A031924 -id:A031924 - cp's OEIS frontend

A031925 Upper prime of a difference of 6 between consecutive primes.

29, 37, 53, 59, 67, 79, 89, 137, 157, 163, 173, 179, 239, 257, 263, 269, 277, 337, 359, 373, 379, 389, 439, 449, 509, 547, 563, 569, 577, 593, 599, 607, 613, 653, 659, 683, 733, 739, 757, 947, 953, 977, 983, 997, 1019, 1039, 1069, 1103, 1109

Offset: 1

Jeff Burch

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between

Cf. A031924.

for i from 1 to 186 do if ithprime(i+1) = ithprime(i) + 6 then print({ithprime(i+1)}); fi; od; # Zerinvary Lajos, Mar 19 2007

is(n)=nextprime(n-5)==n && isprime(n-6) \\ Charles R Greathouse IV, Apr 29 2015

A047948 Smallest of three consecutive primes with a difference of 6: primes p such that p+6 and p+12 are the next two primes.

Original entry on oeis.org

47, 151, 167, 251, 257, 367, 557, 587, 601, 647, 727, 941, 971, 1097, 1117, 1181, 1217, 1361, 1741, 1747, 1901, 2281, 2411, 2671, 2897, 2957, 3301, 3307, 3631, 3727, 4007, 4451, 4591, 4651, 4987, 5101, 5107, 5297, 5381, 5387, 5557, 5801, 6067, 6257, 6311, 6317

Offset: 1

Enoch Haga

Let p(k) be the k-th prime; sequence gives p(k) such that p(k+2) - p(k+1) = p(k+1) - p(k) = 6.

			47 is a term as the next two primes are 53 and 59.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Subsequence of A031924.
A033451 (four consecutive primes with difference 6) is a subsequence.
Cf. A078853, A046789.
Cf. A001223, A033451, A052197, A052198.

ok[p_] := (q = NextPrime[p]) == p+6 && NextPrime[q] == q+6; Select[Prime /@ Range[1000], ok][[;; 45]] (* Jean-François Alcover, Jul 11 2011 *)
Transpose[Select[Partition[Prime[Range[1000]],3,1],Differences[#]=={6,6}&]] [[1]] (* Harvey P. Dale, Apr 25 2014 *)

is_A047948(n)={nextprime(n+1)==n+6 && nextprime(n+7)==n+12 && isprime(n)} \\ Charles R Greathouse IV, Aug 17 2011, simplified by M. F. Hasler, Jan 13 2013

p=2;q=3;forprime(r=5,1e4,if(r-p==12&&q-p==6,print1(p", "));p=q;q=r) \\ Charles R Greathouse IV, Aug 17 2011

Corrected by T. D. Noe, Mar 07 2008

A320701 Indices of primes followed by a gap (distance to next larger prime) of 6.

Original entry on oeis.org

9, 11, 15, 16, 18, 21, 23, 32, 36, 37, 39, 40, 51, 54, 55, 56, 58, 67, 71, 73, 74, 76, 84, 86, 96, 100, 102, 103, 105, 107, 108, 110, 111, 118, 119, 123, 129, 130, 133, 160, 161, 164, 165, 167, 170, 174, 179, 184, 185, 187, 188, 194, 195, 199, 200, 202, 208, 210, 216, 218, 219, 227, 231

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes given in A031924.

Subsequence of indices of sexy primes A023201.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between.

Equals A000720 o A031924.
Row 3 of A174349.
Indices of 6's in A001223.
Cf. A029707, A029709, A320702, A320703, ..., A320720 (analog for gaps 2, 4, 8, 10, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

Position[Differences[Prime[Range[250]]],6]//Flatten (* Harvey P. Dale, Oct 13 2022 *)

A(N=100,g=6,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(A031924(n)).

A320701 = { i > 0 | prime(i+1) = prime(i) + 6 } = A001223^(-1)({6}).

