A045713 -id:A045713 - cp's OEIS frontend

A045711 Primes with first digit 5.

5, 53, 59, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 5003, 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261

Offset: 1

Felice Russo

Subsequence of A000040.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Column k=5 of A262369.

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.

[p: p in PrimesUpTo(5300) | Intseq(p)[#Intseq(p)] eq 5]; // Vincenzo Librandi, Aug 08 2014

Select[Table[Prime[n], {n, 5300}], First[IntegerDigits[#]]==5 &] (* Vincenzo Librandi, Aug 08 2014 *)

from itertools import chain, count, islice
from sympy import primerange
def A045711_gen(): # generator of terms
    return chain.from_iterable(primerange(5*(m:=10**l),6*m) for l in count(0))
A045711_list = list(islice(A045711_gen(),40)) # Chai Wah Wu, Dec 08 2024

from sympy import primepi
def A045711(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x+primepi(min(5*(m:=10**(l:=len(str(x))-1))-1,x))-primepi(min(6*m-1,x))+sum(primepi(5*(m:=10**i)-1)-primepi(6*m-1) for i in range(l))
    return bisection(f,n,n) # Chai Wah Wu, Dec 08 2024

More terms from Erich Friedman.

Leading 5 added by Jaroslav Krizek, Apr 29 2010

A045707 Primes with first digit 1.

Original entry on oeis.org

11, 13, 17, 19, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151

Offset: 1

Felice Russo

Also primes with all divisors starting with digit 1. Complement of A206288 (nonprime numbers with all divisors starting with digit 1) with respect to A206287 (numbers with all divisors starting with digit 1). - Jaroslav Krizek, Mar 04 2012
Cohen and Katz show that the set of primes with first digit 1 has no natural density, but has supernatural/Dirichlet density log_{10} (2) ~= 0.3, the primes with first digit 2 have (supernatural) density log_{10} (3/2) ~= 0.176, ... and the primes with first digit 9 have density log_{10} (10/9) ~= 0.046. This would seem to explain the first digit phenomenon. Note that sum_{k = 1}^9 log_{10} (k+1)/k = 1. - Gary McGuire, Dec 22 2004
Lower density is 1/9, upper density is 5/9. The Dirichlet density, if it exists, is always between the lower and upper density (as it does and is in this case). - Charles R Greathouse IV, Sep 26 2022

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Daniel I. A. Cohen and Talbot M. Katz, Prime numbers and the first digit phenomenon, J. Number Theory 18 (1984), 261-268.

For primes with initial digit d (1 <= d <= 9) see A045707 (1, this sequence), A045708 (2), A045709 (3), A045710 (4), A045711 (5), A045712 (6), A045713 (7), A045714 (8), A045715 (9).
Cf. A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509
Column k=1 of A262369.

[p: p in PrimesUpTo(10^4) | IsOne(Intseq(p)[#Intseq(p)])]; // Bruno Berselli, Jul 19 2014

Select[Table[Prime[n], {n, 500}], First[IntegerDigits[#]] == 1 &]
Flatten[Table[Prime[Range[PrimePi[10^n] + 1, PrimePi[2 * 10^n]]], {n, 3}]] (* Alonso del Arte, Jul 18 2014 *)

list(lim)=my(v=[]); for(d=1,#digits(lim\=1)-1, v=concat(v,primes([10^d,min(lim,2*10^d-1)]))); v \\ Charles R Greathouse IV, Sep 26 2022

from itertools import chain, count, islice
from sympy import primerange
def A045707_gen(): # generator of terms
    return chain.from_iterable(primerange(m:=10**l,(m<<1)) for l in count(0))
A045707_list = list(islice(A045707_gen(),40)) # Chai Wah Wu, Dec 07 2024

from sympy import primepi
def A045707(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x+primepi((m:=10**(l:=len(str(x))-1))-1)-primepi(min((m<<1)-1,x))+sum(primepi((m:=10**i)-1)-primepi((m<<1)-1) for i in range(l))
    return bisection(f,n,n) # Chai Wah Wu, Dec 07 2024

More terms from Erich Friedman.

Cohen-Katz reference from Victor S. Miller, Dec 21 2004

A045708 Primes with first digit 2.

Original entry on oeis.org

2, 23, 29, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221

Offset: 1

Felice Russo

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040.
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
Cf. A000030, subsequence of A208272.
Column k=2 of A262369.

a045708 n = a045708_list !! (n-1)
a045708_list = filter ((== 2) . a000030) a000040_list
-- Reinhard Zumkeller, Mar 16 2012

[p: p in PrimesUpTo(2300) | Intseq(p)[#Intseq(p)] eq 2]; // Vincenzo Librandi, Aug 08 2014

Select[Table[Prime[n], {n, 3000}], First[IntegerDigits[#]]==2 &] (* Vincenzo Librandi, Aug 08 2014 *)

from sympy import isprime
def agen(limit=float('inf')):
  yield 2
  digits, adder = 1, 20
  while True:
    for i in range(1, 10**digits, 2):
      test = adder + i
      if test > limit: return
      if isprime(test): yield test
    digits, adder = digits+1, adder*10
agento = lambda lim: agen(limit=lim)
print(list(agento(2222))) # Michael S. Branicky, Feb 23 2021

from sympy import primepi
def A045708(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x+primepi(min(((m:=10**(l:=len(str(x))-1))<<1)-1,x))-primepi(min(3*m-1,x))+sum(primepi(((m:=10**i)<<1)-1)-primepi(3*m-1) for i in range(l))
    return bisection(f,n,n) # Chai Wah Wu, Dec 07 2024

See A045707 for comments on density of these sequences.

