A102281 -id:A102281 - cp's OEIS frontend

A038627 Number of solutions x to n * pi(x) = x, where pi(x) = number of primes <= x.

0, 4, 3, 3, 6, 7, 6, 6, 3, 9, 1, 18, 11, 12, 21, 3, 10, 33, 31, 32, 24, 8, 13, 32, 35, 4, 15, 9, 15, 26, 22, 24, 9, 3, 14, 55, 36, 3, 65, 52, 33, 139, 42, 2, 85, 25, 7, 96, 16, 33

Offset: 1

Jud McCranie

Equivalently, a(n) is number of solutions x to the equation pi(n*x) = x. - Farideh Firoozbakht, Jan 09 2005 [For example, a(2) = 4 because 1, 2, 3 & 4 are all solutions of pi(2*x) = x and a(11) = 1 because 15927 is the only solution of the equation pi(11*x) = x.]

S. W. Golomb proved that a(n) > 0 for each integer n > 1. - Carlo Sanna, Nov 09 2015

			11*pi(x) = x has only 1 solution, so a(11) = 1.

S. W. Golomb, On the ratio of N to π(N), The American Mathematical Monthly, Vol. 69, No. 1 (Jan 1962), pp. 36-37.
Robert T. Harger and William L. Hightower, An Interesting Property of x/π(x), The College Mathematics Journal Vol. 40, No. 3 (May 2009), pp. 213-214.
Eric Weisstein's World of Mathematics, Prime Counting Function.

Cf. A038623, A038624, A038625, A038626, A102281, A087237.

(* Assumes upper and lower bounds are as defined in A038626. *)
xmin = .5; xmax = 2;
Join[{0},Table[c = 0; x = Floor[2.4*xmin]; x1 = 2.7*xmax + 7;
  xmin = Infinity; xmax = 0; While[x <= x1,
   If[x == PrimePi[n x], c++; xmin = Min[x, xmin];
xmax = Max[x, xmax]]; x++]; c, {n, 2, 15}]] (* Robert Price, Mar 28 2020 *)

One more term from Labos Elemer, Sep 05 2003
a(24)-a(26) from Labos Elemer, Sep 12 2003
Edited by N. J. A. Sloane, Sep 06 2008 at the suggestion of R. J. Mathar
a(27)-a(29) from David Radcliffe, Sep 10 2014
a(29) corrected and a(30)-a(50) obtained from the A038625 values computed by Jan Büthe. - Giovanni Resta, Aug 31 2018

A038626 Smallest positive integer m such that m = pi(nm) = A000720(nm).

Original entry on oeis.org

1, 9, 24, 66, 168, 437, 1051, 2614, 6454, 15927, 40071, 100346, 251706, 637197, 1617172, 4124436, 10553399, 27066969, 69709679, 179992838, 465769802, 1208198523, 3140421715, 8179002095, 21338685402, 55762149023, 145935689357, 382465573481, 1003652347080, 2636913002890, 6935812012540

Offset: 2

Jud McCranie

nonn

Golomb shows that solutions exist for each n>1.
For all known terms, we have 2.4*a(n) < a(n+1) < 2.7*a(n) + 7. A038627(n) gives number of natural solutions of the equation m = pi(n*m). - Farideh Firoozbakht, Jan 09 2005
a(n) grows as exp(n)/n. Thus, a(n+1)/a(n) tends to e=exp(1) as n grows. - Max Alekseyev, Oct 15 2017

			pi(3059) = 437 and 3059/437 = 7, so a(7)=437.

Giovanni Resta, Table of n, a(n) for n = 2..50
S. W. Golomb, On the Ratio of N to pi(N), American Mathematical Monthly, 69 (1962), 36-37.
Eric Weisstein's World of Mathematics, Prime Counting Function.

Cf. A038623, A038624, A038625, A038627, A102281, A087237, A293529.

a(n) = limit of f^(k)(1) as k grows, where f(x)=A000720(n*x). Also, a(n) = f^(A293529(n))(1). - Max Alekseyev, Oct 11 2017

a(n) = A038625(n) / n. - Max Alekseyev, Oct 13 2023

a(24) from Farideh Firoozbakht, Jan 09 2005
Edited by N. J. A. Sloane at the suggestion of Chris K. Caldwell, Apr 08 2008
a(25)-a(32) from Max Alekseyev, Jul 18 2011, Oct 14 2017
a(33)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Aug 31 2018

A087237 a(n) = (Max{x : npi(x) = x} - Min{x : npi(x) = x})/n = A087236(n)/n.

Original entry on oeis.org

3, 2, 6, 6, 21, 5, 8, 86, 18, 0, 50, 17, 55, 143, 2, 269, 454, 308, 286, 335, 36, 338, 219, 113, 4, 92, 36, 72, 296, 296, 327, 23, 2, 168, 658, 770, 90, 1274, 1454, 1514, 3259, 612, 6, 2896, 367, 15, 2011, 287, 1915

Offset: 2

Labos Elemer, Sep 04 2003

nonn

a(n) is the difference between the largest and smallest solutions to n = x/pi(x), divided by n, where pi(x) = A000720(x).

Equivalently, a(n) is the difference between largest solution and smallest natural solution of the equation pi(n*x) = x. - Farideh Firoozbakht, Jan 09 2005 [Max Alekseyev observes that this is trivially equivalent to the first definition.]

			n=22: a(22) = (10246936436-10246935644)/22 = 792/22 = 36.
a(2) = 3 because 1, 2, 3 & 4 are all solutions of pi(2*x) = x and 4 - 1 = 3; a(11) = 0 because 15927 is the only solution of the equation pi(11*x) = x and 15927 - 15927 = 0. - _Farideh Firoozbakht_, Jan 09 2005

Jan Büthe, Table of n, a(n) for n = 2..50

Cf. A038623-A038627, A057809, A087235-A087240.

