A045711 -id:A045711 - 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.

A217398 Numbers starting with 5.

Original entry on oeis.org

5, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542

Offset: 1

Jeremy Gardiner, Oct 02 2012

The lower and upper asymptotic densities of this sequence are 1/45 and 5/27, respectively. - Amiram Eldar, Feb 27 2021

Also numbers such that when the leftmost digit is moved to the unit's place the result is divisible by 5. - Stefano Spezia, Jul 08 2025

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

Cf. A000867, A011535.
Subsequences include: A045711, A077330, A077681, A106415, A106425.
Cf. A131835, A217394, A217395, A217397, A217399, A217400, A217401, A217402.

a217398 n = a217398_list !! (n-1)
a217398_list = filter ((== 5) . a000030) [1..]
-- Reinhard Zumkeller, Mar 13 2014

Select[Range[1000], IntegerDigits[#][[1]] == 5 &] (* T. D. Noe, Oct 02 2012 *)

def A217398(n): return n+(44*10**(len(str(9*n-8))-1))//9 # Chai Wah Wu, Dec 07 2024

A000030(a(n)) = 5; A143473(a(n)) = a(n). - Reinhard Zumkeller, Mar 13 2014

a(n) = n + (44*10^floor(log_10(9*n-8))-8)/9. - Alan Michael Gómez Calderón, May 17 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 *)

A087765 Primes whose reversal is a multiple of 17.

Original entry on oeis.org

43, 71, 109, 137, 193, 631, 911, 997, 1049, 1291, 1543, 1571, 1609, 1637, 1693, 1759, 1787, 1823, 1973, 2053, 2081, 2269, 2297, 2333, 2549, 2699, 2707, 2791, 2857, 3001, 3217, 3581, 3769, 3797, 3833, 4007, 4091, 4129, 4157, 4409, 4493, 4651, 4903, 4931, 5011

Offset: 1

Zak Seidov, Oct 03 2003

Mohammed Yaseen, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Primes whose reversal is a multiple of k: A045711 (k=5), A087762 (k=7), A087764 (k=13), this sequence (k=17), A087766 (k=19), A087767 (k=23).

Select[Prime[Range[700]],Divisible[IntegerReverse[#],17]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 25 2019 *)

forprime(n=2, 6000, if(fromdigits(Vecrev(digits(n)))%17==0, print1(n, ", "))) \\ Mohammed Yaseen, Jul 20 2022

Corrected and extended by Ray Chandler, Oct 05 2003

Corrected and extended by Harvey P. Dale, Aug 25 2019

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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A217398 Numbers starting with 5.

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