A022006 -id:A022006 - cp's OEIS frontend

A245304 Numbers m such that m+1, m+3, m+7, m+9 and m+13 are all primes.

4, 10, 100, 1480, 16060, 19420, 21010, 22270, 43780, 55330, 144160, 165700, 166840, 195730, 201820, 225340, 247600, 268810, 326140, 347980, 361210, 397750, 465160, 518800, 536440, 633460, 633790, 661090, 768190, 795790, 829720, 857950, 876010, 958540

Offset: 1

Reinhard Zumkeller, Jul 18 2014

nonn

W. Sierpiński, 250 Problems in Elementary Number Theory. New York: American Elsevier, 1970. Problem #82, variant.

Reinhard Zumkeller and Jens Kruse Andersen, Table of n, a(n) for n = 1..1000 (first 120 terms from Zumkeller)

Cf. A010051, A022006, A245305, A007811, subsequence of A125855.

a245304 n = a245304_list !! (n-1)
a245304_list = map (pred . head) $ filter (all (== 1) . map a010051') $
   iterate (zipWith (+) [1, 1, 1, 1, 1]) [1, 3, 7, 9, 13]

[n: n in [0..10^6] | IsPrime(n+1) and IsPrime(n+3) and IsPrime(n+7) and IsPrime(n+9) and IsPrime(n+13)]; // Vincenzo Librandi, Jun 15 2015

Select[Range[10^6],AllTrue[#+{1,3,7,9,13},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 07 2015 *)

forprime(p=2, 10^7, m=p-1; if(isprime(m+3)&&isprime(m+7)&&isprime(m+9)&&isprime(m+13), print1(m", "))) \\ Jens Kruse Andersen, Jul 18 2014

a(n) = A022006(n)-1. - Jens Kruse Andersen, Jul 18 2014

A350825 Number of prime 5-tuples with initial member (A086140) between 10^(n-1) and 10^n.

Original entry on oeis.org

2, 2, 1, 4, 12, 44, 256, 1062, 5838

Offset: 1

M. F. Hasler, Mar 01 2022

"Between 10^(n-1) and 10^n" is equivalent to saying "with n digits".
For n = 1 and n = 2, the last term of the last 5-tuple in that range (cf EXAMPLE) has one digit more than the initial term.
Terms a(1)-a(9) computed from b-files a(1..10000) for A022006 and A022007.

			a(1) = 2 because there are just two single-digit primes to start a prime 5-tuple, namely 5 = A022006(1) and 7 = A022007(1).
a(2) = 2 because 11 = A022006(2) and 97 = A022007(2) are the only two two-digit primes to start a prime 5-tuple.
a(3) = 1 because there is only one three-digit prime to start a prime 5-tuple, namely 101 = A022006(3).
Then there are a(4) = 4 four-digit primes, 1481, 1867, 3457 and 5647, which start a prime 5-tuple.

Cf. A086140 (initial members p of prime quintuplets), A022006, A022007 (idem, specifically for patterns (p, p+2, ...) resp. (p, p+4, ...)).

Cf. A350826, A350827, A350828: similar for sextuplets, septuplets and octuplets.

(D(v)=v[^1]-v[^-1])( [setsearch(A086140, 10^n, 1) | n<-[0..9]] ) \\ where A086140 is a vector of at least 7221 terms of that sequence.

A078947 Primes p such that the differences between the 5 consecutive primes starting with p are (2,4,6,6).

Original entry on oeis.org

41, 641, 1091, 4001, 9461, 26681, 26711, 44531, 79811, 103991, 110921, 112571, 172421, 223241, 276821, 289841, 290021, 317771, 373181, 381371, 434921, 450881, 493121, 602081, 678761, 788351, 834131, 907211, 974861, 1076501, 1081121, 1097891, 1200371, 1409531, 1426151

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+2, p+6, p+12 and p+18 are consecutive primes.

			641 is in the sequence since 641, 643 = 641 + 2, 647 = 641 + 6, 653 = 641 + 12 and 659 = 641 + 18 are consecutive primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A078847.

Cf. A001223, A078866, A078867, A078946-A078971, A022006, A022007.

Select[Partition[Prime[Range[50000]], 5, 1], Differences[#] == {2, 4, 6, 6} &][[;;, 1]] (* Amiram Eldar, Feb 21 2025 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 2 && p3 - p2 == 4 && p4 - p3 == 6 && p5 - p4 == 6, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 21 2025

a(n) == 11 (mod 30). - Amiram Eldar, Feb 21 2025

Edited by Dean Hickerson, Dec 20 2002

A201074 Initial primes in prime 5-tuples (p, p+2, p+6, p+8, p+12) preceding the maximal gaps in A201073.

