A055625 -id:A055625 - cp's OEIS frontend

A057620 Initial prime in first sequence of n consecutive primes congruent to 1 modulo 6.

7, 31, 151, 1741, 1741, 1741, 19471, 118801, 148531, 148531, 406951, 2339041, 2339041, 51662593, 51662593, 73451737, 232301497, 450988159, 1444257673, 1444257673, 1444257673, 24061965043, 24061965043, 43553959717, 43553959717

Offset: 1

Robert G. Wilson v, Oct 09 2000

nonn

See A055626 for the variant "exactly n". See A247967 for the indices of these primes. See A057620, A057621 for variants "congruent to 5 (mod 6)", resp. "(mod 3)". - M. F. Hasler, Sep 03 2016

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

			a(6) = 1741 because this number is the first in a sequence of 6 consecutive primes all of the form 3n + 1.

R. K. Guy, "Unsolved Problems in Number Theory", A4

Giovanni Resta, Table of n, a(n) for n = 1..35 (terms < 4*10^14)
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. A057619, A057622, A057624, A055625.

p = 0; Do[a = Table[-1, {n}]; k = Max[1, p]; While[Union[a] != {1}, k = NextPrime[k]; a = Take[AppendTo[a, Mod[k, 3]], -n]]; p = NestList[NextPrime[#, -1] &, k, n]; Print[p[[-2]]]; p = p[[-1]], {n, 1, 18}] (* Robert G. Wilson v, updated by Michael De Vlieger, Sep 03 2016 *)
Table[k = 1; While[Total@ Boole@ Map[Mod[#, 6] == 1 &, NestList[NextPrime, Prime@ k, n - 1]] != n, k++]; Prime@ k, {n, 12}] (* Michael De Vlieger, Sep 03 2016 *)

m=c=o=0; forprime(p=1,, p%6 != 1 && (!c||!c=0) && next; c||o=p; c++>m||next; m++; print1(", ",o)) \\ M. F. Hasler, Sep 03 2016

a(n) <= A055625(n). - Zak Seidov, Aug 29 2016

a(n) = A000040(A247967(n)). a(n) = min { A055625(k); k >= n }. - M. F. Hasler, Sep 03 2016

More terms from Don Reble, Nov 16 2003
More terms from Jens Kruse Andersen, May 30 2006
Definition clarified by Zak Seidov, Jun 19 2017

A055626 First prime starting a chain of exactly n consecutive primes congruent to 5 modulo 6.

Original entry on oeis.org

5, 23, 47, 251, 1889, 7793, 43451, 243161, 726893, 759821, 2280857, 1820111, 10141499, 40727657, 19725473, 136209239, 744771077, 400414121, 1057859471, 489144599, 13160911739, 766319189, 38451670931, 119618704427, 21549657539, 141116164769, 140432294381, 437339303279

Offset: 1

Labos Elemer, Jun 05 2000

nonn

The term "exactly" means that before the first and after the last primes of chain, the immediate primes are not congruent to 5 modulo 6.
a(21)>2^31, a(22)= 766319189. - Hugo Pfoertner, Jul 31 2003
See A057622 for the variant where "exactly" is replaced by "at least". See A055625 for the variant "congruent to 1 (mod 6)". - M. F. Hasler, Sep 03 2016

Giovanni Resta, Table of n, a(n) for n = 1..35 (terms < 4*10^14)
J. K. Andersen, Consecutive Congruent Primes.

Cf. A055623, A055624, A055625, A085516.

pp = Table[{p = Prime[n], Mod[p, 6]}, {n, 10^6}];
sp = Split[pp, Mod[#1[[2]], 6] == Mod[#2[[2]], 6]&];
a[n_] := SelectFirst[sp, Length[#] == n && MatchQ[#, {{_Integer, 5} ..}]& ][[1, 1]];
Table[an = a[n]; Print[n, " ", an]; an, {n, 1, 13}] (* Jean-François Alcover, Nov 21 2018 *)

okchain(n, p) = {if ((precprime(p-1) % 6) == 5, return (0)); for (i=1, n, if ((p % 6) != 5, return (0)); p = nextprime(p+1);); if ((p % 6) == 5, 0, 1);}
a(n) = {p = 5; while (! okchain(n, p), p = nextprime(p+1)); p;} \\ Michel Marcus, Dec 17 2013

a(9)-a(13), including correction of a(9)-a(10) from Reiner Martin, Jul 18 2001
a(14)-a(20) from Hugo Pfoertner, Jul 31 2003
a(21)-a(25) from Jens Kruse Andersen, May 30 2006
a(26) and beyond from Giovanni Resta, Aug 04 2013

A085515 Order of first occurrence of a sequence of exactly n consecutive primes of the form 6*k+1.

Original entry on oeis.org

1, 2, 3, 6, 4, 5, 7, 8, 10, 11, 9, 13, 12, 15, 16, 14, 17, 18, 21, 19, 20, 23, 22, 25, 24, 26, 31, 27, 28, 30, 29, 32, 34, 33, 35

Offset: 1

Hugo Pfoertner, Jul 31 2003

This sequence gives the index of A055625(n) after sorting by increasing magnitude.

			a(3)=3, a(4)=6, a(5)=4 because the first occurrence of exactly 6 consecutive primes of the form 6*k+1 (1741,1747,1753,1759,1777,1783) occurs after the first occurrence of 3 consecutive primes of this form (151,157,163) and before the first occurrence of 4 consecutive "6k+1"-primes (3049,3061,3067,3079).

