A045712 -id:A045712 - cp's OEIS frontend

A073512 Number of primes less than 10^n with initial digit 6.

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

Offset: 1

Shyam Sunder Gupta, Aug 14 2002

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

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. 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[7*10^n] - PrimePi[6*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

A073513 Number of primes less than 10^n with initial digit 5.

Original entry on oeis.org

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

Offset: 1

Shyam Sunder Gupta, Aug 14 2002

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

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

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

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

A073514 Number of primes less than 10^n with initial digit 4.

Original entry on oeis.org

0, 3, 20, 139, 1069, 8747, 74114, 641594, 5661135, 50653546, 458352691, 4185483176, 38510936699, 356622729564, 3320632228693, 31067060521057, 291869049531878, 2752144407792176, 26035873192178041, 247025281876786013, 2349914303292170310, 22407593754131275705

Offset: 1

Shyam Sunder Gupta, Aug 14 2002

			a(2)=3 because there are 3 primes up to 10^2 whose initial digit is 4 (namely 41, 43 and 47).

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. 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[5*10^n] - PrimePi[4*10^n] + f[n - 1]; f[0] = 0; Table[ f[n], {n, 0, 13}]

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

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

A073515 Number of primes less than 10^n with initial digit 3.

Original entry on oeis.org

1, 3, 19, 139, 1097, 8960, 75290, 651085, 5735086, 51247361, 463196868, 4225763390, 38851672813, 359541975662, 3345924530873, 31288310624754, 293820812588401, 2769490109678920, 26191046215879444, 248421640738371325, 2362546444095790527, 22522418647770393663

Offset: 1

Shyam Sunder Gupta, Aug 14 2002

			a(2)=3 because there are 3 primes up to 10^2 whose initial digit is 3 (namely 3, 31 and 37).

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. 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[4*10^n] - PrimePi[3*10^n] + f[n - 1]; f[0] = 1; Table[ f[n], {n, 0, 12}]

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

a(21)-a(22) added by David Baugh, Mar 22 2015

A073516 Number of primes less than 10^n with initial digit 2.

Original entry on oeis.org

1, 3, 19, 146, 1129, 9142, 77025, 664277, 5837665, 52064915, 469864125, 4281198201, 39319600765, 363545360347, 3380562309312, 31590949437540, 296487794277035, 2793170342851930, 26402713858800478, 250324979315879678, 2379753569255122805, 22678735843184786383

Offset: 1

Shyam Sunder Gupta, Aug 14 2002

			a(2)=3 because there are 3 primes up to 10^2 whose initial digit is 2 (namely 2, 23 and 29).

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. 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[3*10^n] - PrimePi[2*10^n] + f[n - 1]; f[0] = 1; Table[ f[n], {n, 0, 13}]

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

a(21)-a(22) added by David Baugh, Mar 21 2015

A045713 Primes with first digit 7.

Original entry on oeis.org

7, 71, 73, 79, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283

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=7 of A262369.

[p: p in PrimesUpTo(7300) | Intseq(p)[#Intseq(p)] eq 7]; // Vincenzo Librandi, Aug 08 2014

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

from itertools import chain, count, islice
from sympy import primerange
def A045713_gen(): # generator of terms
    return chain.from_iterable(primerange(7*(m:=10**l),m<<3) for l in count(0))
A045713_list = list(islice(A045713_gen(),40)) # Chai Wah Wu, Dec 08 2024

from sympy import primepi
def A045713(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(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))
    return bisection(f,n,n) # Chai Wah Wu, Dec 08 2024

More terms from Erich Friedman.
Corrected by Jud McCranie, Jan 03 2001
a(13)=757 added from Vincenzo Librandi, Aug 08 2014

A045715 Primes with first digit 9.

Original entry on oeis.org

97, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 9001, 9007, 9011, 9013, 9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, 9127, 9133, 9137, 9151, 9157, 9161, 9173, 9181, 9187, 9199, 9203, 9209, 9221, 9227, 9239, 9241, 9257

Offset: 1

Felice Russo

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

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=9 of A262369.

[p: p in PrimesUpTo(10^4) | Intseq(p)[#Intseq(p)] eq 9]; // Bruno Berselli, Jul 19 2014

[p: p in PrimesInInterval(9*10^n,10^(n+1)), n in [0..3]]; // Bruno Berselli, Aug 08 2014

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

from itertools import chain, count, islice
def A045715_gen(): # generator of terms
    return chain.from_iterable(primerange(9*(m:=10**l),10*m) for l in count(0))
A045715_list = list(islice(A045715_gen(),40)) # Chai Wah Wu, Dec 08 2024

from sympy import primepi
def A045715(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(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))
    return bisection(f,n,n) # Chai Wah Wu, Dec 08 2024

More terms from Erich Friedman.

