A136349 -id:A136349 - cp's OEIS frontend

A136358 Increasing sequence obtained by union of two sequences {b(n)} and {c(n)}, where b(n) is the smallest odd composite number m such that both m-2 and m+2 are prime and the set of distinct prime factors of m consists of the first n odd primes and c(n) is the smallest composite number m such that both m-1 and m+1 are primes and the set of the distinct prime factors of m consists of the first n primes.

4, 6, 9, 15, 30, 105, 420, 2310, 3465, 15015, 180180, 765765, 4084080, 106696590, 247342095, 892371480, 3011753745, 9704539845, 100280245065, 103515091680, 4412330782860, 29682952539240, 634473110526255, 22514519501013540

Offset: 1

Enoch Haga, Dec 25 2007

This sequence is different from A070826 and A118750.

			a(1)=4 is preceded by 3 and followed by 5, both primes; a(3)=9, preceded by 7 and followed by 11, both primes.

Cf. A136349-A136357, A070826, A118750.

b[n_]:=(d=Product[Prime[k],{k,n}]; For[m=1,!(!PrimeQ[d*m]&&PrimeQ[d*m-1] &&PrimeQ[d*m+1]&&Length[FactorInteger[c*m]]==n),m++ ]; d*m); c[n_]:=(d=Product [Prime[k],{k,2,n+1}]; For[m=1,!(!PrimeQ[d*(2*m-1)]&&PrimeQ[d(2m-1)-2]&&PrimeQ [d(2m-1)+2]&&Length[FactorInteger[d(2m-1)]]==n),m++ ]; d(2m-1)); Take[Union[Table [b[k],{k,24}],Table[c[k],{k,24}]],24] (* Farideh Firoozbakht, Aug 13 2009 *)

10 'A136358, Enoch Haga, Jun 19 2009'
11 'compute and combine input 2 or 3 separately; begin with 4 and 9
20 input "prime, 2 or 3";A
30 if A=2 or A=3 then B=nxtprm(A)
40 print A;B;:R=A*B:print R;:stop
50 if even(R)=1 then if R-1=prmdiv(R-1) and R+1=prmdiv(R+1) then print "*"
60 if even(R)=0 then if R-2=prmdiv(R-2) and R+2=prmdiv(R+2) then print "+"
61 print R:stop
70 B=nxtprm(B):R=B*R
90 print B;R:stop
100 goto 50
- Enoch Haga, Jul 11 2009

Edited, corrected and extended by Farideh Firoozbakht, Aug 13 2009

A088256 Primorial numbers k such that both k-1 and k+1 are prime.

Original entry on oeis.org

6, 30, 2310

Offset: 1

Amarnath Murthy, Sep 27 2003

Conjecture: sequence is finite.
No more terms in the first 300 primorials. - David Wasserman, Jul 25 2005
Search extended to first 700 primorials by Michael De Vlieger, Aug 31 2016
Intersection of A014574 and A002110. - Michel Marcus, Dec 03 2016
Search extended to first 3000 primorials. - Josey Stevens, Aug 10 2021
The first more than 230000 primorial numbers k have been checked for whether k-1 or k+1 or both are primes. See links. If another term k exists, it is over about 10^1400000. - Jeppe Stig Nielsen, Oct 19 2021

			210 = primorial(4) is not a member as 209 is composite.

Chris K. Caldwell, The Top Twenty: Primorial, in the Prime Pages.
PrimeGrid, PRPNet: Primorial Prime Search - Server Statistics.

