A007053 -id:A007053 - cp's OEIS frontend

A077643 Number of squarefree integers in closed interval [2^n, -1 + 2*2^n], i.e., among 2^n consecutive numbers beginning with 2^n.

1, 2, 3, 5, 9, 19, 39, 79, 157, 310, 621, 1246, 2491, 4980, 9958, 19924, 39844, 79672, 159365, 318736, 637457, 1274916, 2549816, 5099651, 10199363, 20398663, 40797299, 81594571, 163189087, 326378438, 652756861, 1305513511, 2611026987, 5222053970, 10444108084

Offset: 0

Labos Elemer, Nov 14 2002

nonn

Number of squarefree numbers with binary expansion of length n, or with n bits. The sum of these numbers is given by A373123. - Gus Wiseman, Jun 02 2024

			For n=4: among the 16 numbers of {16, ..., 31}, nine are squarefree [17, 19, 21, 22, 23, 26, 29, 30, 31], so a(4) = 9.

Amiram Eldar, Table of n, a(n) for n = 0..63 (calculated from the b-file at A143658)

Cf. A077641, A077642.
Partial sums (except first term) are A143658.
Run-lengths of A372475.
The minimum is A372683, delta A373125, indices A372540.
The maximum is A372889 (except at n=1), delta A373126, indices A143658.
Row-sums are A373123.
A005117 lists squarefree numbers, first differences A076259.
A053797 gives nonempty lengths of exclusive gaps between squarefree numbers.
A029837 counts bits, row-lengths of A030190 and A030308.
For primes between powers of 2:
- sum A293697
- length A036378 or A162145
- min A104080 or A014210, delta A092131, indices A372684
- max A014234, delta A013603, indices A007053
For squarefree numbers between primes:
- sum A373197
- length A373198 = A061398 - 1
- min A000040
- max A112925 (delta A240473), opposite A112926 (delta A240474)
Cf. A010036, A029931, A035100, A049093-A049096, A372473 (firsts of A372472), A372541 (firsts of A372433).

Table[Apply[Plus, Table[Abs[MoebiusMu[2^w+j]], {j, 0, 2^w-1}]], {w, 0, 15}]
(* second program *)
Length/@Split[IntegerLength[Select[Range[10000],SquareFreeQ],2]]//Most (* Gus Wiseman, Jun 02 2024 *)

{ a(n) = sum(m=1,sqrtint(2^(n+1)-1), moebius(m) * ((2^(n+1)-1)\m^2 - (2^n-1)\m^2) ) } \\ Max Alekseyev, Oct 18 2008

a(n) = Sum_{j=0..-1+2^n} abs(mu(2^n + j)).

a(n)/2^n approaches 1/zeta(2), so limiting sequence is floor(2^n/zeta(2)), n >= 0. - Wouter Meeussen, May 25 2003

More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 12 2003
More terms from Wouter Meeussen, May 25 2003
a(25)-a(32) from Max Alekseyev, Oct 18 2008
a(33)-a(34) from Amiram Eldar, Jul 17 2024

A059305 a(n) = pi(Mersenne(n)): index of n-th Mersenne prime.

Original entry on oeis.org

2, 4, 11, 31, 1028, 12251, 43390, 105097565, 55890484045084135, 10201730804263125133012340

Offset: 1

Reto Keiser (rkeiser(AT)ee.ethz.ch), Jan 25 2001

Similar to A016027, but gives the number of the n-th Mersenne prime (rather than the number of the prime exponent).

A subsequence of A007053 and A086690.

			Element 2 = 4 because Mersenne2 = (2^3)-1 = 7; 7 is the 4th prime.

