A006880 -id:A006880 - cp's OEIS frontend

A091709 Number of primes less than 10^n having at least one digit 8.

0, 2, 27, 314, 3217, 31699, 308774, 2987107, 28824402, 277779084, 2674980022, 25752370493, 247919235555, 2387154761520

Offset: 1

Enoch Haga, Jan 30 2004

			a(2) = 2 because of the 25 primes less than 10^2, 2 have at least one digit 8.

a(n) + A091642(n) = A006880(n).

Cf. A091644, A091645, A091646, A091647, A090705, A090706, A091707, A091708, A091710.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; c = 0; p = 1; Do[ While[ p = NextPrim[p]; p < 10^n, If[ Position[ IntegerDigits[p], 8] != {}, c++ ]]; Print[c]; p--, {n, 1, 8}] (* Robert G. Wilson v, Feb 02 2004 *)

from sympy import sieve # use slower primerange for larger terms
def a(n): return sum('8' in str(p) for p in sieve.primerange(2, 10**n))
print([a(n) for n in range(1, 8)]) # Michael S. Branicky, Apr 23 2021

Edited and extended by Robert G. Wilson v, Feb 02 2004
3 more terms from Ryan Propper, Aug 22 2005
a(13) from Robert Price, Nov 11 2013
a(14) from Giovanni Resta, Jul 21 2015

A091710 Number of primes less than 10^n having at least one digit 9.

Original entry on oeis.org

0, 6, 60, 542, 4826, 43359, 397093, 3677641, 34316162, 321993007, 3035059323, 28710966351, 272413818120, 2591276923548

Offset: 1

Enoch Haga, Jan 30 2004

			a(2) = 6 because of the 25 primes less than 10^2, 6 have at least one digit 9.

a(n) + A091643(n) = A006880(n).

Cf. A091644, A091645, A091646, A091647, A090705, A090706, A091707, A091708, A091709.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; c = 0; p = 1; Do[ While[ p = NextPrim[p]; p < 10^n, If[ Position[ IntegerDigits[p], 9] != {}, c++ ]]; Print[c]; p--, {n, 1, 8}] (* Robert G. Wilson v, Feb 02 2004 *)

Edited and extended by Robert G. Wilson v, Feb 02 2004
3 more terms from Ryan Propper, Aug 22 2005
a(13) from Robert Price, Nov 11 2013
a(14) from Giovanni Resta, Jul 21 2015

A120048 Number of 7-almost primes less than or equal to 10^n.

Original entry on oeis.org

0, 0, 0, 14, 231, 2973, 35585, 409849, 4600247, 50678212, 550454756, 5913771637, 62981797962, 665997804082, 7001087934965, 73232029374751, 762783057783010, 7916319351632036, 81898808371556517

Offset: 0

Robert G. Wilson v, Feb 07 2006

nonn

			There are 14 seven-almost primes up to 1000: 128, 192, 288, 320, 432, 448, 480, 648, 672, 704, 720, 800, 832 & 972.

Cf. A066265, A014613, A116382.

Cf. A006880, A036352, A109251, A114106, A114453, A120047, A120048, A120049, A120050, A120051, A120052, A120053, A116430.

AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric W. Weisstein, Feb 07 2006 *)
Table[AlmostPrimePi[7, 10^n], {n, 11}]

More terms from Robert G. Wilson v, Jan 07 2007
Example corrected by Harvey P. Dale, Jan 25 2013
a(15)-a(18) from Henri Lifchitz, Mar 18 2025

A120050 Number of 9-almost primes less than or equal to 10^n.

Original entry on oeis.org

0, 0, 0, 2, 47, 671, 8491, 101787, 1180751, 13377156, 148930536, 1636170477, 17787688377, 191742524399, 2052389350029, 21838745177567, 231206458686127, 2437121982958248, 25591920108631224, 267840642082525459

Offset: 0

Robert G. Wilson v, Feb 07 2006

nonn

			There are 2 nine-almost primes up to 1000: 512 & 768.

Cf. A066265, A014613, A116382.

Cf. A006880, A036352, A109251, A114106, A114453, A120047, A120048, A120049, A120050, A120051, A120052, A120053, A116430.

AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric W. Weisstein, Feb 07 2006 *)
Table[AlmostPrimePi[9, 10^n], {n, 12}]

a(13) and a(14) from Robert G. Wilson v, Jan 07 2007

a(15)-a(19) from Henri Lifchitz, Mar 18 2025

A120051 Number of 10-almost primes less than or equal to 10^n.

