A118255
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.
Original entry on oeis.org
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
Offset: 1
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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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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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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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
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# 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
A118256
Concatenation for i=1 to n of A005171(i); also A118255 in base 2.
Original entry on oeis.org
1, 10, 100, 1001, 10010, 100101, 1001010, 10010101, 100101011, 1001010111, 10010101110, 100101011101, 1001010111010, 10010101110101, 100101011101011, 1001010111010111, 10010101110101110, 100101011101011101, 1001010111010111010, 10010101110101110101, 100101011101011101011
Offset: 1
A005171 : 1,0,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,1 ................
a(1)=1, a(2)=10, a(3)=100, a(4)=1001, ...
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Array[FromDigits@ Array[Boole[! PrimeQ@ #] &, #] &, 21] (* or *)
FromDigits@ IntegerDigits[#, 2] & /@ Last@ Transpose@ NestList[{#1 + 1, If[PrimeQ[#1 + 1], 2 #2, 2 #2 + 1]} & @@ # &, {1, 1}, 21] (* Michael De Vlieger, Nov 01 2016, latter after Harvey P. Dale at A118255 *)
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a(n) = sum(k=1, n, !isprime(k)*10^(n-k)); \\ Michel Marcus, Nov 01 2016
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from sympy import isprime
def a(n): return int("".join(str(1-isprime(i)) for i in range(1, n+1)))
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Jan 10 2022
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# 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 = 10 * an + int(not isprime(k))
yield an
print(list(islice(agen(), 21))) # Michael S. Branicky, Jan 10 2022
A139119
Primes whose binary representation shows the distribution of prime numbers using "0" for primes and "1" for nonprime numbers.
Original entry on oeis.org
2, 37, 149, 599, 153437, 39279991, 628479869, 11056334789265976156021, 3263254052013454238294691704608897001027543, 7524551543123483484068003542235060639999919940760883731360687
Offset: 1
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Select[Table[FromDigits[Boole /@ Not /@ PrimeQ /@ Range@k, 2], {k, 1, 100}], PrimeQ] (* Federico Provvedi, Oct 07 2013 *)
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f(n) = fromdigits(vector(n, k, !isprime(k)), 2); \\ A118255
lista(nn) = for (n=1, nn, if (isprime(p=f(n)), print1(p, ", "))); \\ Michel Marcus, Apr 04 2019
A139120
Primes that show the distribution of prime numbers using "0" for primes and "1" for nonprime numbers.
Original entry on oeis.org
10010101, 1001010111010111, 100101011101011101011101111101011111011101011101111101111101011111011101011111011101111101111111011101011101011101111111111111011101111101011111111101011111011111011101111101111, 100101011101011101011101111101011111011101011101111101111101011111011101011111011101111101111111011101011101011101111111111111011101111101011111111101011111011111011101111101111101011111111101011101011111111
Offset: 1
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A118255[n_] := Module[{},
If[n == 1, A118255[1] = 1,
If[PrimeQ[n], A118255[n] = 2 A118255[n - 1],
A118255[n] = 2 A118255[n - 1] + 1]]];
Select[Table[FromDigits[IntegerDigits[A118255[n], 2]], {n, 1, 1000}], PrimeQ] (* Robert Price, Apr 03 2019 *)
A139122
Primes whose binary representation shows the distribution of prime numbers up to some prime minus 1, using "0" for primes and "1" for nonprime numbers.
Original entry on oeis.org
2, 37, 599, 153437, 628479869
Offset: 1
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Select[Table[ sum = 0; For[i = 1, i <= Prime[n] - 1 , i++, sum = sum*2; If[! PrimeQ[i], sum++]]; sum, {n, 1, 1000}], PrimeQ[#] &] (* Robert Price, Apr 03 2019 *)
Module[{nn=500,p,x},p=Table[If[PrimeQ[n],0,1],{n,nn}];x=SequencePosition[p,{1,0}][[All,1]];Join[{2},Select[Table[FromDigits[Take[p,k],2],{k,x}],PrimeQ]]] (* Harvey P. Dale, Jun 15 2022 *)
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f(n) = fromdigits(vector(prime(n)-1, k, !isprime(k)), 2); \\ A139102
lista(nn) = for (n=1, nn, if (isprime(p=f(n)), print1(p, ", ")));
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# uses agen() in A139102
from sympy import isprime
print(list(islice(filter(isprime, agen()), 5))) # Michael S. Branicky, Jan 25 2022
Showing 1-5 of 5 results.
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