Original entry on oeis.org

5, 11, 101, 1481, 22271, 55331, 536441, 661091, 1461401, 1615841, 5527001, 11086841, 35240321, 53266391, 72610121, 92202821, 117458981, 196091171, 636118781, 975348161, 1156096301, 1277816921, 1347962381, 2195593481, 3128295551

Offset: 1

Alexei Kourbatov, Nov 26 2011

nonn

Prime quintuplets (p, p+2, p+6, p+8, p+12) are one of the two types of densest permissible constellations of 5 primes. Maximal gaps between quintuplets of this type are listed in A201073; see more comments there.

			The initial four gaps of 6, 90, 1380, 14580 (starting at p=5, 11, 101, 1481) form an increasing sequence of records. Therefore a(1)=5, a(2)=11, a(3)=101, and a(4)=1481. The next gap is smaller, so a new term is not added.

Alexei Kourbatov, Table of n, a(n) for n = 1..64
Tony Forbes, Prime k-tuplets
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
Alexei Kourbatov, Maximal gaps between prime 5-tuples (graphs/data up to 10^15)
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
Eric W. Weisstein, k-Tuple Conjecture

Cf. A022006 (prime 5-tuples p, p+2, p+6, p+8, p+12), A201073, A233432.

A233432 Primes p in prime 5-tuples (p, p+2, p+6, p+8, p+12) at the end of the maximal gaps in A201073.

Original entry on oeis.org

11, 101, 1481, 16061, 43781, 144161, 633461, 768191, 1573541, 1917731, 5928821, 11664551, 35930171, 54112601, 73467131

Offset: 1

Alexei Kourbatov, Dec 09 2013

nonn

Prime quintuplets (p, p+2, p+6, p+8, p+12) are one of the two types of densest permissible constellations of 5 primes. Maximal gaps between quintuplets of this type are listed in A201073; see more comments there.

			The initial four gaps of 6, 90, 1380, 14580 (ending at p=11, 101, 1481, 16061) form an increasing sequence of records. Therefore a(1)=11, a(2)=101, a(3)=1481 and a(4)=16061. The next gap is not a record, so a new term is not added.

Alexei Kourbatov, Table of n, a(n) for n = 1..64
Tony Forbes, Prime k-tuplets
Alexei Kourbatov, Maximal gaps between prime 5-tuples
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053, 2013.
Eric W. Weisstein, k-Tuple Conjecture

Cf. A022006, A201073, A201074.

A270998 Table read by rows: list of prime 5-tuples of the form (p, p+2, p+6, p+8, p+12).

Original entry on oeis.org

5, 7, 11, 13, 17, 11, 13, 17, 19, 23, 101, 103, 107, 109, 113, 1481, 1483, 1487, 1489, 1493, 16061, 16063, 16067, 16069, 16073, 19421, 19423, 19427, 19429, 19433, 21011, 21013, 21017, 21019, 21023, 22271, 22273, 22277, 22279, 22283, 43781, 43783, 43787, 43789, 43793

Offset: 1

Arkadiusz Wesolowski, Jul 12 2016

A prime 5-tuple is a constellation of five successive primes with distance 12, and is of the form (p, p+2, p+6, p+8, p+12) or (p, p+4, p+6, p+10, p+12).
Initial members p (other than 5) of prime 5-tuples of the form (p, p+2, p+6, p+8, p+12) are congruent to 11 or 101 (mod 210).
Also called prime 5-tuples of the first kind.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
C. K. Caldwell, Top Twenty page, Quintuplet
Eric Weisstein's World of Mathematics, Prime Constellation
Wikipedia, Prime quadruplet
Index entries for primes, gaps between

Cf. A022006, A270999.

lst:=[]; for p in [5..43781 by 2] do if p eq 5 xor p mod 210 in {11, 101} then if IsPrime(p) then t:=[c: c in [p+2..p+12] | IsPrime(c)]; if #t eq 4 then lst:=lst cat [p] cat t; end if; end if; end if; end for; lst;

m = {0, 2, 6, 8, 12}; Union@ Flatten@ Map[# + m &, Select[Prime@ Range[10^4], Times @@ Boole@ PrimeQ[# + m] == 1 &]] (* Michael De Vlieger, Jul 13 2016 *)
Select[Partition[Prime[Range[5000]],5,1],Differences[#]=={2,4,2,4}&]// Flatten (* Harvey P. Dale, Jul 27 2020 *)

a(5*n-4) = A022006(n).

A343635 10^n + a(n) is the least (n+1)-digit prime member of a prime 5-tuple, or a(n) = 0 if no such number exists.

