%I #32 Dec 08 2024 17:19:01

%S 7,71,73,79,701,709,719,727,733,739,743,751,757,761,769,773,787,797,

%T 7001,7013,7019,7027,7039,7043,7057,7069,7079,7103,7109,7121,7127,

%U 7129,7151,7159,7177,7187,7193,7207,7211,7213,7219,7229,7237,7243,7247,7253,7283

%N Primes with first digit 7.

%H Vincenzo Librandi, <a href="/A045713/b045713.txt">Table of n, a(n) for n = 1..5000</a>

%t Select[ Table[ Prime[ n ], {n, 1000} ], First[ IntegerDigits[ # ]]==7& ]

%o (Magma) [p: p in PrimesUpTo(7300) | Intseq(p)[#Intseq(p)] eq 7]; // _Vincenzo Librandi_, Aug 08 2014

%o (Python)

%o from itertools import chain, count, islice

%o from sympy import primerange

%o def A045713_gen(): # generator of terms

%o return chain.from_iterable(primerange(7*(m:=10**l),m<<3) for l in count(0))

%o A045713_list = list(islice(A045713_gen(),40)) # _Chai Wah Wu_, Dec 08 2024

%o (Python)

%o from sympy import primepi

%o def A045713(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(7*(m:=10**(l:=len(str(x))-1))-1,x))-primepi(min((m<<3)-1,x))+sum(primepi(7*(m:=10**i)-1)-primepi((m<<3)-1) for i in range(l))

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

%Y Cf. A000040.

%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=7 of A262369.

%K nonn,base

%O 1,1

%A _Felice Russo_

%E More terms from _Erich Friedman_.

%E Corrected by _Jud McCranie_, Jan 03 2001

%E a(13)=757 added from _Vincenzo Librandi_, Aug 08 2014