A073641 -id:A073641 - cp's OEIS frontend

A073840 Product of the composite numbers between n and 2n (both inclusive).

1, 4, 24, 192, 4320, 51840, 120960, 29030400, 65318400, 145152000, 6706022400, 160944537600, 8717829120000, 6590678814720000, 14122883174400000, 30128817438720000, 2112783322890240000, 2662106986841702400000

Offset: 1

Amarnath Murthy, Aug 13 2002

nonn

a(n) is divisible by central binomial coefficients, A001405[n]

			a(6) = 6*8*9*10*12 = 51840.

Cf. A073839, A073641.

for n from 1 to 50 do l := 1:for j from n to 2*n do if not isprime(j) then l := l*j:fi:od:a[n] := l:od:seq(a[j],j=1..50);

cs[x_] := Flatten[Position[Table[PrimeQ[j], {j, x, 2*x}], False]]+x-1; prcs[x_] := Apply[Times, cs[x]]; Table[prcs[w], {w, 1, 25}]

PARI
```
a(n)=prod(i=n,2*n,i^if(isprime(i),0,1))
```

a(n)=A049614(2n)/A049614(n-1)

More terms from Sascha Kurz and Labos Elemer, Aug 14 2002

A350853 a(1) = 2, a(2) = 3; a(n) is the smallest prime not included earlier such that concatenation of three successive terms is a prime.

Original entry on oeis.org

2, 3, 11, 23, 31, 13, 29, 7, 17, 19, 37, 41, 59, 79, 67, 107, 47, 61, 43, 113, 71, 109, 89, 53, 157, 97, 83, 101, 73, 173, 131, 223, 149, 127, 197, 137, 373, 139, 167, 163, 179, 151, 191, 193, 241, 317, 211, 229, 281, 103, 227, 233, 283, 251

Offset: 1

Haines Hoag, Jan 18 2022

Not a permutation of the primes. 5 never appears, since numbers m mod 10 = 5 are divisible by 5, and concatenation of 2 previous terms and 5 guarantee a composite number. - Michael De Vlieger, Feb 16 2022

			From _Michael De Vlieger_, Feb 16 2022: (Start)
a(3) = 11 since 235 and 237 are composite, but 2311 is prime.
a(4) = 23 since 3115, 3117, 31113, 31117, and 31119 are composite, but 31123 is prime.
a(5) = 31 since 11235, 11237, 112313, 112317, 112319, and 112329 are composite, but 112331 is prime. (End)

Haines Hoag, Table of n, a(n) for n = 1..31499
Michael De Vlieger, Scatterplot of a(n) for n = 1..2^14, showing records in red and local minima (aside from q = 5, which never appears) in blue.

Cf. A073641, A000040.

a[1]=2; a[2]=3; a[n_]:=a[n]=(k=2; While[!PrimeQ[FromDigits@Join[Flatten[IntegerDigits/@{a[n-2],a[n-1]}],IntegerDigits@k]]||MemberQ[Array[a,n-1],k],k=NextPrime@k];k);Array[a,54] (* Giorgos Kalogeropoulos, Jan 19 2022 *)

A083439 a(1) = 3; then a(n+1) = smallest prime not already in the sequence such that the concatenations a(n)a(n+1) and a(n+1)a(n) are both primes.

Original entry on oeis.org

3, 7, 19, 13, 61, 151, 31, 139, 67, 37, 79, 193, 163, 127, 157, 97, 43, 103, 307, 367, 457, 643, 73, 277, 223, 229, 373, 199, 109, 313, 211, 271, 241, 421, 181, 283, 397, 337, 349, 331, 577, 523, 463, 613, 439, 541, 571, 433, 787, 547, 661, 769, 487, 601, 823, 709

Offset: 3

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 30 2003

Cf. A073641, A075609.

{ p=3; S=Set(); while(!setsearch(S,p), S=setunion(S,Set([p])); print1(p,", "); forprime(q=2,10^4, if(setsearch(S,q),next); if( isprime(eval(concat(Str(p),Str(q)))) && isprime(eval(concat(Str(q),Str(p)))), p=q; break))) } \\ Max Alekseyev, Apr 24 2009

Corrected and extended by Max Alekseyev, Apr 24 2009

cp's OEIS Frontend

A073840 Product of the composite numbers between n and 2n (both inclusive).

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A350853 a(1) = 2, a(2) = 3; a(n) is the smallest prime not included earlier such that concatenation of three successive terms is a prime.

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A083439 a(1) = 3; then a(n+1) = smallest prime not already in the sequence such that the concatenations a(n)a(n+1) and a(n+1)a(n) are both primes.

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