A031924 -id:A031924 - cp's OEIS frontend

A248855 a(n) is the smallest positive integer m such that if k >= m then a(k+1,n)^(1/(k+1)) <= a(k,n)^(1/k), where a(k,n) is the k-th term of the sequence {p | p and p+2n are primes}.

1, 1, 1, 1, 3556, 1, 34, 3, 4, 1, 2, 1, 11285, 5, 2, 124, 569, 1, 290, 3, 1, 165, 2, 1, 1, 2, 1, 316, 1, 2, 58957, 1, 3, 58617, 522, 2, 1, 1, 4, 1, 2, 1, 1, 2, 1, 7932, 4, 1, 5875, 1679, 4, 4, 3, 3, 1, 2, 307, 1, 1, 1, 1, 1, 4, 3206, 2, 1, 1, 3, 2, 1, 1, 1, 1, 5, 2, 11170, 1, 2, 4245, 1, 1, 81, 2, 1, 1, 2, 58, 1, 3, 4, 7303, 1, 1, 5, 1, 3, 3, 3, 383, 111408, 1

Offset: 0

Farideh Firoozbakht and Jahangeer Kholdi, Nov 25 2014

nonn

All terms conjecturally are found. Note that according to the definition a(k,0) is the k-th term of the sequence {p | p is prime} namely for every positive integer k, a(k,0) = prime(k). Hence if Firoozbakht's conjecture is true then a(0)=1.

			a(0)=a(1)=a(2)=a(3)=1 conjecturally states that the four sequences A000040, A001359, A023200 and A023201 have this property: For every positive integer n, b(n) exists and b(n+1) < b(n)^(1+1/n). Namely b(n)^(1/n) is a strictly decreasing function of n.
If in the definition instead of the sequence {p | p and p+2n are primes} we set {p | p is prime and nextprime(p)=p+2n} then it seems that except for n=3 all terms of the new sequence {c(n)} are equal to 1 and for n=3, c(3)=7746. Note that c(3)=7746 means that the sequence {p | p is prime and nextprime(p)=p+6} = A031924 has this property: For all k >= 7746, A031924(k+1)^(1/(k+1)) < A031924(k)^(1/k).

Wikipedia, Firoozbakht's conjecture

Cf. A000040, A001359, A023200, A023201, A023202, A023203, A031924.

A346239 Möbius transform of A341512, sigma(n)A003961(n) - nsigma(A003961(n)).

Original entry on oeis.org

0, 1, 2, 10, 2, 33, 4, 74, 44, 55, 2, 278, 4, 115, 116, 490, 2, 613, 4, 498, 242, 169, 6, 1942, 92, 265, 742, 1046, 2, 1591, 6, 3086, 344, 355, 330, 4986, 4, 487, 542, 3570, 2, 3347, 4, 1638, 2326, 737, 6, 12542, 376, 2121, 716, 2546, 6, 9869, 388, 7510, 986, 943, 2, 12894, 6, 1225, 4872, 18970, 630, 5353, 4, 3498, 1492

Offset: 1

Antti Karttunen, Jul 13 2021

nonn

Cf. A000203, A003961, A008683, A341528, A341529, A341512, A346240, A347096, A347124, A347125.

Cf. also the sequences A001359, A029710, A031924 that give the positions of 2's, 4's and 6's in this sequence, or at least subsets of such positions.

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341528(n) = (n*sigma(A003961(n)));
A341529(n) = (sigma(n)*A003961(n));
A341512(n) = (A341529(n)-A341528(n));
A346239(n) = sumdiv(n,d,moebius(n/d)*A341512(d));

a(n) = Sum_{d|n} A008683(n/d) * A341512(d).
a(n) = A341512(n) - A346240(n).
a(n) = A347125(n) - A347124(n). - Antti Karttunen, Aug 25 2021

A382810 Primes p such that p + 6, p + 10 and p + 16 are also primes.