A052380 a(n) = D is the smallest distance (D) between 2 non-overlapping prime twins differing by d=2n; these twins are [p,p+d] or [p+D,p+D+d] and p > 3.

Original entry on oeis.org

6, 6, 6, 12, 12, 12, 18, 18, 18, 24, 24, 24, 30, 30, 30, 36, 36, 36, 42, 42, 42, 48, 48, 48, 54, 54, 54, 60, 60, 60, 66, 66, 66, 72, 72, 72, 78, 78, 78, 84, 84, 84, 90, 90, 90, 96, 96, 96, 102, 102, 102, 108, 108, 108, 114, 114, 114, 120, 120, 120, 126, 126, 126, 132

Offset: 1

Labos Elemer, Mar 13 2000

For d=D the quadruple of primes becomes a triple: [p,p+d],[p+d,p+2d].
Without the p > 3 condition, a(1)=2.
The starter prime p, is followed by a prime d-pattern of [d,D-d,d], where D-d=a(n)-2n is 4,2 or 0; these d-patterns are as follows: [2,4,2], [4,2,4], [6,6], [8,4,8], [10,2,10], [12,12], etc.
All terms of this sequence have digital root 3, 6 or 9. - J. W. Helkenberg, Jul 24 2013
a(n+1) is also the number of the circles added at the n-th iteration of the pattern generated by the construction rules: (i) At n = 0, there are six circles of radius s with centers at the vertices of a regular hexagon of side length s. (ii) At n > 0, draw a circle with center at each boundary intersection point of the figure of the previous iteration. The pattern seems to be the flower of life except at the central area. See illustration. - Kival Ngaokrajang, Oct 23 2015

			n=5, d=2n=10, the minimal distance for 10-twins is 12 (see A031928, d=10) the smallest term in A053323. It occurs first between twins of [409,419] and [421,431]; see 409 = A052354(1) = A052376(1) = A052381(5).

Kival Ngaokrajang, Illustration of initial terms
Sacred Geometry, Flower of life

Cf. A001223, A031924-A031938, A053319-A053331, A052350-A052358, A008620.

Table[2 n + 4 - 2 Mod[n + 2, 3], {n, 66}] (* Michael De Vlieger, Oct 23 2015 *)

vector(200, n, n--; 6*(n\3+1)) \\ Altug Alkan, Oct 23 2015

a(n) = 6*ceiling(n/3) = 6*ceiling(d/6) = D = D(n).
a(n) = 2n + 4 - 2((n+2) mod 3). - Wesley Ivan Hurt, Jun 30 2013
a(n) = 6*A008620(n-1). - Kival Ngaokrajang, Oct 23 2015

A052381 The smallest initial prime of 2 non-overlapping d-twin primes if the distance between pairs (D) is minimal (see A052380).

Original entry on oeis.org

3, 7, 47, 389, 409, 199, 24749, 3373, 20183, 46703, 19867, 16763, 142811, 14563, 69593, 763271, 276637, 255767, 363989, 383179, 247099, 2130809, 15370423, 3565931, 458069, 9401647, 6314393, 20823437, 9182389, 4911251, 15442121

Offset: 1

Labos Elemer, Mar 13 2000

nonn

A prime quadruple (triple), {[p,p+d],[p+D,p+D+d]} is called a "non-overlapping" (disjoint or touching) pair of twins if D = distance >= d = difference "inside" twin.

			If n=23, d=46, min{D}=48 then the first suitable quadruple of primes is [15370423, 15370469, 15370471, 15370517] with difference pattern [46, 2, 46]; if n=3, d=6, min{D}=6 then the first such triple is [47, 53, 53, 59] = [47, 53, 59] with difference pattern [6, 6].

The first 10 terms here appear as initial terms in A052350-A052359.

Smallest p so that [p, p+2n], [p+min{D}, p+2n+min{D}] is a quadruple (or triple if d=min{D}) of consecutive primes.

Corrected by Jud McCranie, Jan 04 2001

a(11) corrected by Sean A. Irvine, Nov 07 2021

A341512 a(n) = A341529(n) - A341528(n) = (sigma(n)A003961(n)) - (nsigma(A003961(n))).