Original entry on oeis.org

4, 1, 1, 481, 5727, 1107, 8851, 18027, 5457, 408807, 57931, 358531, 274587, 256497, 6111627, 67437, 3246567, 1638811, 8224977, 11687221, 24556351, 3129657, 15602131, 571381, 23034391, 110598987, 26716321, 31722117, 39711931, 5046777, 81054327, 1346611, 44656587

Offset: 0

M. F. Hasler, Jul 17 2021

The smallest (n+1)-digit prime 5-tuple is given by 10^n + a(n) + D, with either D = {0, 2, 6, 8, 12} or D = {0, 4, 6, 10, 12}. N = 0 is the only case where the last member of the 5-tuple has one digit more than the first member.
Numerical evidence strongly suggests the conjecture that 0 < a(n) < 10^n for all n > 0, but not even the existence of infinitely many prime 5-tuples is proved.
Some further isolated terms, due to Norman Luhn et al., giving the start of the smallest 500, 600, 700, ..., 1200 digit quintuplets of first or second type:
  a(499) = min(58195471283341, 69672492141807),
  a(599) = min(319491304676641, 12754947401547),
  a(699) = min(2254633393747621, 209264286017367),
  a(799) = min(2117758391972791, 1299258655252617),
  a(899) = min(2365663735968811, 1484244113736867),
  a(999) = min(3554007760224751, 3818999670116007),
  a(1099) = min(26317044823878361, 15720821612555937),
  a(1199) = min(20483870459152351, 7033048489975137).
Terms through a(399) may be determined by taking the minima of those in the linked tables for quintuplets by Norman Luhn et al. - Michael S. Branicky, Jul 24 2021
The first member of the quintuplets of the first type always ends in digit 1 (except for the 5-tuple (5, 7, 11, 13, 17) corresponding to a(0)), for the second type it always ends in digit 7. Therefore all a(n), n > 0, end in a digit 1 or 7, which indicates the type of the 5-tuple, i.e., the set D that has to be added to 10^n + a(n) to get the whole 5-tuple. - M. F. Hasler, Aug 04 2021

			a(0) = 4 because {5, 7, 11, 13, 17} is the smallest prime 5-tuple and it starts with the single-digit prime 10^0 + a(0) = 5 = A022006(1).
a(1) = 1 because 10^1 + 1 = 11 = A022006(2) is the 2-digit prime to start a prime 5-tuple {11, 13, 17, 19, 23}, again of the first type.
a(2) = 1 and a(3) = 481 because 10^2 + 1 = 101 = A022006(3) and 10^3 + 481 = 1481 = A022006(4) are the smallest 3-digit, resp. 4-digit, initial members of a prime 5-tuple, both again of the first type.
a(4) = 5727 because 10^4 + 5727 = 15727 = A022007(6) is the smallest 5-digit initial member of a prime 5-tuple, now of the second type.
It appears that for all n > 0, a(n) < 10^n, so that the primes are of the form 10...0XXX where XXX = a(n) and 0...0 stands for a string of zero or more digits 0.

M. F. Hasler, Table of n, a(n) for n = 0..399 (terms up to a(51) from Michael S. Branicky), Aug 04 2021.
Norman Luhn, Primzahltupel, prime k-tuple: Smallest-n-digit-prime-quintuplets, on mathematikalpha.de, 2020. Tables.

Cf. A022006 and A022007 (initial members of prime 5-tuples of first and second type).

Cf. A343636, A343637 (analog for sextuplets and septuplets).

apply( {A343635(n,q=[1..4],i=0)=forprime(p=10^n,, (q[1+i]+12==q[i++]=p) && return(p-12-10^n); i>3 && i=0)}, [0..15]) \\ Shorter but slightly slower (?)

apply( {A343635(n, i=ispseudoprime, q)=forprime(p=10^n,, i(p+12) && i(p+6) && (p+6 > q=nextprime(p+2)) && i(q+6) && return(p-10^n))}, [0..15])

from sympy import nextprime
def a(n):
    p = nextprime(10**n)
    q = nextprime(p); r = nextprime(q); s = nextprime(r); t = nextprime(s)
    while p < 10**(n+1):
        if t - p == 12: return p - 10**n
        p, q, r, s, t = q, r, s, t, nextprime(t)
    return 0
print([a(n) for n in range(14)]) # Michael S. Branicky, Jul 24 2021

A376136 Primes p_1 where products m of k = 5 consecutive primes p_1..p_k are such that only p_1 < m^(1/k).

Original entry on oeis.org

3229, 3271, 4759, 6173, 6803, 6917, 8389, 8971, 9439, 10433, 11743, 12011, 12853, 12983, 13967, 14107, 14593, 15683, 16033, 16141, 18013, 18097, 19183, 19333, 21283, 21347, 21529, 22573, 22817, 23633, 23719, 25261, 27701, 27919, 28229, 29537, 30593, 31397, 31699

Offset: 1

Michael De Vlieger, Sep 17 2024

nonn

Primes p_1 are such that the difference p_2-p_1 is larger than the sum of the differences p_(j+1)-p_j for j < k.