Original entry on oeis.org

7, 13, 31, 37, 73, 97, 157, 223, 373, 433, 1087, 1291, 1423, 1483, 1543, 1861, 1987, 2341, 2383, 2677, 2683, 3313, 3607, 4441, 4507, 4783, 4993, 5641, 5851, 6037, 6961, 7237, 7867, 8731, 9613, 9733, 10723, 13093, 13681, 14143, 14731, 16057, 16411, 16921, 17377

Offset: 1

Alexander Yutkin, Apr 05 2025

nonn

The four primes need not be consecutive; otherwise we have the sequence A078856.

			p=37: 37+6=43, 37+10=47, 37+16=53 -> prime quartet: (37, 43, 47, 53).

Robert Israel, Table of n, a(n) for n = 1..2500

Cf. A000040, A001223, A023200, A031924.

Cf. A078852 [4, 6, 6], A078856 [6, 4, 6], A078858 [6, 6, 4], A033451 [6, 6, 6].

q:= p-> andmap(i->isprime(p+i), [0, 6, 10, 16]):
select(q, [$2..20000])[];  # Alois P. Heinz, Apr 05 2025

Select[Prime[Range[2000]],AllTrue[#+{6,10,16},PrimeQ]&] (* James C. McMahon, Apr 13 2025 *)

A078562 p, p+6 and p+10 are consecutive primes.

Original entry on oeis.org

31, 61, 73, 157, 271, 373, 433, 607, 733, 751, 1291, 1543, 1657, 1777, 1861, 1987, 2131, 2287, 2341, 2371, 2383, 2467, 2677, 2791, 2851, 3181, 3313, 3607, 3691, 4441, 4507, 4723, 4993, 5407, 5431, 5521, 5563, 5641, 5683, 5851, 6037, 6211, 6571, 6961

Offset: 1

Labos Elemer, Dec 10 2002

nonn

Subsequence of A031924. - R. J. Mathar, Jun 15 2013

			Between p and p+10 the difference-pattern is [64] like e.g. for p=31: 31(6)37(4)41.

R. J. Mathar, Table of n, a(n) for n = 1..1000

Cf. analogous inter-prime d-patterns with d<=6: A022004[24], A022005[42], A049437[26], A049438[62], A078561[46], A078562[64], A047948[66].

Transpose[Select[Partition[Prime[Range[1000]],3,1],#[[3]]-#[[1]]==10&&#[[2]]-#[[1]]==6&]][[1]] (* Harvey P. Dale, Dec 09 2010 *)

A192175 Array of primes determined by distance to next prime, by antidiagonals.

Original entry on oeis.org

2, 3, 7, 5, 13, 23, 11, 19, 31, 89, 17, 37, 47, 359, 139, 29, 43, 53, 389, 181, 199, 41, 67, 61, 401, 241, 211, 113, 59, 79, 73, 449, 283, 467, 293, 1831, 71, 97, 83, 479, 337, 509, 317, 1933, 523, 101, 103, 131, 491, 409, 619, 773, 2113, 1069, 887, 107

Offset: 1

Clark Kimberling, Jun 24 2011

Row 1:  primes p such that p+1 or p+2 is a prime.
Row r>1:  primes p such that the least h for which p+2h is prime is r.
Rows 1-7:  A124588, A023200, A031924, A031926, A031928, A031932, A031924.

			Northwest corner:
  2.....3.....5.....11....17....29....41
  7.....13....19....37....43....67....79
  23....31....47....53....61....73....83
  89....359...389...401...449...479...491
  139...181...241...283...337...409...421
For example, 31 is in row 3 because 31+2*3 is a prime, unlike 31+2*1 and 31+2*2.  Every prime occurs exactly once.  For each row, it is not known whether it is finite.