J. K. Andersen, Consecutive Congruent Primes.
Hugo Pfoertner, Record producing runs of primes of the forms 6k+-1.
Hugo Pfoertner, FORTRAN function ISPRIM to check primality.

Cf. A055625, A085516 (sorted occurrence of first runs of "6k-1" primes).

Fortran
```
! Program given at link.
```

(* Program not suitable to compute a large number of terms. *)
pp = Table[{p = Prime[n], Mod[p, 6]}, {n, 10^7}];
sp = Split[pp, Mod[#1[[2]], 6] == Mod[#2[[2]], 6]&];
b[n_] := SelectFirst[sp, Length[#] == n && MatchQ[#, {{_Integer, 1} ..}]& ][[1, 1]];
Sort[Table[{b[n], n}, {n, 1, 16}]][[All, 2]] (* Jean-François Alcover, Nov 21 2018 *)

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 27 2006

a(32)-a(35) from Giovanni Resta, Aug 04 2013

A181938 Isolated primes = 1 mod 6: sandwiched by primes = 5 mod 6.

Original entry on oeis.org

7, 13, 19, 43, 97, 103, 109, 127, 139, 181, 193, 229, 241, 283, 307, 313, 349, 397, 409, 421, 457, 463, 487, 499, 643, 691, 709, 769, 787, 811, 823, 829, 853, 859, 877, 883, 907, 919, 937, 967, 1021, 1051, 1093, 1153, 1171, 1279, 1303, 1423, 1429, 1447, 1483

Offset: 1

Zak Seidov, Apr 03 2012

nonn

Primes p(m) = 1 mod 6 such that both p(m-1) and p(m+1) are congruent to 5 mod 6.
Corresponding indices m are 4, 6, 8, 14, 25, 27, 29, 31 (A181978).
Note that values of d = p(m+1) - p(m-1) are multiples of 6.

			7 = p(4) = 1 mod 6 and both p(3) = 5 and p(5) = 11 are congruent to 5 mod 6,
13 = p(6) = 1 mod 6 and both p(5) = 11 and p(7) = 17 are congruent to 5 mod 6,
43 = p(14) = 1 mod 6 and both p(13) = 41 and p(15) = 47 are congruent to 5 mod 6.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A002476, A039704, A055625, A181978, A210248.

Select[Prime[Range[2, 300]], Mod[#, 6] == 1 && Mod[NextPrime[#, -1], 6] == 5 && Mod[NextPrime[#, 1], 6] == 5 &] (* T. D. Noe, Apr 04 2012 *)
Transpose[Select[Partition[Prime[Range[250]],3,1],Mod[#[[1]],6] == Mod[#[[3]],6] == 5&&Mod[#[[2]],6]==1&]][[2]] (* Harvey P. Dale, Sep 17 2012 *)

A344093 a(n) is the smallest positive integer not already in the sequence such that a(n) + a(n-1) is the product of two distinct primes, with a(1) = 1.

Original entry on oeis.org

1, 5, 9, 6, 4, 2, 8, 7, 3, 11, 10, 12, 14, 19, 15, 18, 16, 17, 21, 13, 20, 26, 25, 30, 27, 24, 22, 29, 28, 23, 32, 33, 36, 38, 31, 34, 35, 39, 43, 42, 40, 37, 45, 41, 44, 47, 46, 48, 58, 53, 62, 49, 57, 54, 52, 59, 56, 50, 61, 68, 51, 55, 60, 63, 66, 67, 74, 69, 64

Offset: 1

Atticus Stewart, Aug 15 2021

nonn

This sequence is not to be confused with A243625 (similar but with "semiprime" instead of "product of two distinct primes"). This sequence omits squares of primes, whereas semiprimes include them. This sequence is also similar to A055625, in which sums of consecutive terms are primes instead of semiprimes.

Interestingly, the first 9 terms are a permutation of the first 9 positive integers. This is also true for the first 12, 21, 30, and 48 terms, and possibly higher values as well. This suggests that every positive integer occurs, but this is unproved.

			a(4) = 6 because 6 is the smallest k such that a(3) + k is the product of two distinct primes.

Cf. A243625, A055625, A006881.

a[1]=1;a[n_]:=a[n]=(k=1;While[MemberQ[Array[a,n-1],k]||Last/@FactorInteger[a[n-1]+k]!={1,1},k++];k);Array[a,100] (* Giorgos Kalogeropoulos, Aug 16 2021 *)

terms = [1]
previous = 1
def isValid(num):
  counter = 0
  for possibleDiv in range(1, int(math.sqrt(num)) + 1):
    if num % possibleDiv == 0:
      counter += 1
      if num/possibleDiv % possibleDiv == 0 and possibleDiv != 1:
        return False
    if counter > 2:
      return False
  if counter == 2:
    return True
  return False
def generateSequence(numOfTerms):
  for i in range(numOfTerms):
    testNum = 1
    valid = False
    while not valid:
      if testNum not in terms:
        possibleNum = previous + testNum
        if isValid(num):
          valid = True
          terms.append(testNum)
          previous = testNum
      testNum += 1

cp's OEIS Frontend

A057620 Initial prime in first sequence of n consecutive primes congruent to 1 modulo 6.

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A055626 First prime starting a chain of exactly n consecutive primes congruent to 5 modulo 6.

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A085515 Order of first occurrence of a sequence of exactly n consecutive primes of the form 6*k+1.

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A181938 Isolated primes = 1 mod 6: sandwiched by primes = 5 mod 6.

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A344093 a(n) is the smallest positive integer not already in the sequence such that a(n) + a(n-1) is the product of two distinct primes, with a(1) = 1.

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