Does not intersect A022006 or A022007.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A022006, A022007, A376261.

k = 5; s = {1}~Join~Prime[Range[k - 1]]; Reap[Do[s = Append[Rest[s], Prime[i + k - 1]]; r = Surd[Times @@ s, k]; If[Count[s, _?(# < r &)] == 1, Sow[Prime[i]] ], {i, 32000}]][[-1, 1]]

A078948 Primes p such that the differences between the 5 consecutive primes starting with p are (2,6,4,2).

Original entry on oeis.org

29, 59, 269, 1289, 2129, 2789, 5639, 8999, 13679, 14549, 18119, 36779, 62129, 75989, 80669, 83219, 88799, 93479, 113159, 115769, 124769, 132749, 150209, 160079, 163979, 203309, 207509, 223829, 228509, 278489, 282089, 284729, 298679, 312929, 313979, 323369, 337859

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+2, p+8, p+12 and p+14 are consecutive primes.

All terms are congruent to 29 (mod 30). - Muniru A Asiru, Sep 04 2017

			59 is in the sequence since 59, 61 = 59 + 2, 67 = 59 + 8, 71 = 59 + 12 and 73 = 59 + 14 are consecutive primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from R. J. Mathar)
R. J. Mathar, Table of Prime Gap Constellations.

Subsequence of A078848. - R. J. Mathar, Feb 10 2013

Cf. A001223, A078866, A078867, A078946-A078971, A022006, A022007.

K:=26*10^7+1;; # to get all terms <= K.
P:=Filtered([1,3..K],IsPrime);;  I:=[2,6,4,2];;
P1:=List([1..Length(P)-1],i->P[i+1]-P[i]);;
Q:=List(Positions(List([1..Length(P)-Length(I)],i->[P1[i],P1[i+1],P1[i+2],P1[i+3]]),I),i->P[i]); # Muniru A Asiru, Sep 04 2017

for i from 1 to 10^5 do if [ithprime(i+1),ithprime(i+2),ithprime(i+3),ithprime(i+4)] = [ithprime(i)+2,ithprime(i)+8,ithprime(i)+12,ithprime(i)+14] then print(ithprime(i)); fi; od;  # Muniru A Asiru, Sep 04 2017

Select[Partition[Prime[Range[26000]],5,1],Differences[#]=={2,6,4,2}&][[;;,1]] (* Harvey P. Dale, Dec 10 2024 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 2 && p3 - p2 == 6 && p4 - p3 == 4 && p5 - p4 == 2, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 21 2025

Edited by Dean Hickerson, Dec 20 2002

A078949 Primes p such that the differences between the 5 consecutive primes starting with p are (2,6,4,6).

Original entry on oeis.org

71, 431, 2339, 2381, 5849, 6959, 27791, 32561, 41609, 45119, 46439, 48479, 51419, 54401, 63599, 78779, 81551, 106859, 115319, 130631, 138569, 143501, 153269, 166601, 183569, 196169, 204359, 229751, 246929, 266081, 279119, 321311, 326999, 350729, 357659, 362741

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+2, p+8, p+12 and p+18 are consecutive primes.

			71 is in the sequence since 71, 73 = 71 + 2, 79 = 71 + 8, 83 = 71 + 12 and 89 = 71 + 18 are consecutive primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A078848. - R. J. Mathar, Feb 10 2013

Cf. A001223, A078866, A078867, A078946-A078971, A022006, A022007.

Select[Partition[Prime[Range[50000]], 5, 1], Differences[#] == {2, 6, 4, 6} &][[;;, 1]] (* Amiram Eldar, Feb 21 2025 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 2 && p3 - p2 == 6 && p4 - p3 == 4 && p5 - p4 == 6, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 21 2025

From Amiram Eldar, Feb 21 2025: (Start)
a(n) == 5 (mod 6).
a(n) == 11 or 29 (mod 30). (End)

Edited by Dean Hickerson, Dec 20 2002

cp's OEIS Frontend

A245304 Numbers m such that m+1, m+3, m+7, m+9 and m+13 are all primes.

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A350825 Number of prime 5-tuples with initial member (A086140) between 10^(n-1) and 10^n.

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A078947 Primes p such that the differences between the 5 consecutive primes starting with p are (2,4,6,6).

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A201074 Initial primes in prime 5-tuples (p, p+2, p+6, p+8, p+12) preceding the maximal gaps in A201073.

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A233432 Primes p in prime 5-tuples (p, p+2, p+6, p+8, p+12) at the end of the maximal gaps in A201073.

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A270998 Table read by rows: list of prime 5-tuples of the form (p, p+2, p+6, p+8, p+12).

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Magma

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A343635 10^n + a(n) is the least (n+1)-digit prime member of a prime 5-tuple, or a(n) = 0 if no such number exists.

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A376136 Primes p_1 where products m of k = 5 consecutive primes p_1..p_k are such that only p_1 < m^(1/k).

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