Cf. A001097, A002110, A088257, A014574, A119411, A136349.

f:= proc(n)
  local P;
  P:= mul(seq(ithprime(i),i=1..n));
  if isprime(P+1) and isprime(P-1) then P else NULL fi
end proc:
map(f, [$1..300]); # Robert Israel, Aug 31 2016

Select[Times @@ # & /@ Prime@ Range@ Range@ 700, Times @@ Boole@ PrimeQ@ {# - 1, # + 1} == 1 &] (* Michael De Vlieger, Aug 31 2016 *)
Select[FoldList[Times,Prime[Range[20]]],AllTrue[#+{1,-1},PrimeQ]&] (* Harvey P. Dale, Mar 31 2023 *)

lista(nn) = for (n=1, nn, pr = prod(i=1, n, prime(i)); if (isprime(pr-1) && isprime(pr+1), print1(pr, ", "))); \\ Michel Marcus, Aug 31 2016

Corrected by Ray Chandler, Sep 28 2003

A136351 Primorial numbers p# such that p# + 1 is a prime.

Original entry on oeis.org

1, 2, 6, 30, 210, 2310, 200560490130

Offset: 1

Enoch Haga, Dec 25 2007

This sequence is different from A121069 and A002110.

a(8) = A002110(75) has 154 digits and is too long to be listed. - R. J. Mathar, Jul 23 2008

			a(6)=2310 is followed by prime 2311 whereas 30030 is not followed by a prime.

James C. McMahon, Table of n, a(n) for n = 1..10 (a(n) for n = 2..10 by Miles Englezou).

Cf. A002110, A014545, A018239, A121069, A136349, A136350, A136352.

Select[FoldList[Times, 1, Prime[Range[18]]],PrimeQ[#+1]&] (* James C. McMahon, May 08 2025 *)

S=[];for(n=0, 80, k=vecprod(primes(n)); if(isprime(k+1), S=concat(S,k))); S \\ Miles Englezou, Oct 28 2024

{A002110(j): A002110(j)+1 in A000040}. - R. J. Mathar, Jul 23 2008
a(n) = A002110(A014545(n)). - Michel Marcus, Apr 05 2021
a(n) = A018239(n) - 1. - James C. McMahon, May 08 2025

Changed a(1) from 4 to 2 and edited by R. J. Mathar, Jul 23 2008

a(1)=1 inserted by James C. McMahon, May 08 2025

A136352 Primorials P for which neither P-1 nor P+1 is prime.

Original entry on oeis.org

510510, 9699690, 223092870, 6469693230, 7420738134810, 13082761331670030, 614889782588491410, 32589158477190044730, 1922760350154212639070, 117288381359406970983270, 7858321551080267055879090

Offset: 1

Enoch Haga, Dec 25 2007

This sequence is different from A121069 and A002110.

For almost all primes p, p# is in this sequence by Brun's theorem. - Charles R Greathouse IV, Sep 14 2015

			13# = 30030 is preceded by a prime but is not followed by a prime. 17# = 510510 is neither preceded nor followed by a prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..350

Cf. A136349, A136350, A136351, A002110, A121069.

Select[FoldList[Times, 1, Prime[Range[19]]],!PrimeQ[#+1]&&!PrimeQ[#-1]&] (* James C. McMahon, May 08 2025 *)

Compute P = product of the first k primes. If P is neither preceded nor followed by a prime add P to the sequence.

Edited by and more terms from Charles R Greathouse IV, Sep 29 2008

A136350 Primorial numbers #p such that (#p)-1 is composite.

Original entry on oeis.org

210, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 13082761331670030, 614889782588491410, 32589158477190044730, 1922760350154212639070, 117288381359406970983270

Offset: 1

Enoch Haga, Dec 25 2007

This sequence is different from A121069 and A002110.

			a(1)=210 because 209=A002110(4)-1 is not prime.

Cf. A136349, A136351, A136352, A002110, A121069.

Select[FoldList[Times, 1, Prime[Range[18]]],CompositeQ[#-1]&] (* James C. McMahon, May 08 2025 *)

{A002110(j): A002110(j)-1 in A002808}. - R. J. Mathar, Jul 23 2008

Edited and extended by R. J. Mathar, Jul 23 2008

A136354 a(n) is the smallest odd composite number m such that m+2 is prime and the set of distinct prime factors of m consists of the first n odd primes.