Original entry on oeis.org

0, 0, 0, 0, 22, 306, 4016, 49163, 578154, 6618221, 74342563, 823164388, 9011965866, 97765974368, 1052666075366, 11263041623194, 119864659464824, 1269754732725522, 13396817167474205, 140847445420555406

Offset: 0

Robert G. Wilson v, Feb 07 2006

			There are 22 ten-almost primes up to 10000: 1024, 1536, 2304, 2560, 3456, 3584, 3840, 5184, 5376, 5632, 5760, 6400, 6656, 7776, 8064, 8448, 8640, 8704, 8960, 9600, 9728, and 9984.

Cf. A066265, A014613, A116382.

Cf. A006880, A036352, A109251, A114106, A114453, A120047, A120048, A120049, A120050, A120051, A120052, A120053, A116430.

AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric W. Weisstein, Feb 07 2006 *)
Table[AlmostPrimePi[10, 10^n], {n, 12}]

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A120051(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    return int(sum(primepi(10**n//prod(c[1] for c in a))-a[-1][0] for a in g(10**n,0,1,1,10))) # Chai Wah Wu, Nov 03 2024

More terms from Robert G. Wilson v, Jan 07 2007

a(15)-a(19) from Henri Lifchitz, Mar 20 2025

A120053 Number of 12-almost primes less than or equal to 10^n.

Original entry on oeis.org

0, 0, 0, 0, 3, 63, 865, 11068, 133862, 1563465, 17836903, 200051717, 2214357712, 24255601105, 263439785143, 2841076717752, 30457549169277, 324855769153426, 3449587218984911, 36489283363168885

Offset: 0

Robert G. Wilson v, Feb 07 2006

			There are 3 twelve-almost primes up to 10000: 4096, 6144, and 9216.

Cf. A066265, A014613, A116382.

Cf. A006880, A036352, A109251, A114106, A114453, A120047, A120048, A120049, A120050, A120051, A120052, A120053, A116430.

AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric W. Weisstein, Feb 07 2006 *)
Table[AlmostPrimePi[12, 10^n], {n, 11}]

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A120053(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    return int(sum(primepi(10**n//prod(c[1] for c in a))-a[-1][0] for a in g(10**n,0,1,1,12))) # Chai Wah Wu, Aug 23 2024

a(13) and a(14) from Robert G. Wilson v, Jan 07 2007
a(15) from Chai Wah Wu, Aug 24 2024
a(16)-a(19) from Henri Lifchitz, Mar 18 2025

A190802 Gauss' approximation for the number of primes below 10^n.

Original entry on oeis.org

5, 29, 177, 1245, 9629, 78627, 664917, 5762208, 50849234, 455055614, 4118066400, 37607950280, 346065645809, 3204942065691, 29844571475287, 279238344248556, 2623557165610821, 24739954309690414, 234057667376222381, 2220819602783663483

Offset: 1

Nathaniel Johnston, May 25 2011

nonn

The offset logarithmic integral or Eulerian logarithmic integral Li(10^n)-Li(2), i.e., integral(2..x, dt/log(t)), appears in Gauss’s formula for counting prime numbers < 10^n and is sometimes referred to as the "European" definition. - Vladimir Pletser, Mar 17 2013

Jonathan Borwein, David H. Bailey, "Mathematics by Experiment", A. K. Peters, 2004, p. 65 (Table 2.2).

Vladimir Pletser, Table of n, a(n) for n = 1..500
Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, and Stephan Ramon Garcia, The Bateman-Horn Conjecture: Heuristics, History, and Applications, arXiv:1807.08899 [math.NT], 2018-2019. See Table 1 p. 6.

Cf. A006880, A106313.

seq(round(evalf(integrate(1/log(t),t=2..10^n))), n=1..21);

Table[Round[Integrate[1/Log[t],{t,2,10^n}]],{n,20}] (* James C. McMahon, Feb 06 2024 *)

a(n) = round(integral(dt/log(t),t=2..10^n)).

A365474 a(n) = A365339(10^n).

Original entry on oeis.org

1, 7, 34, 193, 1276, 9656, 78562, 664643, 5761519, 50847598

Offset: 0

Chai Wah Wu, Sep 04 2023

The Pollack et al. reference lists a(4)-a(7) and conjectures that A365339(n) = A000720(n)+64 for n >= 31957 which in turn implies the conjecture that a(n) = A006880(n)+64 for n >= 5.

Thomas Bloom, Problem 49, Erdős Problems.
Paul Pollack, Carl Pomerance, and Enrique Treviño, Sets of monotonicity for Euler's totient function, preprint. See M(n).
Paul Pollack, Carl Pomerance, and Enrique Treviño, Sets of monotonicity for Euler's totient function, Ramanujan J. 30 (2013), no. 3, pp. 379-398.
Terence Tao, Monotone non-decreasing sequences of the Euler totient function, arXiv:2309.02325 [math.NT], 2023.
Terence Tao, Erdős problem database, see no. 49.

