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.
For all of {2,5,11,23,47}, i.e. at positions {j}={1,3,5,9,15} a(j)=5. Similarly for indices of all terms in {89,...,5759} a(i)=6. No chains are intelligible with length = 1 because the minimal chain enclose one Sophie Germain and also one safe prime. Dominant values are 0 and 2.
a(11)-a(10) = 21 means that between 1024 and 2048 exactly 21 primes introduce Cunningham chains: {1031, 1049, 1103, 1223, 1229, 1289, 1409, 1451, 1481, 1499, 1511, 1559, 1583, 1601, 1733, 1811, 1889, 1901, 1931, 1973, 2003}.
Their lengths are 2, 3 or 4. Thus the complete chains spread over more than one binary size-zone: {1409, 2819, 5639, 11279}. The primes 1439 and 2879 also form a chain but 1439 is not at the beginning of that chain, 89 is.
c = 0; k = 1; Do[ While[k <= 2^n, If[ PrimeQ[k] && !PrimeQ[(k - 1)/2] && PrimeQ[2k + 1], c++ ]; k++ ]; Print[c], {n, 1, 29}]
from itertools import count, islice
from sympy import isprime, primerange
def c(p): return not isprime((p-1)//2) and isprime(2*p+1)
def agen():
s = 1
for n in count(2):
yield s; s += sum(1 for p in primerange(2**(n-1)+1, 2**n) if c(p))
print(list(islice(agen(), 20))) # Michael S. Branicky, Oct 09 2022
Select[Prime[Range[70]],PrimeOmega[#-1]>2&] (* Harvey P. Dale, May 28 2012 *)
f(n) = forprime(x=2,n,y=bigomega(x-1);if(y>2,print1(x",")))
a(1)=1 counts one prime (the 3) between 2 and 5; a(2)=1 counts one prime (the 5) between 3 and 7; a(5)=6 counts the primes from 23 to 43 between 19 and 53.
isA005385 := proc(n) if isprime(n) then isprime( (n-1)/2 ) ; else false; fi; end: isA059456 := proc(n) if isprime(n) then not isprime( (n-1)/2 ) ; else false; fi; end: A059456 := proc(n) if n = 1 then 2; else for a from procname(n-1)+1 do if isA059456(a) then RETURN(a) ; fi; od: fi; end: A005385 := proc(n) if n = 1 then 5; else for a from procname(n-1)+1 do if isA005385(a) then RETURN(a) ; fi; od: fi; end: A000720 := proc(n) numtheory[pi](n) ; end: A163757 := proc(n) max(0,A000720(A005385(n)-1)-A000720(A059456(n))) ; end: seq(A163757(n),n=1..80) ; # R. J. Mathar, Aug 06 2009
Select[Prime[Range[70]]-1,PrimeOmega[#]!=2&] (* Harvey P. Dale, Jan 14 2015 *)
is(n)=isprime(n+1) && !isprime(n\2) \\ Charles R Greathouse IV, Dec 29 2024
Comments