A077800 -id:A077800 - cp's OEIS frontend

A079164 Twin-primorial numbers: running products of twin primes.

3, 15, 75, 525, 5775, 75075, 1276275, 24249225, 703227525, 21800053275, 893802184275, 38433493923825, 2267576141505675, 138322144631846175, 9820872268861078425, 716923675626858725025, 72409291238312731227525

Offset: 1

Cino Hilliard, Feb 03 2003

nonn

The sum of the reciprocals converges to 0.4154254016622336549103692152614908366885449298862362851444631680740051...

			The first two twin primes are 3 and 5, so the first term is 3 and the second term is 15.  The next two twin primes are 5 and 7, so the third term is 5*15=75 and the fourth term is 75*7=525

Amiram Eldar, Table of n, a(n) for n = 1..309

Partial products of A077800.

Rest[FoldList[Times,1,Flatten[Select[Partition[Prime[Range[30]], 2,1], Last[#]-First[#]==2&]]]] (* Harvey P. Dale, Mar 16 2011 *)

twprfact(n) = {sr=0; tp = vector(10000); k=1; forprime(j = 3,n, if(nextprime(j+1)-j == 2, tp[k] = j; tp[k+1] = j+2; k+=2; ); ); for(j=1,k-1, y=1; for(i = 1,j, y*=tp[i]; ); print1(y", "); sr+=1.0/y; ); print(); print(sr); }

Definition clarified and example provided by Harvey P. Dale, Mar 16 2011

A097490 Primes which are two greater than A097489 terms.

Original entry on oeis.org

5, 17, 167, 302946354048717875530381041444257, 17164738545781348456175905084853738838912866540727619406614703260339837793050935010265073947

Offset: 1

Cino Hilliard, Aug 24 2004

nonn

			a(3) = 167 = (Product_{k=1..3} A001359(k)) + 2 = 3 * 5 * 11 + 2 = A097489(3) + 2. - _Hartmut F. W. Hoft_, Apr 27 2021

Amiram Eldar, Table of n, a(n) for n = 1..6

Cf. A097489, A097491.

Cf. A001359, A077800.

step[{list_, q_}] := Module[{p=NextPrime[q]}, {Join[list, If[PrimeQ[p+2], {{p,p+2}}, {}]], p}]
smallerTwin[n_] := First[Transpose[First[NestWhile[step, {{{3, 5}}, 3}, Length[First[step[#]]]<=n&]]]]
a097489[n_] := Rest[FoldList[Times, 1, smallerTwin[n]]]
a097490[n_] := Select[Map[#+2&, a097489[n]], PrimeQ]
a097490[39] (* Hartmut F. W. Hoft, Apr 27 2021 *)

fp(n) = p=1;for(x=1,n,p*=twinl(x);if(isprime(p+2),print1(p+2", ")))
twinl(n) = { local(c,x); c=0; x=1; while(c

Edited by Don Reble, Apr 16 2007

A097491 Primes which are two greater than the terms of A079164.

Original entry on oeis.org

5, 17, 21800053277, 72409291238312731227527, 86984485062381462583582279727, 21679097826151232817152558557032490897727272048343000297777, 107025222275017133994159705286756083545279583250537082122450588876727

Offset: 1

Cino Hilliard, Aug 24 2004

nonn

A097491(8) = 2948...794027 has 76 digits and A097491(9) = 152400...802327 has 288 digits. - Hartmut F. W. Hoft, Apr 27 2021

			a(3) = 21800053277 = A079164(17) + 2 = 3*5*5*7*11*13*17*19*29*31 + 2. - _Hartmut F. W. Hoft_, Apr 27 2021

Amiram Eldar, Table of n, a(n) for n = 1..9

Cf. A048599, A097490, A097493.

Cf. A037074, A077800, A079164.

step[{list_, q_}] := Module[{p=NextPrime[q]}, {Join[list, If[PrimeQ[p+2], {{p,p+2}}, {}]], p}]
pairList[n_] := First[NestWhile[step, {{{3, 5}}, 3}, Length[First[step[#]]]<=n&]]
a079164[n_] := Rest[FoldList[Times, 1, Take[Flatten[pairList[n]], n]]]
a097491[n_] := Select[Map[#+2&, a079164[n]], PrimeQ]
a097491[39] (* Hartmut F. W. Hoft, Apr 27 2021 *)

ft(n) = p=1;for(x=1,n,p*=twinl(x);if(isprime(p+2),print1(p+2", ")); p*=twinu(x);if(isprime(p+2),print1(p+2", ")))
twinl(n) = { local(c,x); c=0; x=1; while(c

Edited by Don Reble, Apr 16 2007

Name corrected by Hartmut F. W. Hoft, Apr 27 2021

A179067 Orders of consecutive clusters of twin primes.