Original entry on oeis.org

9, 15, 105, 3465, 15015, 765765, 33948915, 334639305, 3234846615, 100280245065, 3710369067405, 1369126185872445, 32706903329175075, 307444891294245705, 211829530101735290745, 961380175077106319535, 762374478836145311391255

Offset: 1

Enoch Haga, Dec 25 2007

This sequence is different from A070826 and A118750.

			a(1)=9 because k=1 with prime factor 3 and 9+2=11, prime

Cf. A136349-A136353, A136355-A136358, A070826, A118750.

a[n_]:=(c=Product[Prime[k],{k,2,n+1}]; For[m=1,!(!PrimeQ[c(2m-1)]&&PrimeQ[c(2m-1)+2]&&Length[FactorInteger[c(2m-1)]]==n),m++ ]; c(2m-1)); Table[a[n],{n,17}] (* Farideh Firoozbakht, Aug 12 2009 *)

Edited, corrected and extended by Farideh Firoozbakht, Aug 12 2009

A136355 Numbers of the form P = product of the first k odd primes where P+2 is composite.

Original entry on oeis.org

1155, 255255, 4849845, 111546435, 152125131763605, 6541380665835015, 16294579238595022365, 58644190679703485491635, 3929160775540133527939545, 278970415063349480483707695, 20364840299624512075310661735, 1608822383670336453949542277065

Offset: 1

Enoch Haga, Dec 25 2007

This sequence is different from A070826 and A118750.

			a(1)=1155 because 1157 is not prime.

Harvey P. Dale, Table of n, a(n) for n = 1..332

Cf. A136349-A136354, A136356-A136358, A070826, A118750.

v=Select[Range[21],!PrimeQ[Product[Prime[k+1],{k,#}]+2]&]; Table[Product[Prime[k+1],{k,v[[t]]}],{t,Length[v]}] (* Farideh Firoozbakht, Aug 12 2009 *)
Select[FoldList[Times,Prime[Range[2,22]]],CompositeQ[#+2]&] (* Harvey P. Dale, Jun 08 2022 *)

Edited with more terms by Farideh Firoozbakht, Aug 12 2009

A136353 First odd composite N divisible by precisely the first n odd primes with N-2 prime.

Original entry on oeis.org

9, 15, 105, 1155, 15015, 255255, 4849845, 111546435, 9704539845, 100280245065, 18551845337025, 152125131763605, 98120709987525225, 7071232499767651215, 16294579238595022365, 33648306127698721183725, 527797716117331369424715

Offset: 1

Enoch Haga, Dec 25 2007

nonn

This sequence is different from A070826 and A118750.

A clearer definition of the sequence: a(n) is the smallest odd composite number m such that m - 2 is prime and the set of the distinct prime factors of m equals the set of the first n odd primes. - Farideh Firoozbakht, Jun 30 2009

			The first odd prime is 3, 3*3-2 = 7 is prime so a(1) = 9.
The product of the first two odd primes is 15, and 15-2 is prime, so a(2) = 15.

Charles R Greathouse IV, Table of n, a(n) for n = 1..200

Cf. A136349, A136350, A136351, A136352, A136354, A136355, A136356, A136357, A136358.

Cf. A070826, A118750.

a[n_]:=(c=Product[Prime[k],{k,2,n+1}];For[m=1,!(!PrimeQ[c (2m-1)]&&PrimeQ[c(2m-1)-2]&&Length[FactorInteger[c(2m-1)]]==n),m++ ];c(2m-1));Table[a[n],{n,20}] (* Enoch Haga, Jul 02 2009 *)

sm(n,x)=forprime(p=2,x, n/=p^valuation(n,p)); n==1
a(n)=my(m=factorback(primes(n+1)[2..n+1]),k,p=prime(n+1)); while(!isprime(k++*m-2) && sm(k,p),); k*m \\ Charles R Greathouse IV, Sep 14 2015

Compute N = product of the first n odd primes. If N-2 is prime, add N to the sequence. Otherwise test 3N, 5N, 7N, 9N, ... until kN - 2 is prime, subject to A006530(k) <= n+1.

