A045714 -id:A045714

A045711

Primes with first digit 5.

+10
29

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Subsequence of A000040.

LINKS

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

nonn,base,easy

AUTHOR

Felice Russo

EXTENSIONS

More terms from Erich Friedman.

Leading 5 added by Jaroslav Krizek, Apr 29 2010

STATUS

approved

A045707

Primes with first digit 1.

+10
27

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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.

MATHEMATICA

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 *)

PROG

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

(PARI) 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

CROSSREFS

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.

KEYWORD

nonn,base,easy

AUTHOR

Felice Russo

EXTENSIONS

More terms from Erich Friedman.

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

STATUS

approved

A045708

Primes with first digit 2.

+10
24

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

FORMULA

See A045707 for comments on density of these sequences.

MATHEMATICA

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

PROG

(Haskell)

a045708 n = a045708_list !! (n-1)

a045708_list = filter ((== 2) . a000030) a000040_list

-- Reinhard Zumkeller, Mar 16 2012

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

(Python)

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

CROSSREFS

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.

KEYWORD

nonn,base,easy

AUTHOR

Felice Russo

EXTENSIONS

More terms from Erich Friedman.

Offset fixed by Reinhard Zumkeller, Mar 15 2012

STATUS

approved

A073517

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

+10
24

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..22.

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)

EXAMPLE

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

MATHEMATICA

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

CROSSREFS

Cf. 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

KEYWORD

base,hard,nonn

AUTHOR

Shyam Sunder Gupta, Aug 14 2002

EXTENSIONS

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

a(20)-a(21) added by David Baugh, Mar 21 2015

a(22) from Chai Wah Wu, Sep 18 2018

STATUS

approved

A045709

Primes with first digit 3.

+10
23

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

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.

KEYWORD

nonn,base,easy

AUTHOR

Felice Russo

EXTENSIONS

More terms from Erich Friedman.

STATUS

approved

A073509

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

+10
23

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Table of n, a(n) for n=1..22.

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)

EXAMPLE

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

MATHEMATICA

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

CROSSREFS

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

KEYWORD

base,hard,nonn

AUTHOR

Shyam Sunder Gupta, Aug 14 2002

EXTENSIONS

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

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

STATUS

approved

A073510

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

+10
23

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..22.

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)

EXAMPLE

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

MATHEMATICA

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

CROSSREFS

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

KEYWORD

base,hard,nonn

AUTHOR

Shyam Sunder Gupta, Aug 14 2002

EXTENSIONS

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

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

STATUS

approved

A073511

Number of primes less than 10^n with initial digit 7.

+10
23

1, 4, 18, 125, 1027, 8435, 71564, 622882, 5516130, 49495432, 448855139, 4106164356, 37838546363, 350849788546, 3270531245684, 30628143485953, 287992070079777, 2717649138419586, 25726964404879666, 244242934202964444, 2324722877951987037, 22178433287546997612

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..22.

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)

EXAMPLE

a(2)=4 because there are 4 primes up to 10^2 whose initial digit is 7 (namely 7, 71, 73 and 79).

MATHEMATICA

f[n_] := f[n] = PrimePi[8*10^n] - PrimePi[7*10^n] + f[n - 1]; f[0] = 1; Table[ f[n], {n, 0, 12}]

CROSSREFS

Cf. 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

KEYWORD

base,hard,nonn

AUTHOR

Shyam Sunder Gupta, Aug 14 2002

EXTENSIONS

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

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

STATUS

approved

A073512

Number of primes less than 10^n with initial digit 6.

+10
23

0, 2, 18, 135, 1013, 8458, 72257, 628206, 5556434, 49815418, 451476802, 4128049326, 38024311091, 352446754137, 3284400373590, 30749731897370, 289066731934716, 2727216210298152, 25812680778645432, 245015325044029789, 2331718909954888809, 22242097596092999144

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..22.

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)

EXAMPLE

a(2)=2 because there are 2 primes up to 10^2 whose initial digit is 2 (namely 61 and 67).

MATHEMATICA

f[n_] := f[n] = PrimePi[7*10^n] - PrimePi[6*10^n] + f[n - 1]; f[0] = 0; Table[ f[n], {n, 0, 12}]

CROSSREFS

Cf. 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

KEYWORD

base,hard,nonn

AUTHOR

Shyam Sunder Gupta, Aug 14 2002

EXTENSIONS

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

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

STATUS

approved

A073513

Number of primes less than 10^n with initial digit 5.

+10
23

1, 3, 17, 131, 1055, 8615, 72951, 633932, 5602768, 50193913, 454577490, 4153943134, 38243708524, 354330372215, 3300752009165, 30892997367352, 290332329192655, 2738477783884855, 25913537508233527, 245923809778144431, 2339944887042508496, 22316931815316988517

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..22.

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)

EXAMPLE

a(2)=3 because there are 3 primes up to 10^2 whose initial digit is 5 (namely 5, 53 and 59).

MATHEMATICA

f[n_] := f[n] = PrimePi[6*10^n] - PrimePi[5*10^n] + f[n - 1]; f[0] = 1; Table[ f[n], {n, 0, 13}]

CROSSREFS