A173427 Decimal value a(n) of the binary number b(n) obtained by starting from 1, sequentially concatenating all binary numbers up to n and then sequentially concatenating all binary numbers from n-1 down to 1.

1, 13, 221, 7069, 451997, 28931485, 1851651485, 237010810269, 60674754606493, 15532737233548701, 3976380732916495773, 1017953467644930815389, 260596087717395474544029, 66712598455657932715586973, 17078425204648505835166758301, 8744153704780027821877938484637

Offset: 1

Author

Umut Uludag, Feb 18 2010

Keywords

• base
• nonn

Comments

a(2) = 13 and a(4) = 7069 are primes. What other terms are primes? - N. J. A. Sloane, Feb 18 2023

a(38) is the next prime. - Michael S. Branicky, Feb 18 2023

Examples

			a(1)=binary_to_decimal(1)=1, a(2)=binary_to_decimal(1101)=13, a(3)=binary_to_decimal(11011101)=221, a(4)=binary_to_decimal(1101110011101)=7069 etc.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..239
• Index entries for sequences related to Most Wanted Primes video

Crossrefs

Cf. A359149 (binary representations).

Programs

Maple

a:= n-> Bits[Join](map(x-> Bits[Split](x)[], [$1..n, n-i$i=1..n-1])):
seq(a(n), n=1..16);  # Alois P. Heinz, Feb 18 2023

PARI

a(n)=sum(i=1,#n=concat(vector(n*2-1,k,binary(min(k, n*2-k)))),n[i]<<(#n-i))

PARI

A173427(n)={my(s=0,s1=0,t=0,b=0);for(k=1,n-1,s1+=k<>b&&b++;s=s<>b&&b++;(s<M. F. Hasler, Aug 06 2015

Python

from itertools import count, islice
def agen(): # generator of terms
    sl, sr, sk = "", "", "1"
    for k in count(1):
        sk = bin(k)[2:]
        sl += sk
        yield int(sl + sr, 2)
        sr = sk + sr
print(list(islice(agen(), 16))) # Michael S. Branicky, Feb 18 2023

Formula

a(n) = binary_to_decimal(concatenate(1,10,11,..., binary(n-2), binary(n-1), binary(n), binary(n-1), binary(n-2),..., 11, 10, 1))