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A118255
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a(1)=1, then a(n)=2*a(n-1) if n is prime, a(n)=2*a(n-1)+1 if n not prime.
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9
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1, 2, 4, 9, 18, 37, 74, 149, 299, 599, 1198, 2397, 4794, 9589, 19179, 38359, 76718, 153437, 306874, 613749, 1227499, 2454999, 4909998, 9819997, 19639995, 39279991, 78559983, 157119967, 314239934, 628479869, 1256959738, 2513919477, 5027838955, 10055677911
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history;
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internal format)
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OFFSET
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1,2
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COMMENTS
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In base 2 a(n) is the concatenation for i=1 to n of A005171(i).
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LINKS
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Michael S. Branicky, Table of n, a(n) for n = 1..3322 (terms 1..1000 from Harvey P. Dale)
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FORMULA
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a(n) = floor(k * 2^n) where k = 0.585317... = 1 - A051006. [Charles R Greathouse IV, Dec 27 2011]
From Ridouane Oudra, Aug 26 2019: (Start)
a(n) = 2^n - 1 - (1/2)*(pi(n) + Sum_{i=1..n} 2^(n-i)*pi(i)), where pi = A000720
a(n) = A000225(n) - A072762(n). (End)
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EXAMPLE
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a(2) = 2*1 = 2 as 2 is prime;
a(3) = 2*2 = 4 as 3 is prime;
a(4) = 2*4+1 = 9 as 4 is composite;
a(5) = 2*9 = 18 as 5 is prime.
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MAPLE
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f:=proc(n) option remember; if n=1 then RETURN(1); fi; if isprime(n) then 2*f(n-1) else 2*f(n-1)+1; fi; end; # N. J. A. Sloane
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MATHEMATICA
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nxt[{n_, a_}]:={n+1, If[PrimeQ[n+1], 2a, 2a+1]}; Transpose[NestList[nxt, {1, 1}, 40]][[2]] (* Harvey P. Dale, Jan 22 2015 *)
Array[FromDigits[#, 2] &@ Array[Boole[! PrimeQ@ #] &, #] &, 34] (* Michael De Vlieger, Nov 01 2016 *)
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PROG
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(Python)
from sympy import isprime, prime
def a(n): return int("".join(str(1-isprime(i)) for i in range(1, n+1)), 2)
print([a(n) for n in range(1, 35)]) # Michael S. Branicky, Jan 10 2022
(Python) # faster version for initial segment of sequence
from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
an = 0
for k in count(1):
an = 2 * an + int(not isprime(k))
yield an
print(list(islice(agen(), 34))) # Michael S. Branicky, Jan 10 2022
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CROSSREFS
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Cf. A000225, A005171, A051006, A072762, A118256, A118257.
Sequence in context: A152537 A182028 A081253 * A206927 A019299 A052932
Adjacent sequences: A118252 A118253 A118254 * A118256 A118257 A118258
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KEYWORD
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nonn
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AUTHOR
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Pierre CAMI, Apr 19 2006
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EXTENSIONS
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Corrected by Omar E. Pol, Nov 08 2007
Corrections verified by N. J. A. Sloane, Nov 17 2007
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STATUS
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approved
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