Original entry on oeis.org

1, 3, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Franz Vrabec, Jun 27 2010

For k>=1, 2k+4 consecutive primes P1, P2, ..., P2k+4 defining a cluster of twin primes of order k iff P2-P1 <> 2, P4-P3 = P6-P5 = ... = P2k+2 - P2k+1 = 2, P2k+4 - P2k+3 <> 2.

Also the lengths of maximal runs of terms differing by 2 in A029707 (leading index of twin primes), complement A049579. - Gus Wiseman, Dec 05 2024

			The twin prime cluster ((101,103),(107,109)) of order k=2 stems from the 2k+4 = 8 consecutive primes (89, 97, 101, 103, 107, 109, 113, 127) because 97-89 <> 2, 103-101 = 109-107 = 2, 127-113 <> 2.
From _Gus Wiseman_, Dec 05 2024: (Start)
The leading indices of twin primes are:
  2, 3, 5, 7, 10, 13, 17, 20, 26, 28, 33, 35, 41, 43, 45, 49, 52, ...
with maximal runs of terms differing by 2:
  {2}, {3,5,7}, {10}, {13}, {17}, {20}, {26,28}, {33,35}, {41,43,45}, {49}, {52}, ...
with lengths a(n).
(End)

Robert Israel, Table of n, a(n) for n = 1..10000
Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

Cf. A077800.
Cf. A001359, A111950, A087641.
Cf. A035789, A035790, A035791, A035792, A035793, A035794, A035795.
A000040 lists the primes, differences A001223 (run-lengths A333254, A373821).
A006512 gives the greater of twin primes.
A029707 gives the leading index of twin primes, complement A049579.
A038664 finds the first prime gap of length 2n.
A046933 counts composite numbers between primes.
A000720, A006560, A006562, A014574, A037201, A107770, A122535, A155752, A175632, A251092.

R:= 1: count:= 1: m:= 0:
q:= 5: state:= 1:
while count < 100 do
 p:= nextprime(q);
 if state = 1 then
    if p-q = 2 then state:= 2; m:= m+1;
    else
      if m > 0 then R:= R,m; count:= count+1; fi;
      m:= 0
    fi
 else state:= 1;
 fi;
 q:= p
od:
R; # Robert Israel, Feb 07 2023

Length/@Split[Select[Range[2,100],Prime[#+1]-Prime[#]==2&],#2==#1+2&] (* Gus Wiseman, Dec 05 2024 *)

a(n)={my(o,P,L=vector(3));n++;forprime(p=o=3,,L=concat(L[2..3],-o+o=p);L[3]==2||next;L[1]==2&&(P=concat(P,p))&&next;n--||return(#P);P=[p])} \\ M. F. Hasler, May 04 2015

More terms from M. F. Hasler, May 04 2015

A274792 a(n) = smallest prime p(1) in a symmetrical constellation of n consecutive twin primes: p(1), p(1)+2, ..., p(n), p(n)+2.

Original entry on oeis.org

3, 5, 5, 663569, 3031329797, 17479880417, 1855418882807417, 2640138520272677

Offset: 1

Natalia Makarova, Jul 07 2016

			The list of two consecutive twin primes (5, 7, 11, 13) is symmetrical because 5+13 = 7+11. Thus a(2) = 5.
The list of three consecutive twin primes (5, 7, 11, 13, 17, 19) is symmetrical because 5+19 = 7+17 = 11+13. Thus a(3) = 5.

N. Makarova and C. Rivera, Symmetrical compositions of consecutive twin primes.

Cf. A077800.

a(7)-a(8) from Dmitry Petukhov, Jul 07 2016

A097493 Primes which are two greater than A097492 terms.

Original entry on oeis.org

7, 37, 457, 8647, 51315414607

Offset: 1

Cino Hilliard, Aug 24 2004

nonn

The next term (17866..79237) has 186 digits.

			a(4) = 8647 = (Product_{k=1..4} A006512(k)) + 2 = 5*7*13*19 + 2 = A097492(4) + 2. - _Hartmut F. W. Hoft_, Apr 27 2021

Amiram Eldar, Table of n, a(n) for n = 1..7

Cf. A097490, A097491, A097492.

