mod ( 5+ 7) =  36 mod 12 = 0,
a(4) = ( 7*11+1) mod ( 7+11) =  78 mod 18 = 6,
a(5) = (11*13+1) mod (11+13) = 144 mod 24 = 0.

Cf. A000040, A212769, A029707 (indices of 0's).

Cf. A001043, A023523.

a:= n-> ((p, q)-> irem(p*q+1, p+q))(map(ithprime, [n, n+1])[]):
seq(a(n), n=1..63);  # Alois P. Heinz, Jul 03 2021

Mod[#1*#2 + 1, #1 + #2] & @@@ Partition[Select[Range[300], PrimeQ], 2, 1] (* Amiram Eldar, Jun 16 2021 *)

a(n)=my(p=prime(n), q=nextprime(p+1)); (p*q+1)%(p+q)

from sympy import nextprime
def aupton(nn):
    alst, p, q = [], 2, 3
    while len(alst) < nn: alst.append((p*q+1)%(p+q)); p, q = q, nextprime(q)
    return alst
print(aupton(62)) # Michael S. Branicky, Jun 16 2021

require 'prime'
values = []
Prime.first(21).each_cons(2) do |a, b|
    values << (a * b + 1) % (a + b)
end
p values

a(n) = A023523(n+1) mod A001043(n). - Michel Marcus, Jun 17 2021

A373828 Run-sums (differing by 0) of run-lengths (differing by 2) of odd primes.

Original entry on oeis.org

3, 4, 1, 2, 1, 2, 2, 2, 1, 2, 4, 4, 3, 4, 4, 6, 2, 2, 1, 2, 3, 2, 1, 2, 2, 2, 3, 2, 10, 4, 4, 2, 7, 2, 4, 2, 3, 2, 2, 2, 1, 2, 2, 2, 18, 6, 2, 2, 2, 2, 17, 4, 1, 4, 2, 2, 6, 2, 9, 2, 3, 2, 1, 2, 1, 2, 1, 2, 8, 2, 3, 2, 2, 4, 15, 2, 1, 2, 4, 2, 1, 2, 1, 2, 7, 2

Offset: 1

Gus Wiseman, Jun 23 2024

nonn

Run-sums of A251092.

			The odd primes are:
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...
with runs:
{3,5,7}, {11,13}, {17,19}, {23}, {29,31}, {37}, {41,43}, {47}, {53}, ...
with lengths:
3, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, ...
with runs:
{3}, {2,2}, {1}, {2}, {1}, {2}, {1,1}, {2}, {1}, {2}, {1,1,1,1}, {2,2}, ...
with sums a(n).

Run-sums of A251092.
The run-lengths (instead of run-sums) are A373819, firsts A373825, A373824.
A000040 lists the primes.
A001223 gives first differences of primes.
A027833 gives antirun-lengths of primes > 3 (prepended run-lengths A373820).
A046933 counts composite numbers between primes.
A071148 gives partial sums of odd primes.
A333254 gives run-lengths of first differences of primes.
A373821 gives run-lengths of run-lengths of first differences of odd primes.
Cf. A006560, A006562, A029707, A054265, A067774, A373403, A373404, A373406.

Total/@Split[Length /@ Split[Select[Range[3,10000],PrimeQ], #1+2==#2&]//Most]//Most

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A280941 Least integer k such that prime(k+1) - prime(k) = 2 and prime(k+2) - prime(k+1) = 2n, or 0 if no such k exists.

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A305558 If (p1,p2) is the n-th twin prime pair and p the prime before p1 and q the prime after p2 then a(n) = p + q - (p1 + p2).

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

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A345366 a(n) = (p*q+1) mod (p+q) where p=prime(n) and q=prime(n+1).

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A373828 Run-sums (differing by 0) of run-lengths (differing by 2) of odd primes.

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