A031925 -id:A031925 - cp's OEIS frontend

A173198 Number of pairs of sexy consecutive primes between (A031924(n))^2 and A031924(n)*A031925(n).

10, 10, 12, 8, 11, 14, 12, 15, 18, 19, 21, 21, 25, 31, 19, 23, 32, 29, 27, 28, 43, 36, 36, 35, 42, 51, 52, 46, 43, 53, 45, 55, 41, 55, 51, 46, 71, 52, 66, 60, 54, 62, 75, 66, 56, 67, 91, 65, 78, 75, 77, 97, 62, 80, 90, 81, 68, 78, 89, 99, 86, 90, 98, 98, 106, 96, 90, 84, 105, 89

Offset: 1

Jaspal Singh Cheema, Feb 12 2010

nonn

If you graph a(n) versus n, a clear pattern emerges.
As you go farther along the n-axis, greater are the number of consecutive sexy primes, on average, within each interval obtained.
If one could prove that there is at least one consecutive sexy prime within each interval, this would imply that consecutive sexy primes are infinite.
I suspect all numbers in the sequence are > 0.

			The first sexy prime pair with consecutive primes is (23,29) = A031924(1) and A031925(1). Square the first term, you get 529, then take the product of the two primes, you get 667.
Between these two numbers, namely (529,667), there are ten consecutive sexy primes: (541,547), (557,563), (563,569),
(571,577), (587,593), (593,599), (601,607), (607,613), (647,653), and (653 659).
Hence the very first term of the sequence is 10.

J. S. Cheema, Table of n, a(n) for n = 1..1762
Rick Aster, Prime number sieve SAS prime sieve program
Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021].
Wikipedia, Sexy Primes

Cf. A023201

isA031924 := proc(p) return (isprime(p) and (nextprime(p)-p) = 6 ); end proc:
A031924 := proc(n) local p; if n = 1 then 23; else p := nextprime(procname(n-1)) ; while not isA031924(p) do p := nextprime(p) ; end do ; return p ; end if ; end proc:
A031925 := proc(n) A031924(n)+6 ; end proc:
A173198 := proc(n) local ulim,llim,a,i ; llim := A031924(n)^2 ; ulim := A031924(n)*A031925(n) ; a := 0 ; for i from llim to ulim-6 do if isA031924(i) then a := a+1 ; end if; end do ; a ; end proc:
seq(A173198(n),n=1..80) ; # R. J. Mathar, Feb 15 2010

Comments condensed by R. J. Mathar, Feb 15 2010

A052247 Maximal value of smallest prime divisors of the 5 composite numbers between A031924(n) and A031925(n).

Original entry on oeis.org

5, 5, 7, 5, 5, 7, 5, 7, 5, 7, 13, 5, 5, 11, 7, 5, 5, 5, 5, 7, 13, 5, 19, 5, 5, 5, 13, 5, 5, 19, 5, 5, 13, 11, 5, 7, 17, 11, 5, 23, 13, 7, 11, 5, 5, 17, 11, 7, 5, 19, 7, 7, 29, 23, 5, 5, 5, 5, 5, 29, 37, 5, 31, 5, 7, 5, 7, 5, 5, 11, 5, 17, 7, 13, 5, 11, 23, 5, 11, 5, 5, 5, 47, 5, 29, 5, 5, 13, 5

Offset: 1

Labos Elemer, Feb 01 2000

nonn

			The 5 composites between A031924(1) = 23 and A031925(1) = 29 are 24, 25, 26, 27, 28; their smallest prime divisors are 2 5 2 3 2; the maximal value is 5; so a(1) = 5.

A031924, A031925, A052158.

A031924 Primes followed by a gap of 6, i.e., next prime is p + 6.

Original entry on oeis.org

23, 31, 47, 53, 61, 73, 83, 131, 151, 157, 167, 173, 233, 251, 257, 263, 271, 331, 353, 367, 373, 383, 433, 443, 503, 541, 557, 563, 571, 587, 593, 601, 607, 647, 653, 677, 727, 733, 751, 941, 947, 971, 977, 991, 1013, 1033, 1063, 1097, 1103, 1117, 1123, 1181

Offset: 1

Jeff Burch

nonn

Original name: Lower prime of a difference of 6 between consecutive primes.

Conjecture: The sequence is infinite and for every n >= 7746, a(n+1) < a(n)^(1+1/n). Namely for n >= 7746, a(n)^(1/n) is a strictly decreasing function of n (See comment lines of the sequence A248855). - Jahangeer Kholdi and Farideh Firoozbakht, Nov 29 2014

			23 is a term as the next prime 29 = 23 + 6.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
OEIS wiki, Consecutive primes in arithmetic progression: CPAP with given gap, updated Jan. 2020
Index entries for primes, gaps between

Cf. A001359, A023201, A031925; A031924 and A007529 together give A023201.

