A276414 -id:A276414 - cp's OEIS frontend

A270190 Numbers n for which prime(n+1)-prime(n) is a multiple of three.

9, 11, 15, 16, 18, 21, 23, 32, 36, 37, 39, 40, 46, 47, 51, 54, 55, 56, 58, 67, 71, 73, 74, 76, 84, 86, 91, 96, 97, 99, 100, 102, 103, 105, 107, 108, 110, 111, 114, 118, 119, 121, 123, 129, 130, 133, 139, 160, 161, 164, 165, 167, 168, 170, 174, 179, 180, 184, 185, 187, 188, 194, 195, 197, 199, 200, 202, 203, 205, 208, 210

Offset: 1

Antti Karttunen, Mar 16 2016

nonn

Numbers n for which A001223(n) = 0 modulo 3.
See comments in A270189 and A269364.
Equivalently, numbers n for which prime(n+1)-prime(n) is a multiple of six. See A276414 for runs of increasing length of consecutive integers. - M. F. Hasler, Sep 03 2016

			9 is present as the difference between A000040(9+1) = 29 and A000040(9) = 23 is 6, a multiple of three.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Terence Tao, Biases between consecutive primes, blog entry March 14, 2016

Complement: A270189.
Positions of zeros in A137264.
Left inverse: A269850.
Cf. also A001223, A269364, A270191, A270192, A269399.

Select[Range@ 210, Divisible[Prime[# + 1] - Prime@ #, 3] &] (* Michael De Vlieger, Mar 17 2016 *)
PrimePi/@Select[Partition[Prime[Range[350]],2,1],Divisible[#[[2]]-#[[1]], 3]&][[All,1]] (* Harvey P. Dale, Jul 11 2017 *)

isok(n) = ((prime(n+1) - prime(n)) % 3) == 0; \\ Michel Marcus, Mar 17 2016

Other identities. For all n >= 1:
a(n) = A269399(n) + 6.
A269850(a(n)) = n.

A247967 a(n) is the smallest k such that prime(k+i) (mod 6) takes successively the values 5, 5, ... for i = 0, 1, ..., n-1.

Original entry on oeis.org

3, 9, 15, 54, 290, 987, 4530, 21481, 58554, 60967, 136456, 136456, 673393, 1254203, 1254203, 7709873, 21357253, 21357253, 25813464, 25813464, 39500857, 39500857, 947438659, 947438659, 947438659, 5703167678, 5703167678, 16976360924, 68745739764, 117327812949

Offset: 1

Michel Lagneau, Sep 28 2014

nonn

Weakening the definition to prime(k+i) == 2 (mod 3) yields a(1) = 1, but all other terms are unchanged. See also A247816 (residue 5) or A276414 (equal residues, all 1 or all -1). - M. F. Hasler, Sep 02 2016

			a(1)= 3 => prime(3) == 5 (mod 6).
a(2)= 9 => prime(9) == 5 (mod 6), prime(10) == 5 (mod 6).
a(3)= 15 => prime(15) == 5 (mod 6), prime(16) == 5 (mod 6), prime(17) == 5 (mod 6).
From _Michel Marcus_, Sep 30 2014: (Start)
The resulting primes are:
  5;
  23, 29;
  47, 53, 59;
  251, 257, 263, 269;
  1889, 1901, 1907, 1913, 1931;
  7793, 7817, 7823, 7829, 7841, 7853;
  43451, 43457, 43481, 43487, 43499, 43517, 43541;
  243161, 243167, 243197, 243203, 243209, 243227, 243233, 243239;
  ... (End)

Amiram Eldar, Table of n, a(n) for n = 1..35

Cf. A057622, A247816, A276414.

N = 2*10^8; % to use primes up to N
P = mod(primes(N),6);
P5 = find(P==5);
n5 = numel(P5);
a(1) = P5(1);
for k = 2:100
  r = find(P5(k:n5) == P5(1:n5+1-k)+k-1,1,'first');
  if numel(r) == 0
     break
  end
  a(k) = P5(r);
end
a % Robert Israel, Sep 02 2016

for n from 1 to 22 do :
ii:=0:
   for k from 3 to 10^5 while (ii=0)do :
     s:=0:
      for i from 0 to n-1 do:
        r:=irem(ithprime(k+i),6):
        if r = 5
        then
        s:=s+1:
        else
        fi:
      od:
       if s=n and ii=0
       then
       printf ( "%d %d \n",n,k):ii:=1:
       else
       fi:
    od:
od:

Table[k = 1; While[Times @@ Boole@ Map[Mod[Prime[k + #], 6] == 5 &, Range[0, n - 1]] == 0, k++]; k, {n, 10}] (* Michael De Vlieger, Sep 02 2016 *)

a(n) = {k = 1; ok = 0; while (!ok, nb = 0; for (i=0, n-1, if (prime(k+i) % 6 == 5, nb++, break);); if (nb == n, ok=1, k++);); k;} \\ Michel Marcus, Sep 28 2014

m=c=i=0;forprime(p=1,, i++;p%6!=5&&(!c||!c=0)&&next; c++>m||next; print1(1+i-m=c,",")) \\ M. F. Hasler, Sep 02 2016

a(n) = primepi(A057622(n)). - Michel Marcus, Oct 01 2014

a(11)-a(22) from A057622 by Michel Marcus, Oct 03 2014
a(23)-a(25) from Jinyuan Wang, Jul 08 2019
a(26)-a(30) added using A057622 by Jinyuan Wang, Apr 15 2020

A054679 First of n consecutive primes which differ by a multiple of 6.

