A235860 -id:A235860 - cp's OEIS frontend

A236547 Numbers k such that (2^(2k + 1) - 2)/3 + k - 1 is prime.

1, 2, 4, 10, 14, 28, 54, 62, 124, 592, 1088, 3154, 3920, 5564, 26510, 28674, 65124

Offset: 1

Robin Garcia, Jan 28 2014

nonn

These primes are represented in the numeral system described in A235860 with the digit "2" followed to the right with k-1 "12" strings.

			For n = 2, (2^5 - 2)/3 + 1 = 11, which is prime, so 2 is in the sequence.

Cf. A235860.

isok(n) = isprime((2^(2*n+1) -2)/3 + n-1); \\ Michel Marcus, Jan 28 2014

a(1) inserted and a(15)-a(17) from Michael S. Branicky, Jul 29 2024

A237816 k such that either 2^k + k - 3 or 2^k + k - 2 is prime.

Original entry on oeis.org

2, 4, 6, 10, 70, 82, 143, 150, 220, 413, 426, 816, 5497, 6649, 7429, 7728, 7891, 8248, 14567, 15522, 17935, 24942, 37415, 123773

Offset: 1

Robin Garcia, Feb 13 2014

Numbers of this form can be represented in the numeral system described in A235860 with k - 1 ones followed to the right by k - 1 twos or k ones followed to the right by k - 1 twos, like this: 1, 12, 112, 1122, 11122, 111222, 1111222, ... (1, 3, 4, 8, 9, 17, 18, ... in decimal) and are the least numbers that need one more digit to be represented than any of their predecessors.

The corresponding sequence of primes starts 3, 17, 67, 1031, 1180591620717411303491, ...

Cf. A235860, A236547.

 fQ[n_] := PrimeQ[2^n + n - If[ OddQ@ n, 2, 3]]; Select[ Range@ 30000, fQ]

isok(n) = isprime(2^n + n - 3) || isprime(2^n + n - 2); \\ Michel Marcus, Feb 13 2014

a(21)-a(22) from Robert G. Wilson v, Mar 03 2014
a(23) from Michael S. Branicky, May 01 2023
a(24) from Michael S. Branicky, Jul 30 2024

A237454 Minimal representation (considered minimal in any canonical base b > 3) of n in a binary system with two distinct digits "1" and "3", not allowing zeros, where a digit d in position p (p = 1,2,3,...,n) represents the value d^p.

Original entry on oeis.org

1, 11, 3, 13, 113, 1113, 11113, 111113, 1111113, 31, 131, 33, 133, 1133, 11133, 111133, 1111133, 11111133, 111111133, 1111111133, 11111111133, 111111111133, 1111111111133, 11111111111133, 111111111111133, 1111111111111133, 11111111111111133, 111111111111111133, 311, 1311, 313, 1313, 11313, 111313, 1111313, 11111313, 331, 1331, 333, 1333, 11333

Offset: 1

Robin Garcia, Feb 08 2014

If digit "1" exists, the digits used in these numeral systems do not need to be consecutive.

			a(11) = 131 because 1^3 + 3^2 + 1^1 = 11.

Cf. A235860.

A237662 Primes of the form 2^(k+l+m+1) - 2^(l+m+1) + 2^(m+1) + l - 2.

Original entry on oeis.org

3, 7, 11, 17, 23, 31, 37, 47, 59, 67, 73, 101, 127, 131, 191, 223, 229, 239, 251, 257, 383, 401, 457, 479, 503, 521, 577, 991, 997, 1019, 1031, 1153, 1601, 1993, 2039, 2053, 2069, 3583, 3593, 3851, 3967, 4079, 4091, 4099, 4111, 4133, 6143, 6211

Offset: 1

Robin Garcia, Feb 11 2014

nonn

These prime numbers can be written in the numeral system described in A235860 (perhaps not minimally) this way : 2..21..12..2 (or 1..12..2) k twos followed to the right by l ones, followed to the right by m twos.
k can be zero, with the arbitrary limitation, when k = 0, l <= m.
When k = m = 1 we get primes of the form 2^(l + 2) + l + 2.
It must be noted these primes include the Mersenne primes 3, 7, 31, 127, 8191, ... as well as the Fermat primes 3, 5, 17, 257, 65537, with the exception of 5. Mersenne primes can be represented by a one followed to the right by an even number of twos (if the number of twos is odd, the Mersenne number is divisible by 3) with the exception of 3 represented as 12. The Fermat numbers can be represented with three ones followed to the right by a Mersenne number of twos (2^t - 1 (t = 0, 1, 2, 3, 4, 5,...)) :  3 = 111 instead of shorter 12, 5 = 1112 instead of shorter  21, 17 = 111222, 257 = 1112222222, 65537 = 111222222222222222. The composite (divisible by 641) 2^32 + 1 : three ones followed to the right by thirty one twos. The second Fermat prime: 5, appears in this sequence if we let l > m and l <= 3 when k = 0.
By A235860 3, 7 , 17 and 31 can be represented as 12, 122, 111222, 12222 cases when k=0 (primes of the form 2^(m+1) + l - 2: 31 = 2^5 +1 -2). And 11, 73, 191 as 212, 211122, 2122222 (73 = 2^7 - 2^6 + 2^3 + 3 - 2).

			For k=l=m=1, 2^(k+l+m+1) - 2^(l+m+1) + 2^(m+1) + l - 2 = 2^4 - 2^3 + 2^2 + 1 - 2 = 16 - 8 + 4 + 1 - 2 = 11, so 11 is in the sequence.

Cf. A235860, A236547, A007931, A129962, A237816

n=10^5;e=89;a=1;if(a%2==0,a=a+1);b=ceil(log(n)/log(2));i=0;d=floor(b^(2.5));v=vector(d);for(n=0,b,for(p=a,b,if(n==0,x=p,x=b);forstep(m=a,x,2,c=2^(n+m+p+1)-2^(m+p+1)+2^(p+1)+m-2;if(isprime(c),i++;v[i]=c))));w=vecsort(v,,8);u=vector(#(w)-1);for(j=1,#(w)-1,u[j]=w[j+1]);if(e>#(u),e=#(u));s=vector(e);for(k=1,e,s[k]=u[k];print1(s[k], ", "))

cp's OEIS Frontend

A236547 Numbers k such that (2^(2k + 1) - 2)/3 + k - 1 is prime.

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A237816 k such that either 2^k + k - 3 or 2^k + k - 2 is prime.

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A237454 Minimal representation (considered minimal in any canonical base b > 3) of n in a binary system with two distinct digits "1" and "3", not allowing zeros, where a digit d in position p (p = 1,2,3,...,n) represents the value d^p.

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A237662 Primes of the form 2^(k+l+m+1) - 2^(l+m+1) + 2^(m+1) + l - 2.

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