Andrew R. Booker, The Nth Prime Page
C. K. Caldwell, Mersenne Primes
M. Deleglise and J. Rivat, Computing pi(x): the Meissel, Lehmer, Lagarias, Miller, Odlyzko method, Math. Comp., 65 (1996), 235-245.
Xavier Gourdon and Pascal Sebah, Counting primes
Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x)

Cf. A000043 Mersenne exponents, A000668 Mersenne primes, A016027 pi(mersenne exponents), A001348 Mersenne numbers.

Array[PrimePi[2^MersennePrimeExponent[#] - 1] &, 8] (* Michael De Vlieger, Apr 21 2019 *)

LL(e) = if(e==2, return(1)); my(n, h); n = 2^e-1; h = Mod(2, n); for (k=1, e-2, h=2*h*h-1); return(0==h) \\ after Joerg Arndt in A000043
forprime(p=1, , if(LL(p), print1(primepi(2^p-1), ", "))) \\ Felix Fröhlich, Apr 19 2019

a(n) = A000720(A000668(n))
a(n) = A007053(A000043(n))
A000668(n) = A000040(a(n)). - Omar E. Pol, Jun 29 2012

Revised by Max Alekseyev, Jul 20 2007

a(10) from David Baugh, Oct 08 2020

A028505 Number of primes <= 100*n.

Original entry on oeis.org

25, 46, 62, 78, 95, 109, 125, 139, 154, 168, 184, 196, 211, 222, 239, 251, 266, 278, 290, 303, 317, 327, 342, 357, 367, 378, 393, 407, 419, 430, 442, 452, 463, 478, 489, 503, 516, 528, 539, 550, 565, 574, 590, 599, 610, 622, 634, 646, 654, 669, 681, 692, 702

Offset: 1

Chris Cole (chris(AT)questrel.com)

nonn

Fred W. Helenius (fredh(AT)ix.netcom.com), posting to newsgroup rec.puzzles, Dec. 1997.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000
J. Betrema, Table et repartition des nombres premiers inferieurs a 10000
A. Booker, The Nth Prime Page
Thomas Ehrenborg, Gauss Circle Primes, arXiv:2502.06804 [math.GM], 2025. See Table 2 p. 4.
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. A000720, A038801, A038812, A038813, A038814, A038815, A038816, A038817, A038818, A038819, A038820, A038821, A080122, A080123, A080124, A080125, A080126, A080127, A080128, A080129.

for n:1 through 50 do print(prime_pi(100*n));

Table[ PrimePi[ n*100 ], {n, 55}] (* Robert G. Wilson v, Jan 29 2003 *)

a(n)=primepi(100*n) \\ Charles R Greathouse IV, Apr 28 2015

A073517 Number of primes less than 10^n with initial digit 1.

Original entry on oeis.org

0, 4, 25, 160, 1193, 9585, 80020, 686048, 6003530, 53378283, 480532488, 4369582734, 40063566855, 369893939287, 3435376839800, 32069022099022, 300694113015105, 2830466318006780, 26735673312004455, 253315661161665338, 2406763761677705769, 22923886160712831134, 218839439542390117580

Offset: 1

Shyam Sunder Gupta, Aug 14 2002

			a(2)=4 because there are 4 primes up to 10^2 whose initial digit is 1 (11, 13, 17 and 19).

Chris K. Caldwell, How Many Primes Are There?
Xavier Gourdon & Pascal Sebah, Counting the number of primes
Henri Lifchitz, Parity of Pi(n)
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. A000720 (pi), 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[2*10^n] - PrimePi[10^n] + f[n - 1]; f[0] = 0; Table[ f[n], {n, 0, 13}]

a(n,d=1)=sum(k=0, n-1, primepi((d+1)*10^k-1) - primepi(d*10^k-1)) \\ Andrew Howroyd, Dec 15 2024

a(n) = Sum_{k=0..n-1} pi(2*10^k-1) - pi(10^k-1). - Andrew Howroyd, Dec 15 2024

Edited and extended by Robert G. Wilson v, Aug 29 2002
a(21)-a(22) added by David Baugh, Mar 21 2015
a(23) from Chai Wah Wu, Sep 18 2018
Offset corrected by Andrew Howroyd, Dec 15 2024

A038801 Number of primes less than 10n.