P:=Filtered([1..1200],IsPrime);;
List(Filtered([1..Length(P)-1],i->P[i+1]-P[i]=6),k->P[k]); # Muniru A Asiru, Jan 30 2019

[p: p in PrimesUpTo(1200) | NextPrime(p)-p eq 6]; // Bruno Berselli, Apr 09 2013

A031924 := proc(n)
    option remember;
    if n = 1 then
        return 23;
    else
        p := nextprime(procname(n-1)) ;
        q := nextprime(p) ;
        while q-p <> 6 do
            p := q ;
            q := nextprime(p) ;
        end do:
        return p;
    end if;
end proc: # R. J. Mathar, Jan 23 2013

Transpose[Select[Partition[Prime[Range[200]], 2, 1], Last[#] - First[#] == 6 &]][[1]] (* Bruno Berselli, Apr 09 2013 *)

is(n)=isprime(n)&&nextprime(n+1)-n==6 \\ Charles R Greathouse IV, Mar 21 2013

apply( A031924(n,p=2,show=0,g=6)={forprime(q=p+1,, p+g!=(p=q) || (show&&print1(p-g",")) || n-- || return(p-g))}, [1..99]) \\ Use nxt(p)=A031924(1,p) to get the term following p, use show=1 to print all a(1..n), g to select a different gap. - M. F. Hasler, Jan 02 2020

New name from M. F. Hasler, Jan 02 2020

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

A134100 Primes p > 3 such that neither p-2 nor p-4 are prime.

Original entry on oeis.org

29, 37, 53, 59, 67, 79, 89, 97, 127, 137, 149, 157, 163, 173, 179, 191, 211, 223, 239, 251, 257, 263, 269, 277, 293, 307, 331, 337, 347, 359, 367, 373, 379, 389, 397, 409, 419, 431, 439, 449, 457, 479, 487, 499, 509, 521, 541, 547, 557, 563, 569, 577, 587

Offset: 1

Enoch Haga, Oct 08 2007

Upper primes after a prime gap of 6 or larger (Union of A031925, A031927, A031929, ...) - R. J. Mathar, Mar 15 2012

			29 is a term because 29 follows the odd nonprime 27 which in turn follows the odd nonprime 25.

Bill McEachen, Table of n, a(n) for n = 1..10000

Cf. A007510, A083370, A124582, A134099, A134101, A073830.

Select[Range[5,1000,2],PrimeQ[#]&&!PrimeQ[#-2]&&!PrimeQ[#-4]&] (* Vladimir Joseph Stephan Orlovsky, Feb 03 2012 *)

forprime(p=5,600,if(!isprime(p-2) && !isprime(p-4), print1(p,", "))); \\ Joerg Arndt, Oct 27 2021

list(lim)=my(v=List(),p=23); forprime(q=29,lim, if(q-p>4, listput(v,q)); p=q); Vec(v) \\ Charles R Greathouse IV, Oct 27 2021

a(n) ~ n log n. - Charles R Greathouse IV, Oct 27 2021

Name corrected by Michel Marcus and Amiram Eldar, Oct 27 2021

A031931 Upper prime of a difference of 12 between consecutive primes.

Original entry on oeis.org

211, 223, 479, 521, 631, 673, 809, 1009, 1213, 1249, 1319, 1471, 1511, 1523, 1543, 1721, 1801, 1823, 1901, 2081, 2111, 2309, 2411, 2459, 2591, 2633, 2789, 2939, 3061, 3079, 3181, 3203, 3271, 3343, 3359, 3511, 3571, 3671, 3943, 4001, 4091, 4111, 4409, 4421

Offset: 1

Jeff Burch

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for primes, gaps between

Cf. A031925 (difference of 6).

Select[Partition[Prime[Range[700]],2,1],#[[2]]-#[[1]]==12&][[All,2]] (* Harvey P. Dale, Dec 28 2018 *)

isok(p) = isprime(p) && (p-precprime(p-1) == 12); \\ Michel Marcus, Jun 24 2022

Corrected and extended by Harvey P. Dale, Dec 28 2018

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

A173198 Number of pairs of sexy consecutive primes between (A031924(n))^2 and A031924(n)*A031925(n).

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A052247 Maximal value of smallest prime divisors of the 5 composite numbers between A031924(n) and A031925(n).

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A031924 Primes followed by a gap of 6, i.e., next prime is p + 6.

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

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A134100 Primes p > 3 such that neither p-2 nor p-4 are prime.

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A031931 Upper prime of a difference of 12 between consecutive primes.

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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.