More terms from Erich Friedman.

Offset fixed by Reinhard Zumkeller, Mar 15 2012

A045709 Primes with first digit 3.

Original entry on oeis.org

3, 31, 37, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229

Offset: 1

Felice Russo

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A000040.
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.
Column k=3 of A262369.

[p: p in PrimesUpTo(3300) | Intseq(p)[#Intseq(p)] eq 3]; // Vincenzo Librandi, Aug 08 2014

Select[Table[Prime[n], {n, 4000}], First[IntegerDigits[#]]==3 &] (* Vincenzo Librandi, Aug 08 2014 *)

isok(n) = isprime(n) && (digits(n, 10)[1] == 3) \\ Michel Marcus, Jun 08 2013

from itertools import chain, count, islice
from sympy import primerange
def A045709_gen(): # generator of terms
    return chain.from_iterable(primerange(3*(m:=10**l),m<<2) for l in count(0))
A045709_list = list(islice(A045709_gen(),40)) # Chai Wah Wu, Dec 07 2024

from sympy import primepi
def A045709(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x+primepi(min(3*(m:=10**(l:=len(str(x))-1))-1,x))-primepi(min((m<<2)-1,x))+sum(primepi(3*(m:=10**i)-1)-primepi((m<<2)-1) for i in range(l))
    return bisection(f,n,n) # Chai Wah Wu, Dec 07 2024

More terms from Erich Friedman.

A045714 Primes with first digit 8.

Original entry on oeis.org

83, 89, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 8009, 8011, 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117, 8123, 8147, 8161, 8167, 8171, 8179, 8191, 8209, 8219, 8221, 8231, 8233, 8237, 8243

Offset: 1

Felice Russo

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

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.

Column k=8 of A262369.

[p: p in PrimesUpTo(10^4) | Intseq(p)[#Intseq(p)] eq 8]; // Bruno Berselli, Jul 19 2014

Flatten[Table[Prime[Range[PrimePi[8 * 10^n] + 1, PrimePi[9 * 10^n]]], {n, 3}]] (* Alonso del Arte, Jul 19 2014 *)

from itertools import chain, count, islice
from sympy import primerange
def A045714_gen(): # generator of terms
    return chain.from_iterable(primerange((m:=10**l)<<3,9*m) for l in count(0))
A045714_list = list(islice(A045714_gen(),40)) # Chai Wah Wu, Dec 08 2024

from sympy import primepi
def A045714(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x+primepi(min(((m:=10**(l:=len(str(x))-1))<<3)-1,x))-primepi(min(9*m-1,x))+sum(primepi(((m:=10**i)<<3)-1)-primepi(9*m-1) for i in range(l))
    return bisection(f,n,n) # Chai Wah Wu, Dec 08 2024

More terms from Erich Friedman.

A073517 Number of primes less than 10^n with initial digit 1.

Original entry on oeis.org

0, 4, 25, 160, 1193, 9585, 80020, 686048, 6003530, 53378283, 480532488, 4369582734, 40063566855, 369893939287, 3435376839800, 32069022099022, 300694113015105, 2830466318006780, 26735673312004455, 253315661161665338, 2406763761677705769, 22923886160712831134, 218839439542390117580

Offset: 1

Shyam Sunder Gupta, Aug 14 2002

			a(2)=4 because there are 4 primes up to 10^2 whose initial digit is 1 (11, 13, 17 and 19).

Chris K. Caldwell, How Many Primes Are There?
Xavier Gourdon & Pascal Sebah, Counting the number of primes
Henri Lifchitz, Parity of Pi(n)
Thomas R. Nicely, Some Results of Computational Research in Prime Numbers [See local copy in A007053]
Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x)

Cf. A000720 (pi), A073509 to A073517, their sum is A006880.

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.

f[n_] := f[n] = PrimePi[2*10^n] - PrimePi[10^n] + f[n - 1]; f[0] = 0; Table[ f[n], {n, 0, 13}]

a(n,d=1)=sum(k=0, n-1, primepi((d+1)*10^k-1) - primepi(d*10^k-1)) \\ Andrew Howroyd, Dec 15 2024

a(n) = Sum_{k=0..n-1} pi(2*10^k-1) - pi(10^k-1). - Andrew Howroyd, Dec 15 2024

Edited and extended by Robert G. Wilson v, Aug 29 2002
a(21)-a(22) added by David Baugh, Mar 21 2015
a(23) from Chai Wah Wu, Sep 18 2018
Offset corrected by Andrew Howroyd, Dec 15 2024

A045710 Primes with first digit 4.