Original entry on oeis.org

0, 1, 2, 11, 2, 36, 4, 85, 46, 58, 2, 324, 4, 120, 120, 575, 2, 693, 4, 566, 248, 172, 6, 2340, 94, 270, 788, 1176, 2, 1800, 6, 3661, 348, 358, 336, 5967, 4, 492, 548, 4210, 2, 3744, 4, 1820, 2490, 744, 6, 15372, 380, 2271, 720, 2826, 6, 11304, 392, 8760, 992, 946, 2, 15480, 6, 1232, 5164, 22631, 636, 5904, 4, 3866

Offset: 1

Antti Karttunen, Feb 22 2021

Cf. A000203, A000720, A001223, A003961, A003973, A286385, A341528, A341529, A341530.

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

Array[#2 DivisorSigma[1, #1] - #1 DivisorSigma[1, #2] & @@ {#, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1]} &, 68] (* Michael De Vlieger, Feb 22 2021 *)

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));

a(n) = A341529(n) - A341528(n) = (sigma(n)*A003961(n)) - (n*sigma(A003961(n))).
For all primes p, a(p) = (q*(p+1)) - (p*(q+1)) = (pq + q) - (pq + p) = q - p = A001223(A000720(p)), where q = nextprime(p) = A003961(p).
And in general, a(p^e) = (q^e * (p^(e+1)-1)/(p-1)) - ((p^e) * (q^(e+1)-1)/(q-1)), where q = A003961(p).
Thus, a(p^2) = (p + 1)*q^2 - p^2*q - p^2,
      a(p^3) = (p^2 + p + 1)*q^3 - p^3*q^2 - p^3*q - p^3,
      a(p^4) = (p^3 + p^2 + p + 1)*q^4 - p^4*q^3 - p^4*q^2 - p^4*q - p^4,
      etc.

A049438 p, p+6 and p+8 are all primes (A046138) but p+2 is not.

Original entry on oeis.org

23, 53, 131, 173, 233, 263, 563, 593, 653, 1013, 1223, 1283, 1601, 1613, 2333, 2543, 2963, 3323, 3533, 3761, 3911, 3923, 4013, 4211, 4253, 4643, 4793, 5003, 5273, 5471, 5843, 5861, 6263, 6353, 6563, 6653, 6863, 7121, 7451, 7481, 7541, 7583

Offset: 1

Labos Elemer

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

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

Cf. A007530, A023202, A031926, A046134, A046138, A049436, A049437.

Select[Prime@ Range[10^3], MatchQ[Boole@ PrimeQ@ {# + 2, # + 6, # + 8}, {0, 1, 1}] &] (* Michael De Vlieger, Feb 05 2017 *)

isok(p) = isprime(p) && !isprime(p+2) && isprime(p+6) && isprime(p+8); \\ Michel Marcus, Dec 13 2013

A098974 Primes p such that q-p = 24, where q is the next prime after p.

Original entry on oeis.org

1669, 2179, 4177, 4523, 4759, 5237, 6173, 6397, 6737, 7079, 7369, 7793, 8123, 8329, 9067, 11003, 11633, 11839, 12073, 12119, 13009, 13267, 16033, 16193, 16453, 16763, 16787, 17053, 17683, 17989, 18593, 18637, 19183, 19507, 20483, 22409, 22877, 23227

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Oct 23 2004

Lower prime of a difference of 24 between consecutive primes.

23 successive numbers after prime number p are composite. - Artur Jasinski, Jan 15 2007

Remi Eismann, Table of n, a(n) for n = 1..10000
K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc., 44 (2007), 1-18.
Index entries for primes, gaps between

Cf. A000040, A001223, A054541, A075526, A001359, A054799, A063091, A096292, A015913, A023200, A029710, A031934, A031936, A031938.