Cf. A001359, A077800, A097489.

step[{list_, q_}] := Module[{p=NextPrime[q]}, {Join[list, If[PrimeQ[p+2], {{p,p+2}}, {}]], p}]
largerTwin[n_] := Last[Transpose[First[NestWhile[step, {{{3, 5}}, 3}, Length[First[step[#]]]<=n&]]]]
a097492[n_] := Rest[FoldList[Times, 1, largerTwin[n]]]
a097493[n_] := Select[Map[#+2&, a097492[n]], PrimeQ]
a097493[68] (* Hartmut F. W. Hoft, Apr 27 2021 *)

fu(n) = p=1;for(x=1,n,p*=twinu(x);if(isprime(p+2),print1(p+2", ")))
twinu(n) = { local(c,x); c=0; x=1; while(c

Edited by Don Reble, Apr 16 2007

A104225 Decimal expansion of -x, where x is the real root of f(x) = 1 + Sum_{n} (twin_prime(n))*x^n.

Original entry on oeis.org

6, 6, 5, 0, 7, 0, 0, 4, 8, 7, 6, 4, 8, 5, 2, 2, 9, 2, 0, 4, 3, 4, 8, 7, 1, 4, 3, 2, 8, 0, 8, 7, 1, 4, 5, 8, 9, 4, 2, 2, 8, 1, 0, 5, 2, 6, 1, 3, 6, 4, 6, 0, 6, 0, 4, 2, 4, 0, 2, 8, 5, 9, 0, 6, 0, 9, 4, 1, 2, 3, 4, 0, 3, 7, 0, 7, 2, 8, 4, 1, 9, 5, 9, 0, 0, 9, 1, 0, 1, 5, 6, 4, 6, 4, 0, 0, 6, 4, 9, 8

Offset: 0

Jonathan Vos Post, Apr 01 2005

			-0.665070048764852292...

S. R. Finch, "Kalmar's Composition Constant", Section 5.5 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 292-295, 2003.
Martin Gardner, "Patterns in Primes are a Clue to the Strong Law of Small Numbers." Sci. Amer. 243, 18-28, Dec. 1980.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.

S. R. Finch, Backhouse's constant. 1995 [Cached copy, with permission]
Philippe Flajolet, in response to the previous document from S. R. Finch, Backhouse's constant, 1995
S. R. Finch, Kalmar's Composition Constant, Section 5.5 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 292-295, 2003. [Cached copy, with permission]
Eric Weisstein's World of Mathematics, Backhouse's Constant.
Eric Weisstein's World of Mathematics, Twin Primes.

Cf. A072508, A074269, A077800, A088751.

ps={}; Do[If[PrimeQ[n]&&PrimeQ[n+2], AppendTo[ps, {n, n+2}]], {n, 3, 40001, 2}];
ps=Flatten[ps];
RealDigits[ -x /. FindRoot[0==1+Sum[x^n ps[[n]], {n, 1000}], {x, -0.665}, WorkingPrecision->100]][[1]] (* T. D. Noe *)

Decimal expansion of -x where x is the real root of f(x) = 1 + 3x + 5x^2 + 5x^3 + 7x^4 + 11x^5 + 13x^6 + 17x^7 + 19x^8 + 29x^9 + 31x^10 + 41x^11 + 43x^12 + 59x^13 + 61x^14 + 71x^15 + 73x^16 + ... where for n>0 the coefficient of x^n is the n-th twin prime.

Offset corrected by Sean A. Irvine, May 24 2025

a(99) corrected by Michael De Vlieger, May 24 2025

A129950 Indicator function of twin primes: 1 if n is a twin prime member, 0 if not prime, -1 else (isolated prime or 2).

Original entry on oeis.org

0, -1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0

Offset: 1

Cino Hilliard, Jun 10 2007

sign

Absolute values are the same as A010051.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A001097, A001359, A006512, A010051, A077800, A164292.

Array[If[PrimeQ@ #, If[Or[PrimeQ[# - 2], PrimeQ[# + 2]], 1, -1], 0] &, 100] (* Michael De Vlieger, Jan 03 2019 *)

istwin(n) = local(p1, p2); if(n==5,return(2));if(isprime(n),p1=n-2;p2=n+2; if(isprime(p1),return(1));if(isprime(p2),return(-1));return(0))
t(x) = if(abs(istwin(x))==1||x==5,1,if(isprime(x),-1,0))
for(j=1,100,print1(t(j)","))

a(n) = if(isprime(n), (-1)^(!isprime(n-2) && !isprime(n+2)), 0); \\ Typos corrected by Antti Karttunen, Jan 03 2019

a(n) = 2*A164292(n) - A010051(n). - Antti Karttunen, Jan 03 2019

Definition and a(67) corrected by M. F. Hasler, Jan 20 2012

A182482 6nA182481(n)-1.

