A045708 -id:A045708 - 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.

A217394 Numbers starting with 2.

Original entry on oeis.org

2, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242

Offset: 1

Jeremy Gardiner, Oct 02 2012

The lower and upper asymptotic densities of this sequence are 1/18 and 10/27, respectively. - Amiram Eldar, Feb 27 2021

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

Cf. A006092, A011532.
Subsequences include: A045708, A045726, A077327, A077678, A106412, A106422.
Cf. A131835, A217395, A217397, A217398, A217399, A217400, A217401, A217402.

Select[Range[300], IntegerDigits[#][[1]] == 2 &] (* T. D. Noe, Oct 02 2012 *)

def agen():
  yield 2
  digits, adder = 1, 20
  while True:
    for i in range(10**digits): yield adder + i
    digits, adder = digits+1, adder*10
g = agen()
print([next(g) for i in range(54)]) # Michael S. Branicky, Feb 20 2021

def A217394(n): return n+(17*10**(len(str(9*n-8))-1))//9 # Chai Wah Wu, Dec 07 2024

a(n) = n + (17*10^floor(log_10(9*n-8))-8)/9. - Alan Michael Gómez Calderón, May 15 2023

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 *)

A208272 Primes containing a digit 2.

Original entry on oeis.org

2, 23, 29, 127, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 421, 521, 523, 727, 821, 823, 827, 829, 929, 1021, 1123, 1129, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1321, 1327

Offset: 1

Jaroslav Krizek, Mar 04 2012

Supersequence of A045708. Subsequence of A011532.
Complement of A208273 with respect to A011532.
Also primes p whose divisors d_k (k = 1, 2; 1 = d_1 < d_2 = p) contain digit equal to number k.
Complement of A208275 with respect to A208274.
Primes with at least one digit equal to 2. - Harvey P. Dale, Aug 29 2012

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

Cf. A208273 (composites containing a digit 2), A011532 (numbers containing a digit 2).

Select[Range[2000], PrimeQ[#] && MemberQ[IntegerDigits[#], 2] &] (* T. D. Noe, Mar 06 2012 *)
Select[Prime[Range[300]],DigitCount[#,10,2]>0&] (* Harvey P. Dale, Aug 29 2012 *)

a(n) ~ n log n. - Charles R Greathouse IV, Nov 01 2022

cp's OEIS Frontend

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

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

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

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

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

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A045713 Primes with first digit 7.

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A045715 Primes with first digit 9.

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A217394 Numbers starting with 2.

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