A045715 - OEIS

%I #29 Dec 08 2024 17:19:17

%S 97,907,911,919,929,937,941,947,953,967,971,977,983,991,997,9001,9007,

%T 9011,9013,9029,9041,9043,9049,9059,9067,9091,9103,9109,9127,9133,

%U 9137,9151,9157,9161,9173,9181,9187,9199,9203,9209,9221,9227,9239,9241,9257

%N Primes with first digit 9.

%H Vincenzo Librandi, <a href="/A045715/b045715.txt">Table of n, a(n) for n = 1..1000</a>

%t Flatten[Table[Prime[Range[PrimePi[9 * 10^n] + 1, PrimePi[10^(n + 1)]]], {n, 3}]] (* _Alonso del Arte_, Jul 19 2014 *)

%o (Magma) [p: p in PrimesUpTo(10^4) | Intseq(p)[#Intseq(p)] eq 9]; // _Bruno Berselli_, Jul 19 2014

%o (Magma) [p: p in PrimesInInterval(9*10^n,10^(n+1)), n in [0..3]]; // _Bruno Berselli_, Aug 08 2014

%o (Python)

%o from itertools import chain, count, islice

%o def A045715_gen(): # generator of terms

%o     return chain.from_iterable(primerange(9*(m:=10**l),10*m) for l in count(0))

%o A045715_list = list(islice(A045715_gen(),40)) # _Chai Wah Wu_, Dec 08 2024

%o (Python)

%o from sympy import primepi

%o def A045715(n):

%o     def bisection(f,kmin=0,kmax=1):

%o         while f(kmax) > kmax: kmax <<= 1

%o         while kmax-kmin > 1:

%o             kmid = kmax+kmin>>1

%o             if f(kmid) <= kmid:

%o                 kmax = kmid

%o             else:

%o                 kmin = kmid

%o         return kmax

%o     def f(x): return n+x+primepi(min(9*(m:=10**(l:=len(str(x))-1))-1,x))-primepi(min(10*m-1,x))+sum(primepi(9*(m:=10**i)-1)-primepi(10*m-1) for i in range(l))

%o     return bisection(f,n,n) # _Chai Wah Wu_, Dec 08 2024

%Y For primes with initial digit d (1 <= d <= 9) see A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715; A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509.

%Y Column k=9 of A262369.

%K nonn,base,easy

%O 1,1

%A _Felice Russo_

%E More terms from _Erich Friedman_.