A262369 A(n,k) is the n-th prime whose decimal expansion begins with the decimal expansion of k; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

11, 2, 13, 3, 23, 17, 41, 31, 29, 19, 5, 43, 37, 211, 101, 61, 53, 47, 307, 223, 103, 7, 67, 59, 401, 311, 227, 107, 83, 71, 601, 503, 409, 313, 229, 109, 97, 89, 73, 607, 509, 419, 317, 233, 113, 101, 907, 809, 79, 613, 521, 421, 331, 239, 127

Offset: 1

Alois P. Heinz, Sep 20 2015

			Square array A(n,k) begins:
:  11,   2,   3,  41,   5,  61,   7,  83, ...
:  13,  23,  31,  43,  53,  67,  71,  89, ...
:  17,  29,  37,  47,  59, 601,  73, 809, ...
:  19, 211, 307, 401, 503, 607,  79, 811, ...
: 101, 223, 311, 409, 509, 613, 701, 821, ...
: 103, 227, 313, 419, 521, 617, 709, 823, ...
: 107, 229, 317, 421, 523, 619, 719, 827, ...
: 109, 233, 331, 431, 541, 631, 727, 829, ...

Alois P. Heinz, Antidiagonals n = 1..200, flattened
Index entries for primes involving decimal expansion of n

Columns k=1-9 give: A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715.
Row n=1 gives A018800.
Main diagonal gives A077345.
Cf. A077344, A262365.

u:= (h, t)-> select(isprime, [seq(h*10^t+k, k=0..10^t-1)]):
A:= proc(n, k) local l, p;
      l:= proc() [] end; p:= proc() -1 end;
      while nops(l(k))

u[h_, t_] := Select[Table[h*10^t + k, {k, 0, 10^t - 1}], PrimeQ];
A[n_, k_] := Module[{l, p}, l[] = {}; p[] = -1; While[Length[l[k]] < n, p[k] = p[k]+1; l[k] = Join[l[k], u[k, p[k]]]]; l[k][[n]]];
Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Dec 06 2019, from Maple *)

A217399 Numbers starting with 6.

Original entry on oeis.org

6, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624, 625, 626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638, 639, 640, 641, 642

Offset: 1

Jeremy Gardiner, Oct 02 2012

The lower and upper asymptotic densities of this sequence are 1/54 and 10/63, respectively. - Amiram Eldar, Feb 27 2021

Jeremy Gardiner, Table of n, a(n) for n = 1..1111
Index entries for 10-automatic sequences.

Cf. A000870, A011536.
Subsequences include: A045712, A077331, A077682, A106416, A106426.
Cf. A131835, A217394, A217395, A217397, A217398, A217400, A217401, A217402.

Flat(List([0..2],n->List([0..10^n-1],k->6*10^n+k))); # Muniru A Asiru, Nov 21 2018

[n: n in [1..1600] | Intseq(n)[#Intseq(n)] eq 6]; // Vincenzo Librandi, Nov 24 2018

seq(seq(6*10^n+k, k=0..10^n-1),n=0..3); # Robert Israel, May 08 2017

Select[Range[1000], IntegerDigits[#][[1]] == 6 &] (* T. D. Noe, Oct 02 2012 *)

isok(n) = digits(n)[1] == 6; \\ Michel Marcus, May 08 2017

def A217399(n): return n+(53*10**(len(str(9*n-8))-1))//9 # Chai Wah Wu, Dec 07 2024

a(n) = n + (53*10^floor(log_10(9*n-8))-8)/9. - Alan Michael Gómez Calderón, May 17 2023

A065680 Number of primes <= prime(n) which begin with a 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25

Offset: 1

Reinhard Zumkeller, Nov 13 2001

Considering the frequency of all decimal digits in leading position of prime numbers (A065681 - A065687), we cannot apply Benford's Law. But we observe at 10^e - levels that the frequency for 0 to 9 decreases monotonically, at least in the small range until 10^7.
The "begins with 9" sequence is too dull to include. - N. J. A. Sloane
Note that the primes do not satisfy Benford's law (see A000040). - N. J. A. Sloane, Feb 08 2017

			13 is the second prime beginning with 1: A000040(6) = 13, therefore a(6) = 2. a(664579) = 80020 (A000040(664579) = 9999991 is the largest prime < 10^7).

Harry J. Smith, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Benford's Law
Index entries for sequences related to Benford's law

Cf. A065681, A065682, A065683, A065684, A065685, A065686, A065687, 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.

Accumulate[If[First[IntegerDigits[#]]==1,1,0]&/@Prime[Range[80]]] (* Harvey P. Dale, Jan 22 2013 *)

lista(n) = { my(a=[p\10^logint(p,10)==1 | p<-primes(n)]); for(i=2, #a, a[i]+=a[i-1]); a} \\ Harry J. Smith, Oct 26 2009

cp's OEIS Frontend

A073512 Number of primes less than 10^n with initial digit 6.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A073513 Number of primes less than 10^n with initial digit 5.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A073514 Number of primes less than 10^n with initial digit 4.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A073515 Number of primes less than 10^n with initial digit 3.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A073516 Number of primes less than 10^n with initial digit 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A045713 Primes with first digit 7.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Python

Python

Extensions

A045715 Primes with first digit 9.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Magma

Mathematica

Python

Python

Extensions

A262369 A(n,k) is the n-th prime whose decimal expansion begins with the decimal expansion of k; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Views

Author

Keywords