Cf. A000230, A023200, A031924, A031926, A031928, A031930, A031932, A061779.

a = {}; Do[If[Prime[x + 1] - Prime[x] == 24, AppendTo[a, Prime[x]]], {x, 1, 10000}]; a (* Artur Jasinski, Jan 15 2007 *)

Entry revised by N. J. A. Sloane, Feb 13 2007

A126784 Primes p such that q-p = 32, where q is the next prime after p.

Original entry on oeis.org

5591, 10799, 27701, 27851, 33647, 39047, 41081, 41687, 43721, 44417, 45989, 47459, 50789, 52457, 55259, 55547, 61781, 62351, 64817, 66239, 67307, 69959, 73907, 79907, 80567, 82307, 84089, 88037, 94169, 94961, 99191, 99929, 100559, 102611

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Feb 24 2007

Lower prime of a difference of 32 between consecutive primes.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001359, A029710, A031924, A031926, A031928, A031930, A031932, A031934, A031936, A031938, A061779, A098974, A124594, A124595, A124596, A000230, A050434.

lista(nn) = {p = 2; while (p < nn, q = nextprime(p+1); if (q - p == 32, print1(p, ", ")); p = q;);} \\ Michel Marcus, Jul 17 2013

A160370 Smaller member p of a pair (p,p+6) of consecutive primes in different centuries.

Original entry on oeis.org

1097, 2897, 3797, 4597, 5297, 5897, 9397, 11497, 11897, 12197, 12497, 12697, 15797, 16097, 18797, 19597, 21997, 24097, 24197, 28597, 28697, 29297, 30097, 30197, 30697, 32497, 35597, 36997, 39097, 40897, 41597, 41897, 42397, 45497, 47297

Offset: 1

Ki Punches, May 11 2009

nonn

Note that the smaller member of a pair of sexy primes with the same constraint on centuries defines a different sequence, since members of a sexy prime pair do not need to be *consecutive* primes.
The larger member in the pair is obtained by adding 6 to an entry.
Every a(n)+3 is a multiple of 100 such that neither a(n)+2 nor a(n)+4 are primes. It appears that every integer occurs as the difference round((a(n+1)-a(n))/100); all numbers 1..333 occur as these differences for a(n) < 1000000000. - Hartmut F. W. Hoft, May 18 2017

			30097 + 6 = 30103.

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A158277, A046117, A023201.

Transpose[Select[Partition[Prime[Range[5000]],2,1],#[[2]]-#[[1]] == 6 && Floor[#[[1]]/100]!=Floor[#[[2]]/100]&]][[1]] (* Harvey P. Dale, Apr 28 2012 *)
a160370[n_] := Select[Range[97, n, 100], AllTrue[# + {0, 6}, PrimeQ] && NoneTrue[# + {2, 4}, PrimeQ]&]
a160370[49000] (* data *) (* Hartmut F. W. Hoft, May 18 2017 *)

{A031924(n): [A031924(n)/100] <> [A031925(n)/100]} where [..]=floor(..).

Edited by R. J. Mathar, May 14 2009

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A031925 Upper prime of a difference of 6 between consecutive primes.

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A047948 Smallest of three consecutive primes with a difference of 6: primes p such that p+6 and p+12 are the next two primes.

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A320701 Indices of primes followed by a gap (distance to next larger prime) of 6.

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A052380 a(n) = D is the smallest distance (D) between 2 non-overlapping prime twins differing by d=2n; these twins are [p,p+d] or [p+D,p+D+d] and p > 3.

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A052381 The smallest initial prime of 2 non-overlapping d-twin primes if the distance between pairs (D) is minimal (see A052380).

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A341512 a(n) = A341529(n) - A341528(n) = (sigma(n)*A003961(n)) - (n*sigma(A003961(n))).

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A049438 p, p+6 and p+8 are all primes (A046138) but p+2 is not.

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A098974 Primes p such that q-p = 24, where q is the next prime after p.

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A341512 a(n) = A341529(n) - A341528(n) = (sigma(n)A003961(n)) - (nsigma(A003961(n))).