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A101909 sorted and duplicates removed.
+20
5
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73
PROG
(PARI) bet2n4n(n) = { local(c, c1, x, y); a=vector(5001); for(x=1, n, c=0; forprime(y=2*x+1, 4*x-1, c++; ); a[x] = c; ); b=vecsort(a); for(x=1, 5000, if(b[x]>0, if(b[x]<>b[x+1], print1(b[x]", ") ) ); ) }
(PARI) s=0; v=vectorsmall(10^6, n, s+=isprime(4*n-1)+isprime(4*n-3)-isprime(2*n-1)); v=vecsort(v, , 8); vecextract(v, Str("1..", #v\2)) \\ Charles R Greathouse IV, Mar 12 2012
Numbers that do not occur in A101909 (= number of primes between 2n and 4n).
+20
3
11, 79, 134, 184, 186, 215, 245, 262, 284, 305, 387, 544, 694, 700, 706, 776, 814, 881, 939, 974, 1002, 1027, 1079, 1104, 1133, 1146, 1184, 1193, 1207, 1354, 1387, 1415, 1441, 1495, 1574, 1587, 1608, 1662, 1690, 1801, 1915, 1987, 2054, 2067, 2104, 2170
EXAMPLE
11 is the first number that does not equal a count of primes between 2n and 4n for some n.
MATHEMATICA
f[n_] := PrimePi[4n] - PrimePi[2n]; t = Union[ Table[ f[n], {n, 12000}]]; Complement[ Range[ t[[ -1]]], t] (* Robert G. Wilson v, Feb 10 2005 *)
PROG
(PARI) bet2n4n(n)={ my( b=vecsort(vector(n, x, my(c=0); forprime(y=2*x+1, 4*x-1, c++); c))); for(x=1, n-2, while(b[x+1]-b[x]>1, print1(b[x]++, ", ")))} \\ It's probably faster to use A101909 instead of forprime(...). Edited and corrected by M. F. Hasler, Sep 29 2019 (PARI) primecount(a, b)=primepi(b)-primepi(a);
v=vector(20000);
for(k=1, oo, j=primecount(2*k, 4*k); if(j>#v, break, v[j]++));
for(k=1, 2170, if(v[k]==0, print1(k, ", "))) \\ Hugo Pfoertner, Sep 29 2019
Numbers that occur exactly once in A101909 (= count of primes between 2n and 4n).
+20
2
1, 3, 5, 8, 22, 36, 37, 46, 47, 48, 53, 63, 83, 98, 99, 101, 105, 108, 113, 114, 127, 135, 139, 148, 150, 155, 158, 171, 172, 173, 174, 175, 177, 178, 188, 205, 210, 218, 219, 220, 221, 226, 231, 240, 246, 254, 277, 282, 297, 298, 301, 303, 327, 333, 334, 339
EXAMPLE
There are 5 primes between 16 and 32 and nowhere else between 2n and 4n.
PROG
(PARI) bet2n4n(n)={ my(b=vecsort(vector(n, x, my(c=0); forprime(y=2*x+1, 4*x-1, c++); c))); print1(1", "); for(x=1, n-2, if(b[x+1]>b[x] && b[x+1]<b[x+2], print1(b[x+1]", ")))} \\ Edited and corrected, following a suggestion by Don Reble. - M. F. Hasler, Sep 29 2019
Number of primes between n and 2n exclusive.
+10
58
0, 1, 1, 2, 1, 2, 2, 2, 3, 4, 3, 4, 3, 3, 4, 5, 4, 4, 4, 4, 5, 6, 5, 6, 6, 6, 7, 7, 6, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 9, 10, 9, 9, 10, 10, 9, 9, 10, 10, 11, 12, 11, 12, 13, 13, 14, 14, 13, 13, 12, 12, 12, 13, 13, 14, 13, 13, 14, 15, 14, 14, 13, 13, 14, 15, 15
COMMENTS
The number of partitions of 2n+2 into exactly two parts where the first part is a prime strictly less than 2n+1. - Wesley Ivan Hurt, Aug 21 2013
REFERENCES
M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer NY 2001.
EXAMPLE
a(35)=8 since eight consecutive primes (37,41,43,47,53,59,61,67) are located between 35 and 70.
MAPLE
a := proc(n) local counter, i; counter := 0; from i from n+1 to 2*n-1 do if isprime(i) then counter := counter +1; fi; od; return counter; end:
with(numtheory); seq(pi(2*k-1)-pi(k), k=1..100); # Wesley Ivan Hurt, Aug 21 2013
PROG
(PARI) { for (n=1, 1000, write("b060715.txt", n, " ", primepi(2*n - 1) - primepi(n)); ) } \\ Harry J. Smith, Jul 10 2009 (Haskell)
(Magma) [0] cat [#PrimesInInterval(n+1, 2*n-1): n in [2..80]]; // Bruno Berselli, Sep 05 2012 (Python) from sympy import primerange as pr
EXTENSIONS
Corrected by Dug Eichelberger (dug(AT)mit.edu), Jun 04 2001
More terms from Larry Reeves (larryr(AT)acm.org), Jun 05 2001
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