Cf. A000010, A000720, A006880, A365339.

from bisect import bisect
from sympy import totient
def A365474(n):
    m = 10**n
    plist, qlist, c = tuple(totient(i) for i in range(1,m+1)), [0]*(m+1), 0
    for i in range(m):
        qlist[a:=bisect(qlist,plist[i],lo=1,hi=c+1,key=lambda x:plist[x])]=i
        c = max(c,a)
    return c

a(n) = A006880(n)+64 for n >= 5 (conjectured).

A040014 Number of primes < e^n.

Original entry on oeis.org

0, 1, 4, 8, 16, 34, 79, 183, 429, 1019, 2466, 6048, 14912, 37128, 93117, 234855, 595341, 1516233, 3877186, 9950346, 25614562, 66124777, 171141897, 443963543, 1154106844, 3005936865, 7842921261, 20496470801, 53645077679, 140599114669, 368973074565, 969455391690, 2550043255883

Offset: 0

Jud McCranie

nonn

a(n) = A000720(A000149(n)). - Reinhard Zumkeller, Mar 17 2015

David Baugh, Table of n, a(n) for n = 0..59 (terms n = 0..39 from Giovanni Resta, terms n = 40..59 found using Kim Walisch's primecount program).
Index entries for sequences related to numbers of primes in various ranges

Cf. A006880 and A007053.

Cf. A000720, A000149.

a040014 = a000720 . a000149  -- Reinhard Zumkeller, Mar 17 2015

Mathematica
```
Table[PrimePi[Exp[n]], {n, 0, 33}]
```

a(27)-a(29) from Robert G. Wilson v, Jun 09 2000

a(30)-a(32) from Seiichi Manyama, May 04 2016

A073532 Number of n-digit primes with all digits distinct.

Original entry on oeis.org

4, 20, 97, 510, 2529, 10239, 33950, 90510, 145227, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Zak Seidov, Aug 29 2002

For any base b the number of distinct-digit primes is finite. For base 10, the maximal distinct-digit prime is 987654103; for any larger prime at least two digits coincide. The number of distinct-digit integers is also finite, see A073531.

No such primes exist with 10 or more distinct decimal digits, so a(n) = 0 for n >= 10. - Labos Elemer, Oct 25 2004; Robert G. Wilson v, Jul 25 2008

			a(3)=97 because there are 97 three-digit primes with distinct digits: 103, 107, 109, 127, 137, 139, 149, 157, 163, 167, 173, 179, 193, 197,239, 241, 251, 257, 263, 269, 271, 281, 283, 293,307, 317, 347, 349, 359, 367, 379, 389, 397, 401, 409, 419, 421, 431, 439, 457, 461, 463, 467, 479, 487, 491, 503, 509, 521, 523, 541, 547, 563, 569, 571, 587, 593, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 673, 683, 691, 701, 709, 719, 739, 743, 751, 761, 769, 809, 821, 823, 827, 829, 839, 853, 857, 859, 863, 907, 937, 941, 947, 953, 967, 971, 983.

Cf. A073531, A006880, A006879, A098224-A098227, A140532.

lst = {}; Do[p = Prime@ n; If[ Union[Length /@ Split@ Sort@ IntegerDigits@ p] == {1}, AppendTo[lst, p]], {n, PrimePi[10^9]}]; Table[ Length@ Select[lst, 10^n < # < 10^(n + 1) &], {n, 0, 9}] (* Robert G. Wilson v, Jul 25 2008 *)

from itertools import permutations
from sympy import isprime, primerange
def distinct_digs(n): s = str(n); return len(s) == len(set(s))
def a(n):
  if n >= 10: return 0
  return sum(isprime(int("".join(p))) for p in permutations("0123456789", n) if p[0] != '0')
print([a(n) for n in range(1, 31)]) # Michael S. Branicky, Apr 20 2021

Edited by N. J. A. Sloane, Aug 14 2007

Entries checked by Robert G. Wilson v, Jul 25 2008

cp's OEIS Frontend

A091709 Number of primes less than 10^n having at least one digit 8.

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A091710 Number of primes less than 10^n having at least one digit 9.

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A120048 Number of 7-almost primes less than or equal to 10^n.

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A120050 Number of 9-almost primes less than or equal to 10^n.

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A120051 Number of 10-almost primes less than or equal to 10^n.

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A120053 Number of 12-almost primes less than or equal to 10^n.

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A190802 Gauss' approximation for the number of primes below 10^n.

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A365474 a(n) = A365339(10^n).

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