More terms, better title, and Mathematica program from Farideh Firoozbakht received Jun 30 2009. - Enoch Haga, Jul 02 2009

Further editing by Charles R Greathouse IV, Oct 05 2009

A136356 Increasing sequence obtained by union of two sequences A136353 and {b(n)}, where b(n) is the smallest composite number m such that m-1 is prime and the set of distinct prime factors of m consists of the first n primes.

Original entry on oeis.org

4, 6, 9, 15, 30, 105, 420, 1155, 2310, 15015, 30030, 255255, 1021020, 4849845, 19399380, 111546435, 669278610, 9704539845, 38818159380, 100280245065, 601681470390, 14841476269620, 18551845337025, 152125131763605

Offset: 1

Enoch Haga, Dec 25 2007

This sequence is different from A070826 and A118750.

			a(4)=15 because k=2 and prime factors are 3 and 5; 15 is odd and n-2=13, prime.

Cf. A136349-A136355, A136357-A136358, A070826, A118750.

a[n_]:=(c=Product[Prime[k],{k,n}]; For[m=1,!(!PrimeQ[c*m]&&PrimeQ[c*m-1]&&Length[FactorInteger[c*m]]==n),m++ ]; c*m);
b[n_]:=(c=Product[Prime[k],{k,2,n+1}]; For[m=1,!(!PrimeQ[c(2m-1)]&&PrimeQ[c(2m-1)-2]&&Length[FactorInteger[c(2*m-1)]]==n),m++ ]; c(2m-1));
Take[Union[Table[a[k],{k,24}],Table[b[k],{k,24}]],24] (* Farideh Firoozbakht, Aug 13 2009 *)

Edited, corrected and extended by Farideh Firoozbakht, Aug 13 2009

A136357 Increasing sequence obtained by union of two sequences A136354 and {b(n)}, where b(n) is the smallest composite number m such that m+1 is prime and the set of distinct prime factors of m consists of the first n primes.

Original entry on oeis.org

4, 6, 9, 15, 30, 105, 210, 2310, 3465, 15015, 120120, 765765, 4084080, 33948915, 106696590, 334639305, 892371480, 3234846615, 71166625530, 100280245065, 200560490130, 3710369067405, 29682952539240, 1369126185872445

Offset: 1

Enoch Haga, Dec 25 2007

This sequence is different from A070826 and A118750.

			a(4)=15 because k=2 with prime factors 3 and 5 and 15 is followed by 17, prime;
a(5)=30 because k=3 with prime factors 2, 3, 5 and 30 is followed by 31, prime.

Cf. A136349-A136356, A136358, A070826, A118750.

a[n_]:=(c=Product[Prime[k],{k,n}]; For[m=1,!(!PrimeQ[c*m]&&PrimeQ[c*m+1]&& Length[FactorInteger[c*m]]==n),m++ ]; c*m);
b[n_]:=(c=Product[Prime[k],{k,2, n+1}]; For[m=1,!(!PrimeQ[c(2*m-1)]&&PrimeQ[c(2*m-1)+2]&&Length[FactorInteger [c(2*m-1)]]==n),m++ ]; c(2*m-1));
Take[Union[Table[a[k],{k,24}],Table[b[k],{k, 24}]],24] (* Farideh Firoozbakht, Aug 13 2009 *)

Edited, corrected and extended by Farideh Firoozbakht, Aug 13 2009

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A088256 Primorial numbers k such that both k-1 and k+1 are prime.

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A136351 Primorial numbers p# such that p# + 1 is a prime.

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A136352 Primorials P for which neither P-1 nor P+1 is prime.

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A136350 Primorial numbers #p such that (#p)-1 is composite.

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A136354 a(n) is the smallest odd composite number m such that m+2 is prime and the set of distinct prime factors of m consists of the first n odd primes.

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A136355 Numbers of the form P = product of the first k odd primes where P+2 is composite.

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