Cf. A192176, A192177, A192178, A192179.

z = 5000; (* z=number of primes used *)
row[1] = (#1[[1]] &) /@ Cases[Array[{#1,
      PrimeQ[1 + Prime[#1]] || PrimeQ[2 + Prime[#1]]} &, {z}], {_, True}];
Do[row[x] = Complement[(#1[[1]] &) /@ Cases[Array[{#1, PrimeQ[2 x + Prime[#1]]} &, {z}], {_, True}], Flatten[Array[row, {x - 1}]]], {x, 2, 16}]; TableForm[Array[Prime[row[#]] &, {10}]] (* A192175 array *)
Flatten[Table[ Prime[row[k][[n - k + 1]]], {n, 1, 11}, {k, 1, n}]] (* A192175 sequence *)
(* Peter J. C. Moses, Jun 20 2011 *)

A093738 Number of pairs of consecutive prime (p,q) with q-p=6 and q < 10^n.

Original entry on oeis.org

0, 7, 44, 299, 1940, 13549, 99987, 768752, 6089791, 49392723, 408550278, 3435528229, 29289695650, 252672394234, 2201981901415, 19360330918473, 171550299264139, 1530609037414453

Offset: 1

Enoch Haga, Apr 15 2004

Note that one has to be careful to distinguish between pairs of consecutive primes (p,q) with q-p = 6 (A031924), and pairs of primes (p,q) with q-p = 6 (A023201). Here we consider the former, whereas A080841 considers the latter. - N. J. A. Sloane, Mar 07 2021

			a(2) = 7 because there are 7 prime gaps of 6 below 10^2.

Siegfried "Zig" Herzog, Frequency of Occurrence of Prime Gaps
T. Oliveira e Silva, S. Herzog, and S. Pardi, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4.10^18, Math. Comp., 83 (2014), 2033-2060.

Cf. A007508, A080841, A093737, A093739.

A288021 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 4, and p1, p2 are in different decades.

Original entry on oeis.org

7, 19, 37, 67, 79, 97, 109, 127, 229, 277, 307, 349, 379, 397, 439, 457, 487, 499, 739, 757, 769, 859, 877, 907, 937, 967, 1009, 1087, 1279, 1297, 1429, 1447, 1489, 1549, 1567, 1579, 1597, 1609, 1867, 1999, 2137, 2239, 2269, 2347, 2377, 2389, 2437, 2539, 2617, 2659, 2689, 2707, 2749, 2797, 2857

Offset: 1

Hartmut F. W. Hoft, Jun 04 2017

nonn

The unit digits of the numbers in the sequence are 7's or 9's.

			7 is in this sequence since pair (7,11) is the first with difference 4 spanning a multiple of 10.

Cf. A001359, A023201, A023203, A031924, A031925, A031928, A046117, A046132, A158277, A158861, A160370, A160440, A160500, A287049, A287050, A060229, A288022, A288024.

a288021[n_] := Map[Last, Select[Map[{NextPrime[#, 1], NextPrime[#, -1]}&, Range[10, n, 10]], First[#]-Last[#]==4&]]
a288021[3000] (* data *)

A288022 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 6, and p1, p2 are in different decades.

Original entry on oeis.org

47, 157, 167, 257, 367, 557, 587, 607, 647, 677, 727, 947, 977, 1097, 1117, 1187, 1217, 1367, 1657, 1747, 1777, 1907, 1987, 2207, 2287, 2417, 2467, 2677, 2837, 2897, 2957, 3307, 3407, 3607, 3617, 3637, 3727, 3797, 4007, 4357, 4457, 4507, 4597, 4657, 4937, 4987

Offset: 1

Hartmut F. W. Hoft, Jun 04 2017

nonn

The unit digits of the numbers in the sequence are 7's.

Number of terms < 10^k: 0, 0, 1, 13, 81, 565, 4027, 30422, 237715, ... - Muniru A Asiru, Jan 09 2018

			47 is in the sequence since pair (47,53) is the first with difference 6 spanning a multiple of 10.

Muniru A Asiru, Table of n, a(n) for n = 1..10000

Cf. A001359, A023201, A023203, A031924, A031925, A031928, A046117, A046132, A158277, A158861, A160370, A160440, A160500, A287049, A287050, A060229, A288021, A288024.

