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A045710 Primes with first digit 4.

%I A045710 #28 Dec 08 2024 17:18:17
%S A045710 41,43,47,401,409,419,421,431,433,439,443,449,457,461,463,467,479,487,
%T A045710 491,499,4001,4003,4007,4013,4019,4021,4027,4049,4051,4057,4073,4079,
%U A045710 4091,4093,4099,4111,4127,4129,4133,4139,4153,4157,4159,4177,4201,4211
%N A045710 Primes with first digit 4.
%H A045710 Vincenzo Librandi, <a href="/A045710/b045710.txt">Table of n, a(n) for n = 1..5000</a>
%t A045710 Select[Table[Prime[n], {n, 1000}], First[IntegerDigits[#]] == 4 &]
%t A045710 Flatten[Table[Prime[Range[PrimePi[4 * 10^n] + 1, PrimePi[5 * 10^n]]], {n, 3}]] (* _Alonso del Arte_, Jul 19 2014 *)
%o A045710 (Magma) [p: p in PrimesUpTo(10^4) | Intseq(p)[#Intseq(p)] eq 4]; // _Bruno Berselli_, Jul 19 2014
%o A045710 (Python)
%o A045710 from itertools import chain, count, islice
%o A045710 from sympy import primerange
%o A045710 def A045710_gen(): # generator of terms
%o A045710     return chain.from_iterable(primerange((m:=10**l)<<2,5*m) for l in count(0))
%o A045710 A045710_list = list(islice(A045710_gen(),40)) # _Chai Wah Wu_, Dec 08 2024
%o A045710 (Python)
%o A045710 from sympy import primepi
%o A045710 def A045710(n):
%o A045710     def bisection(f,kmin=0,kmax=1):
%o A045710         while f(kmax) > kmax: kmax <<= 1
%o A045710         while kmax-kmin > 1:
%o A045710             kmid = kmax+kmin>>1
%o A045710             if f(kmid) <= kmid:
%o A045710                 kmax = kmid
%o A045710             else:
%o A045710                 kmin = kmid
%o A045710         return kmax
%o A045710     def f(x): return n+x+primepi(min(((m:=10**(l:=len(str(x))-1))<<2)-1,x))-primepi(min(5*m-1,x))+sum(primepi(((m:=10**i)<<2)-1)-primepi(5*m-1) for i in range(l))
%o A045710     return bisection(f,n,n) # _Chai Wah Wu_, Dec 08 2024
%Y A045710 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 A045710 Column k=4 of A262369.
%K A045710 nonn,base,easy
%O A045710 1,1
%A A045710 _Felice Russo_
%E A045710 More terms from _Erich Friedman_.