Original entry on oeis.org

41, 43, 47, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211

Offset: 1

Felice Russo

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

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.

Column k=4 of A262369.

[p: p in PrimesUpTo(10^4) | Intseq(p)[#Intseq(p)] eq 4]; // Bruno Berselli, Jul 19 2014

Select[Table[Prime[n], {n, 1000}], First[IntegerDigits[#]] == 4 &]
Flatten[Table[Prime[Range[PrimePi[4 * 10^n] + 1, PrimePi[5 * 10^n]]], {n, 3}]] (* Alonso del Arte, Jul 19 2014 *)

from itertools import chain, count, islice
from sympy import primerange
def A045710_gen(): # generator of terms
    return chain.from_iterable(primerange((m:=10**l)<<2,5*m) for l in count(0))
A045710_list = list(islice(A045710_gen(),40)) # Chai Wah Wu, Dec 08 2024

from sympy import primepi
def A045710(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    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))
    return bisection(f,n,n) # Chai Wah Wu, Dec 08 2024

More terms from Erich Friedman.

A045712 Primes with first digit 6.

Original entry on oeis.org

61, 67, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229

Offset: 1

Felice Russo

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

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.

Column k=6 of A262369.

[p: p in PrimesUpTo(10^4) | Intseq(p)[#Intseq(p)] eq 6]; // Bruno Berselli, Jul 19 2014

Flatten[Table[Prime[Range[PrimePi[6 * 10^n] + 1, PrimePi[7 * 10^n]]], {n, 3}]] (* Alonso del Arte, Jul 19 2014 *)
Select[Table[Prime[n],{n, 7000}], First[IntegerDigits[#]]==6 &] (* Vincenzo Librandi, Aug 08 2014 *)

from itertools import chain, count, islice
from sympy import primerange
def A045712_gen(): # generator of terms
    return chain.from_iterable(primerange(6*(m:=10**l),7*m) for l in count(0))
list(islice(A045712_gen(),40)) # Chai Wah Wu, Dec 08 2024

from sympy import primepi
def A045712(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x+primepi(min(6*(m:=10**(l:=len(str(x))-1))-1,x))-primepi(min(7*m-1,x))+sum(primepi(6*(m:=10**i)-1)-primepi(7*m-1) for i in range(l))
    return bisection(f,n,n) # Chai Wah Wu, Dec 08 2024

More terms from Erich Friedman.

A073509 Number of primes less than 10^n with initial digit 9.

Original entry on oeis.org

0, 1, 15, 127, 1006, 8230, 70320, 614821, 5453140, 48982456, 444608278, 4070532710, 37535715441, 348245215460, 3247889171908, 30429496751905, 286235215995588, 2702000272361599, 25586688305447928, 242978340446949438, 2313264023790027111, 22074118786158858975

Offset: 1

Shyam Sunder Gupta, Aug 14 2002

			a(2) = 1 because there is 1 prime less than 100 whose initial digit is 9, i.e., 97.

Thomas R. Nicely, Some Results of Computational Research in Prime Numbers [See local copy in A007053]
Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x)

A006880(n) = A073509(n)+ ... + A073516(n)+A073517(n-1).

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

f[n_] := f[n] = PrimePi[10^(n + 1)] - PrimePi[9*10^n] + f[n - 1]; f[0] = 0; Table[f[n], {n, 0, 12}]

Edited and extended by Robert G. Wilson v, Aug 29 2002

a(20)-a(22) added by David Baugh, Mar 22 2015

A073510 Number of primes less than 10^n with initial digit 8.

Original entry on oeis.org

0, 2, 17, 127, 1003, 8326, 71038, 618610, 5481646, 49221187, 446590932, 4087194991, 37677478288, 349465615584, 3258501713644, 30522628848972, 287059041039078, 2709339704446862, 25652489700275636, 243571629996128384, 2318640708958531064, 22123070798400775157

Offset: 1

Shyam Sunder Gupta, Aug 14 2002

			a(2)=2 because there are 2 primes up to 10^2 whose initial digit is 8 (namely 83 and 89).

Thomas R. Nicely, Some Results of Computational Research in Prime Numbers [See local copy in A007053]
Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x)

A006880(n) = A073509(n)+ ... + A073516(n)+A073517(n-1).

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

f[n_] := f[n] = PrimePi[9*10^n] - PrimePi[8*10^n] + f[n - 1]; f[0] = 0; Table[ f[n], {n, 0, 12}]

Edited and extended by Robert G. Wilson v, Aug 29 2002

a(20)-a(22) added by David Baugh, Mar 22 2015

cp's OEIS Frontend

A045711 Primes with first digit 5.

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A045707 Primes with first digit 1.

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A045708 Primes with first digit 2.

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A045709 Primes with first digit 3.

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A045714 Primes with first digit 8.

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A073517 Number of primes less than 10^n with initial digit 1.

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A045710 Primes with first digit 4.