Original entry on oeis.org

5, 11, 17, 71, 29, 71, 41, 191, 107, 59, 197, 71, 311, 419, 179, 191, 101, 107, 227, 239, 881, 659, 137, 431, 149, 311, 809, 2687, 347, 179, 1301, 191, 197, 1019, 419, 431, 1997, 227, 1871, 239, 1229, 2267, 1031, 1319, 269, 827, 281, 1151, 881, 599

Offset: 1

Vladimir Shevelev, May 01 2012

nonn

By the construction of A182481, every term is lesser of twin primes.
Every lesser more than 3 of twin primes appears in the sequence.
Number m=(a(n)+1)/6 is the place of the last appearance of a(n); m is multiple of all previous places of the appearance of a(n), if they exist.
In particular, a(n) appears only once, if (a(n)+1)/6 is 1 or prime (in this case n is 1 or prime and A182481(n)=1). Conversely is not true. For example, a(10)=59 appears only once, although 10 is not prime.

			All places where 71 appears are 4,6,12. "Thus" 12 is multiple of 4 and 6.
Since (101+1)/6=17 is prime, then 101 appears only once.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A182481, A001359, A006512, A014574, A001097, A077800, A002822.

A253624 Initial members of prime sextuples (p, p+2, p+12, p+14, p+24, p+26).

Original entry on oeis.org

5, 17, 1277, 4217, 21587, 91127, 103967, 113147, 122027, 236867, 342047, 422087, 524957, 560477, 626597, 754967, 797567, 909317, 997097, 1322147, 1493717, 1698857, 1748027, 1762907, 2144477, 2158577, 2228507, 2398157, 2580647, 2615957

Offset: 1

Karl V. Keller, Jr., Jan 06 2015

nonn

This sequence is primes p for which there exist three twin prime pairs (p, p+2), (p+12, p+14) and (p+24, p+26).
Excluding 5, this is a subsequence of each of the following: A128468 (a(n)=30n+17). A039949 (Primes of the form 30n-13), A181605 (twin primes ending in 7).
Note that not in all cases (p, p+2, p+12, p+14, p+24, p+26) are consecutive primes; the first p's for which (p, p+2, p+12, p+14, p+24, p+26) are consecutive primes are 4217, 21587, 91127, 103967, 236867, 342047, 422087, 560477, 797567, 909317, 1322147, 1493717, 1748027, 1762907, 2144477, 2158577, 2228507, 2615957 (not in OEIS). - Zak Seidov, May 16 2017

			For p = 17, the numbers 17, 19, 29, 31, 41, 43 are primes.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes
Wikipedia, Twin prime

Cf. A077800 (twin primes), A128468, A039949, A181605.

select(t -> andmap(isprime, [t,t+2,t+12,t+14,t+24,t+26]),
[5, seq(30*k+17,k=0..10^5)]); # Robert Israel, Jan 07 2015

Select[Prime@ Range[2*10^5], Times @@ Boole@ PrimeQ[# + {2, 12, 14, 24, 26}] == 1 &] (* Michael De Vlieger, May 16 2017 *)
Select[Prime[Range[200000]],AllTrue[#+{2,12,14,24,26},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 15 2021 *)

from sympy import isprime
for n in range(1,10000001,2):
  if isprime(n) and isprime(n+2) and isprime(n+12) and isprime(n+14) and isprime(n+24) and isprime(n+26): print(n,end=', ')

from sympy import isprime, primerange
def aupto(limit):
  alst = []
  for p in primerange(2, limit+1):
    if all(map(isprime, [p+2, p+12, p+14, p+24, p+26])): alst.append(p)
  return alst
print(aupto(3*10**6)) # Michael S. Branicky, May 17 2021

cp's OEIS Frontend

A079164 Twin-primorial numbers: running products of twin primes.

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A097490 Primes which are two greater than A097489 terms.

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A097491 Primes which are two greater than the terms of A079164.

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A179067 Orders of consecutive clusters of twin primes.

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A274792 a(n) = smallest prime p(1) in a symmetrical constellation of n consecutive twin primes: p(1), p(1)+2, ..., p(n), p(n)+2.

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A097493 Primes which are two greater than A097492 terms.

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A104225 Decimal expansion of -x, where x is the real root of f(x) = 1 + Sum_{n} (twin_prime(n))*x^n.

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A182482 6nA182481(n)-1.