A073511
Number of primes less than 10^n with initial digit 7.
Original entry on oeis.org
1, 4, 18, 125, 1027, 8435, 71564, 622882, 5516130, 49495432, 448855139, 4106164356, 37838546363, 350849788546, 3270531245684, 30628143485953, 287992070079777, 2717649138419586, 25726964404879666, 244242934202964444, 2324722877951987037, 22178433287546997612
Offset: 1
a(2)=4 because there are 4 primes up to 10^2 whose initial digit is 7 (namely 7, 71, 73 and 79).
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
A073512
Number of primes less than 10^n with initial digit 6.
Original entry on oeis.org
0, 2, 18, 135, 1013, 8458, 72257, 628206, 5556434, 49815418, 451476802, 4128049326, 38024311091, 352446754137, 3284400373590, 30749731897370, 289066731934716, 2727216210298152, 25812680778645432, 245015325044029789, 2331718909954888809, 22242097596092999144
Offset: 1
a(2)=2 because there are 2 primes up to 10^2 whose initial digit is 2 (namely 61 and 67).
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
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
a(2)=3 because there are 3 primes up to 10^2 whose initial digit is 5 (namely 5, 53 and 59).
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
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
a(2)=3 because there are 3 primes up to 10^2 whose initial digit is 4 (namely 41, 43 and 47).
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
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
a(2)=3 because there are 3 primes up to 10^2 whose initial digit is 3 (namely 3, 31 and 37).
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
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
a(2)=3 because there are 3 primes up to 10^2 whose initial digit is 2 (namely 2, 23 and 29).
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
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
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(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
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
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(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
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
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, ...
-
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 *)
A208270
Primes containing a digit 1.
Original entry on oeis.org
11, 13, 17, 19, 31, 41, 61, 71, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 241, 251, 271, 281, 311, 313, 317, 331, 401, 419, 421, 431, 461, 491, 521, 541, 571, 601, 613, 617, 619, 631, 641, 661
Offset: 1
Cf.
A208271 (nonprimes containing a digit 1),
A011531 (numbers containing a digit 1).
-
[p: p in PrimesUpTo(400) | 1 in Intseq(p)]; // Vincenzo Librandi, Apr 29 2019
-
Select[Prime[Range[124]], MemberQ[IntegerDigits[#], 1] &](* Jayanta Basu, Apr 01 2013 *)
Select[Prime[Range[200]],DigitCount[#,10,1]>0&] (* Harvey P. Dale, Dec 15 2020 *)
-
forprime(p=2,1e3,s=vecsort(eval(Vec(Str(p))),,8);if(s[1]==1||(s[1]==0&&s[2]==1),print1(p", "))) \\ Charles R Greathouse IV, Mar 04 2012
-
is_A208270(n)=isprime(n)&setsearch(Set(Vec(Str(n))),1) \\ M. F. Hasler, Mar 05 2012
Comments