This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a[n_] := 2^(2^n-1) + 1; Array[a,9,0] (* Stefano Spezia, Oct 14 2024 *)
apply( {A332203(n)=1<<(1<
Regarding 32607: - the binary expansion of 32607 is "111111101011111", - the corresponding run lengths are: 7, 1, 1, 1, 5, - in binary: "111", "1", "1", "1", "101", - after concatenation: "111111101", - as "111111101011111" starts with "111111101", 32607 belongs to this sequence.
See Links section.
from itertools import groupby
def ok(n):
if n == 0: return True
b = bin(n)[2:]
c = "".join(bin(len(list(g)))[2:] for k, g in groupby(b))
return b.startswith(c)
print(list(filter(ok, range(2**17)))) # Michael S. Branicky, Oct 02 2021
a(1)=2 because 2*2^(2^1-1)-1 = 2*2^1-1 = 3 which is prime. - _Sean A. Irvine_, May 25 2022 a(4)=4 because 4*2^(2^4-1)-1 = 4*2^15-1 = 4*32768-1 = 131071 which is prime.
svk[n_]:= Module[{k = 1, c = 2^(2^n-1)}, While[!PrimeQ[k*c-1],k++];k]; Join[{2}, svk /@ Range[17]] (* Harvey P. Dale, Feb 03 2021, adjusted for new offset by Michael De Vlieger, May 25 2022 *)
a(n) = my(k=1); while (!isprime(k*2^(2^n-1)-1), k++); k; \\ Michel Marcus, May 27 2022
from sympy import isprime
def a(n):
k, c = 1, 2**(2**n-1)
while not isprime(k*c - 1): k += 1
return k
print([a(n) for n in range(1, 12)]) # Michael S. Branicky, May 25 2022
Table[Nest[FromDigits@ ConstantArray[1, #] &, n, 2], {n, 0, 3}] (* Michael De Vlieger, Jul 30 2017 *)
a002275(n) = (10^n-1)/9 a(n) = a002275(a002275(n))
a(0) = 1 because 1*2^(2^0) - 2*1 + 1 = 1 is a nonprime and 1*2^(2^0) + 2*1 - 1 = 3 is a prime. a(1) = 4 because 4*2^(2^1) - 2*4 + 1 = 9 is a composite and 4*2^(2^1) + 2*4 - 1 = 23 is a prime. a(2) = 1 because 1*2^(2^2) - 2*1 + 1 = 15 is a composite and 1*2^(2^2) + 2*1 - 1 = 17 is a prime. a(3) = 1 because 1*2^(2^3) - 2*1 + 1 = 255 is a composite and 1*2^(2^3) + 2*1 - 1 = 257 is a prime. a(4) = 1 because 1*2^(2^4) - 2*1 + 1 = 65535 is a composite and 1*2^(2^4) + 2*1 - 1 = 65537 is a prime.
Table[Module[{k=1},While[Total[Boole[PrimeQ[k*2^(2^n)+{2k-1,-2k+1}]]]!=1,k++];k],{n,0,14}] (* Harvey P. Dale, Jun 03 2025 *)
a(n) = {my(k=1, m=2^2^n); while(ispseudoprime(k*m-2*k+1)-ispseudoprime(k*m+2*k-1)==0, k++); k; } \\ Jinyuan Wang, Feb 26 2020
For n = 10 a(10) = 2037 because:
Size | Block pair (l,m)(m,r) to merge | Left over block | Encoding
-----+--------------------------------+------------------+-----------
1 | ((0, 0), (0, 1)) | | 1
1 | ((2, 2), (2, 3)) | | 11
1 | ((4, 4), (4, 5)) | | 111
1 | ((6, 6), (6, 7)) | | 1111
1 | ((8, 8), (8, 9)) | | 11111
2 | ((0, 1), (1, 3)) | | 111111
2 | ((4, 5), (5, 7)) | | 1111111
2 | | ((8, 9), (9, 9)) | 11111110
4 | ((0, 3), (3, 7)) | | 111111101
4 | | ((8, 9), (9, 9)) | 1111111010
8 | ((0, 7), (7, 9)) | | 11111110101
11111110101 in base 10 is 2037.
For n=10, the merges performed on 1,...,10 begin with pairs of "blocks" of length 1 each,
1 2 3 4 5 6 7 8 9 10
\--/ \--/ \--/ \--/ \---/
1 1 1 1 1 encoding
[1 2] [3 4] [5 6] [7 8] [9 10]
\---------/ \---------/
1 1 0 encoding
Similarly on the resulting 3 blocks
[1 2 3 4] [5 6 7 8] [9 10]
\-----------------/
1 0 encoding
Then a merge of the resulting 2 blocks to a single sorted block.
[1 2 3 4 5 6 7 8] [9 10]
\----------------------/
1 encoding
These encodings are then a(10) = binary 11111 110 10 1 = 2037.
def a(n):
if n == 1: return 0
s, t, n1 = 1, 0, (n - 1)
while s < n:
d = s << 1
for l in range(0, n, d):
m,r = min(l - 1 + s, n1), min(l - 1 + d, n1)
t = (t << 1) + int(m < r)
s = d
return t
print([a(n) for n in range (1,22)])
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