P:=Filtered([1..20000], IsPrime);
P1:=List(Filtered(Filtered(List([1..Length(P)-1],n->[P[n],P[n+1]]),i->i[2]-i[1]=6),j->j[1] mod 5=2),k->k[1]); # Muniru A Asiru, Jul 08 2017

for n from 1 to 2000 do if [ithprime(n+1)-ithprime(n), ithprime(n) mod 5] = [6,2] then print(ithprime(n)); fi; od; # Muniru A Asiru, Jan 19 2018

a288022[n_] := Map[Last, Select[Map[{NextPrime[#, 1], NextPrime[#, -1]}&, Range[10, n, 10]], First[#]-Last[#]==6&]]
a288022[3000] (* data *)

A288024 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 8, and p1, p2 are in different decades.

Original entry on oeis.org

89, 359, 389, 449, 479, 683, 719, 743, 929, 983, 1109, 1163, 1193, 1373, 1439, 1523, 1559, 1733, 1823, 1979, 2003, 2153, 2213, 2243, 2273, 2459, 2609, 2663, 2699, 2843, 2879, 2909, 3209, 3449, 3623, 3719, 4289, 4349, 4583, 4943, 5189, 5399, 5573, 5693, 5783, 5813

Offset: 1

Hartmut F. W. Hoft, Jun 04 2017

nonn

The unit digits of the numbers in the sequence are 3's or 9's.

			89 is in the sequence since pair (89,97) is the first with difference 8 spanning a multiple of 10.

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

Cf. A001359, A023201, A023203, A031924, A031925, A031928, A046117, A046132, A158277, A158861, A160370, A160440, A160500, A287049, A287050, A060229, A288021, A288022.

a288024[n_] := Map[Last, Select[Map[{NextPrime[#, 1], NextPrime[#, -1]}&, Range[10, n, 10]], First[#]-Last[#]==8&]]
a288024[6000] (* data *)
Select[Partition[Prime[Range[800]],2,1],#[[2]]-#[[1]]==8&&IntegerDigits[#[[1]]][[-2]]!= IntegerDigits[ #[[2]]][[-2]]&][[;;,1]] (* Harvey P. Dale, Jan 09 2024 *)

A052158 Lower prime of a difference of 6 (G-minor-6 primes) between consecutive primes of 6k+1 form.

Original entry on oeis.org

31, 61, 73, 151, 157, 271, 331, 367, 373, 433, 541, 571, 601, 607, 727, 733, 751, 991, 1033, 1063, 1117, 1123, 1231, 1291, 1321, 1453, 1543, 1621, 1657, 1741, 1747, 1753, 1777, 1861, 1987, 2011, 2131, 2281, 2287, 2341, 2371, 2383, 2467, 2551, 2671, 2677

Offset: 1

Labos Elemer, Jan 25 2000

nonn

The corresponding larger primes (G-major-6 primes) are also of the form 6k+1.

			a(1)=31 since a(1) + 6 = 37 is the next prime and 31 = 6*5 + 1.

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

Cf. A031924, A031925.

Transpose[Select[Partition[Prime[Range[400]],2,1],Last[#]-First[#] == 6 && Mod[First[#],6]==1&]][[1]] (* Harvey P. Dale, Oct 01 2013 *)

A031924(n) == 1 (mod 6).

cp's OEIS Frontend

A248855 a(n) is the smallest positive integer m such that if k >= m then a(k+1,n)^(1/(k+1)) <= a(k,n)^(1/k), where a(k,n) is the k-th term of the sequence {p | p and p+2n are primes}.

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A346239 Möbius transform of A341512, sigma(n)*A003961(n) - n*sigma(A003961(n)).

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PARI

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A382810 Primes p such that p + 6, p + 10 and p + 16 are also primes.

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A078562 p, p+6 and p+10 are consecutive primes.

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A192175 Array of primes determined by distance to next prime, by antidiagonals.

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Mathematica

A093738 Number of pairs of consecutive prime (p,q) with q-p=6 and q < 10^n.

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UBASIC

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A288021 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 4, and p1, p2 are in different decades.

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A288022 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 6, and p1, p2 are in different decades.

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Maple

A346239 Möbius transform of A341512, sigma(n)A003961(n) - nsigma(A003961(n)).