A118750 -id:A118750 - 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

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

A118751 Smallest prime >= 3*n.

Original entry on oeis.org

2, 5, 7, 11, 13, 17, 19, 23, 29, 29, 31, 37, 37, 41, 43, 47, 53, 53, 59, 59, 61, 67, 67, 71, 73, 79, 79, 83, 89, 89, 97, 97, 97, 101, 103, 107, 109, 113, 127, 127, 127, 127, 127, 131, 137, 137, 139, 149, 149, 149, 151, 157, 157, 163, 163, 167, 173, 173, 179, 179, 181

Offset: 0

Jonathan Vos Post, Apr 29 2006

Analogous to A060264 = first prime after 2n.

Cf. A060308, A060264, A118750, A118752.

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

A118749 Largest prime <= 3*n.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 23, 29, 31, 31, 37, 41, 43, 47, 47, 53, 53, 59, 61, 61, 67, 71, 73, 73, 79, 83, 83, 89, 89, 89, 97, 101, 103, 107, 109, 113, 113, 113, 113, 113, 127, 131, 131, 137, 139, 139, 139, 149, 151, 151, 157, 157, 163, 167, 167, 173, 173, 179, 181

Offset: 1

Jonathan Vos Post, Apr 29 2006

Analogous to A060308 largest prime <= 2*k.

Cf. A007917 (largest prime <= n), A008585 (3n).
Cf. A060308, A118750, A118751, A118752.
Essentially the same as A081259.

[NthPrime(#PrimesUpTo(3*n)): n in [1..100]]; // Vincenzo Librandi, Nov 25 2015

Table[Max[FactorInteger[(3 n)!/(n!)^3]], {n, 1, 70}] (* Vincenzo Librandi, Nov 25 2015 *)
NextPrime[3*Range[70]+1,-1] (* Harvey P. Dale, Nov 12 2017 *)

vector(100, n, precprime(3*n)) \\ Altug Alkan, Nov 25 2015

a(n) = A007917(A008585(n)). - Michel Marcus, Nov 25 2015

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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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A118751 Smallest prime >= 3*n.

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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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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.

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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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A118749 Largest prime <= 3*n.

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