A136354 -id:A136354 - cp's OEIS frontend

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.

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

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

A309005 Odd squarefree composite numbers m such that m+2 is prime.

Original entry on oeis.org

15, 21, 35, 39, 51, 57, 65, 69, 77, 87, 95, 105, 111, 129, 155, 161, 165, 177, 195, 209, 221, 231, 237, 249, 255, 267, 291, 305, 309, 329, 335, 345, 357, 365, 371, 377, 381, 395, 399, 407, 417, 429, 437, 447, 455, 465, 485, 489, 497, 501, 519, 545, 555, 561, 591, 597, 611

Offset: 1

David James Sycamore, Jul 05 2019

nonn

The squarefree terms of A241809 and A136354 are in this sequence.

			15 = 3*5 is the smallest squarefree composite number m such that m+2 is prime; 15+2=17.

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

Cf. A024556, A241809, A136354.

[n: n in [2..611] | IsPrime(n+2) and  not IsPrime(n) and IsSquarefree(n)]; // Vincenzo Librandi, Jul 07 2019

with(NumberTheory):
N := 500;
for n from 2 to N do
if IsSquareFree(n) and not mod(n, 2) = 0 and not isprime(n) and isprime(n+2) then print(n);
end if:
  end do:

Select[Range[15, 611, 2], And[CompositeQ@ #, SquareFreeQ@ #, PrimeQ[# + 2]] &] (* Michael De Vlieger, Jul 08 2019 *)
Select[Prime[Range[2,150]]-2,SquareFreeQ[#]&&CompositeQ[#]&] (* Harvey P. Dale, Dec 03 2022 *)

isok(n) = isprime(n+2) && (n%2) && (n>1) && !isprime(n) && issquarefree(n); \\ Michel Marcus, Jul 05 2019

cp's OEIS Frontend

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.

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

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A309005 Odd squarefree composite numbers m such that m+2 is prime.

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