Original entry on oeis.org

2, 23, 47, 251, 1741, 1741, 19471, 118801, 148531, 148531, 406951, 1820111, 2339041, 19725473, 19725473, 73451737, 232301497, 400414121, 489144599, 489144599, 766319189, 766319189, 21549657539, 21549657539, 21549657539, 140432294381, 140432294381, 437339303279, 1552841185921, 1552841185921, 1552841185921

Offset: 1

Jeff Burch, Apr 18 2000

nonn

See A276414 for the indices of these primes. - M. F. Hasler, Sep 02 2016

The sequence is infinite, by Shiu's theorem. - Jonathan Sondow, Jun 22 2017

Giovanni Resta, Table of n, a(n) for n = 1..35 (terms < 4*10^14) (corrected by Ray Chandler, Jan 19 2019)
J. K. Andersen, Consecutive Congruent Primes.
D. K. L. Shiu, Strings of Congruent Primes, J. Lond. Math. Soc. 61 (2) (2000) 359-373 [MR1760689]

Cf. A054678, A054680; A276414.

a(n) = A000040(A276414(n)). - M. F. Hasler, Sep 02 2016

a(n) = min(A057620(n), A057621(n)) for all n >= 1. - M. F. Hasler, Sep 03 2016

More terms from Larry Reeves (larryr(AT)acm.org), Nov 09 2000
More terms from Jens Kruse Andersen, May 30 2006
Initial term and a(27)-a(31) added and name edited by M. F. Hasler, Sep 02 2016

A247816 a(n) is the smallest k such that prime(k+i) = 1 (mod 6) for i = 0, 1,...,n-1.

Original entry on oeis.org

4, 11, 36, 271, 271, 271, 2209, 11199, 13717, 13717, 34369, 172146, 172146, 3094795, 3094795, 4308948, 12762142, 23902561, 72084956, 72084956, 72084956, 1052779161, 1052779161, 1857276773, 1857276773, 19398320447, 57446769091, 57446769091, 57446769091

Offset: 1

Michel Lagneau, Sep 28 2014

Equivalently, "mod 6" can be replaced by "mod 3". See A247967 for the variant "= 5 (mod 6)" and A276414 for runs of primes congruent to each other (mod 3). - M. F. Hasler, Sep 02 2016

The sequence is infinite, by Shiu's theorem. - Jonathan Sondow, Jun 22 2017

			a(1)= 4 => prime(4) (mod 6)= 1;
a(2)= 11 => prime(11)(mod 6)= 1, prime(12)(mod 6) = 1;
a(3)= 36 => prime(36)(mod 6)= 1, prime(37)(mod 6)= 1, prime(38)(mod 6)= 1.
The resulting primes are:
7;
31, 37;
151, 157, 163;
1741, 1747, 1753, 1759;
1741, 1747, 1753, 1759, 1777;
1741, 1747, 1753, 1759, 1777, 1783;
19471, 19477, 19483, 19489, 19501, 19507, 19531;
... - _Michel Marcus_, Sep 29 2014

Giovanni Resta, Table of n, a(n) for n = 1..35
D. K. L. Shiu, Strings of Congruent Primes, J. Lond. Math. Soc. 61 (2) (2000) 359-373 [MR1760689]

Cf. A057620, A247967; A276414.

for n from 1 to 22 do :
ii:=0:
   for k from 3 to 10^5 while (ii=0)do :
     s:=0:
      for i from 0 to n-1 do:
        r:=irem(ithprime(k+i),6):
        if r = 1
        then
        s:=s+1:
        else
        fi:
      od:
       if s=n and ii=0
       then
       printf ( "%d %d \n",n,k):ii:=1:
       else
       fi:
    od:
od:

With[{m6=If[Mod[#,6]==1,1,0]&/@Prime[Range[5*10^6]]},Flatten[Table[SequencePosition[ m6,PadRight[{},n,1],1],{n,16}],1]][[;;,1]] (* Harvey P. Dale, May 07 2023 *)

m=c=i=0;forprime(p=1,, i++;p%6!=1&&(!c||!c=0)&&next; c++>m||next; print1(1+i-m=c,",")) \\ M. F. Hasler, Sep 02 2016

a(n) = primepi(A057620(n)). - Michel Marcus, Sep 30 2014

a(12)-a(21) from A057620 by Michel Marcus, Oct 03 2014

a(22)-a(29) from Giovanni Resta, Oct 03 2018

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A270190 Numbers n for which prime(n+1)-prime(n) is a multiple of three.

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A247967 a(n) is the smallest k such that prime(k+i) (mod 6) takes successively the values 5, 5, ... for i = 0, 1, ..., n-1.

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A054679 First of n consecutive primes which differ by a multiple of 6.

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A247816 a(n) is the smallest k such that prime(k+i) = 1 (mod 6) for i = 0, 1,...,n-1.

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