Original entry on oeis.org

4, 8, 10, 12, 15, 17, 19, 22, 24, 25, 29, 30, 31, 34, 35, 37, 39, 41, 42, 46, 46, 47, 50, 52, 53, 55, 57, 59, 61, 62, 63, 66, 66, 68, 70, 72, 73, 75, 77, 78, 80, 81, 82, 85, 87, 88, 91, 92, 93, 95, 97, 97, 99, 99, 101, 102, 104, 106, 107, 109, 111, 114, 114, 115, 118, 120

Offset: 1

Jeff Burch

nonn

For almost all n, that is for a set of density 1, a(n) = a(n-1), and in all cases a(n) <= a(n-1) + 4. - Charles R Greathouse IV, Aug 27 2014

Robert G. Wilson v, Table of n, a(n) for n = 1..10000
Andrew R. Booker, The Nth Prime Page
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. A028505, A038812, A038813, A038814, A038815, A038816, A038817, A038818, A038819, A038820, A038821, A080122, A080123, A080124, A080125, A080126, A080127, A080128, A080129.

Table[ PrimePi[ n*10 ], {n, 1, 66}] (* Robert G. Wilson v *)

a(n)=primepi(10*n) \\ Charles R Greathouse IV, Aug 27 2014

Edited and corrected by Robert G. Wilson v, Jan 29 2003

A073509 Number of primes less than 10^n with initial digit 9.

Original entry on oeis.org

0, 1, 15, 127, 1006, 8230, 70320, 614821, 5453140, 48982456, 444608278, 4070532710, 37535715441, 348245215460, 3247889171908, 30429496751905, 286235215995588, 2702000272361599, 25586688305447928, 242978340446949438, 2313264023790027111, 22074118786158858975

Offset: 1

Shyam Sunder Gupta, Aug 14 2002

			a(2) = 1 because there is 1 prime less than 100 whose initial digit is 9, i.e., 97.

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)

A006880(n) = A073509(n)+ ... + A073516(n)+A073517(n-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

f[n_] := f[n] = PrimePi[10^(n + 1)] - PrimePi[9*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

A073510 Number of primes less than 10^n with initial digit 8.

Original entry on oeis.org

0, 2, 17, 127, 1003, 8326, 71038, 618610, 5481646, 49221187, 446590932, 4087194991, 37677478288, 349465615584, 3258501713644, 30522628848972, 287059041039078, 2709339704446862, 25652489700275636, 243571629996128384, 2318640708958531064, 22123070798400775157

Offset: 1

Shyam Sunder Gupta, Aug 14 2002

			a(2)=2 because there are 2 primes up to 10^2 whose initial digit is 8 (namely 83 and 89).

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)

A006880(n) = A073509(n)+ ... + A073516(n)+A073517(n-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

f[n_] := f[n] = PrimePi[9*10^n] - PrimePi[8*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

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

Shyam Sunder Gupta, Aug 14 2002

			a(2)=4 because there are 4 primes up to 10^2 whose initial digit is 7 (namely 7, 71, 73 and 79).

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[8*10^n] - PrimePi[7*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(20)-a(22) added by David Baugh, Mar 22 2015

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

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

cp's OEIS Frontend

A077643 Number of squarefree integers in closed interval [2^n, -1 + 2*2^n], i.e., among 2^n consecutive numbers beginning with 2^n.

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A059305 a(n) = pi(Mersenne(n)): index of n-th Mersenne prime.

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A028505 Number of primes <= 100*n.

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A073517 Number of primes less than 10^n with initial digit 1.

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A038801 Number of primes less than 10n.

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A073509 Number of primes less than 10^n with initial digit 9.

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A073510 Number of primes less than 10^n with initial digit 8.

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