Cf. A102281.

MapIndexed[(Last@ #1 - First@ #1)/(First@ #2 + 1) &, Values@ Rest@ KeySort@ PositionIndex@ Table[n/PrimePi[n] /. k_ /; Not@ IntegerQ@ k -> 0, {n, 2, 10^6}]] (* Michael De Vlieger, Mar 25 2017, Version 10 *)

a(n) = A102281(n) - A038626(n). - Farideh Firoozbakht, Jan 09 2005

Edited by N. J. A. Sloane, Oct 28 2008 at the suggestion of R. J. Mathar

24 additional terms from Jan Büthe, Jan 16 2015

A245071 a(n) = 12n - prime(n).

Original entry on oeis.org

10, 21, 31, 41, 49, 59, 67, 77, 85, 91, 101, 107, 115, 125, 133, 139, 145, 155, 161, 169, 179, 185, 193, 199, 203, 211, 221, 229, 239, 247, 245, 253, 259, 269, 271, 281, 287, 293, 301, 307, 313, 323, 325, 335, 343, 353, 353, 353, 361, 371, 379, 385, 395, 397, 403, 409, 415, 425

Offset: 1

Freimut Marschner, Jul 21 2014

Prime(n) > n for n > 0. Let prime(n) = k*n with k as an even integer constant, for example, k = 12; then a(n) = k*n - prime(n) is a sequence of odd integers that are positive as long as k*n > prime(n). This is the case up to a(40072) = 11. If k*n < prime(n) then a(n) < 0, a(40073) = -5 up to a(40083) = -5. From a(40084) = 5 up to a(40121) = 5, a(n) > 0 again, but a(n) < 0 for n >= 40122. For k = 12 the table shows this result compared with floor(prime(n)/n) and (prime(n) mod n) <= (prime(n+1) mod (n+1)) for n >= 1. Observations:
(1) If k > floor(prime(n)/n) then a(n) is positive.
(2) If k <= floor(prime(n)/n) and (prime(n) mod n) < (prime(n+1) mod (n+1)) and n > 1 then a(n) is negative.
(3) If k <= floor(prime(n)/n) and (prime(n) mod n) > (prime(n+1) mod (n+1)) then a(n) is positive.
.
n     prime(n) floor(prime(n)/n) (prime(n) mod n)  a(n)
40072 480853        12                 5            11
40073 480881        12                23            -5
40083 481001        11             40079            -5
40084 481003        11             40074             5
40121 481447        12                 5             5
40122 481469        12                13            -5

			a(3) = 12*3 - prime(3) = 36 - 5 = 31.

Freimut Marschner, Table of n, a(n) for n = 1..100000

A000040 (prime(n)), A038605 (floor(prime(n)/n)), A004648 (prime(n) mod n), A038606 (Least k such that k-th prime > n * k), A038607 (the smallest prime number k such that k > n*pi(k)), A102281 (the largest number m such that m = pi(n*m)).

Table[12n - Prime[n], {n, 60}] (* Alonso del Arte, Jul 27 2014 *)

PARI
```
vector(133, n, 12*n-prime(n) )
```

a(n) = 12*n - prime(n).

A102279 Numbers n such that n = phi(d_1)!phi(d_2)! ... *phi(d_k)! where d_1 d_2 ... d_k is the decimal expansion of n and assume that phi(0)=0.

Original entry on oeis.org

1, 48, 720, 17280, 17915904000, 479219999055934390272000000000

Offset: 1

Farideh Firoozbakht, Jan 08 2005

All terms are of the form 2^i*3^j*5^t.

No more terms < 10^100. - David Wasserman, Apr 03 2008

			17280 is in the sequence because 17280 = phi(1)!*phi(7)!*phi(2)!*phi(8)!*phi(0)!.

Cf. A097655, A102280, A102281.

A245650 Primes in the sequence 12*n - prime(n), (A245071).

Original entry on oeis.org

31, 41, 59, 67, 101, 107, 139, 179, 193, 199, 211, 229, 239, 269, 271, 281, 293, 307, 313, 353, 353, 353, 379, 397, 409, 431, 439, 449, 449, 457, 467, 479, 491, 499, 509, 521, 547, 563, 599, 607, 617, 641, 659, 673, 709, 719, 739, 751, 761, 769, 809, 811, 821, 827, 859, 863, 881, 883, 911, 911, 919, 929

Offset: 1

Freimut Marschner, Jul 30 2014

As the arithmetic derivative of prime numbers is [prime(n)]' = 1 the comparison of the first arithmetic derivative of (A245071)' = A245649 leads to a selection of 13.386 prime numbers out of 100.000, where some prime numbers are repeated. Remark: The sign of prime(n) here used is +, so prime(n) is distributed relative to 12 = floor(prime(n)/n).

Since 12*i - prime(i) < 0 for i > A102281(12) = 40121, this sequence is finite. It has 13386 members, with 1793 distinct primes, the largest being 16369 = a(4713). - Robert Israel, Jul 30 2014

Robert Israel, Table of n, a(n) for n = 1..13386(full sequence)

Cf. A102281, A245071, A245649.

A:=select(isprime, [seq(12*n - ithprime(n), n=1..40121)]); # Robert Israel, Jul 30 2014

for(n=1,10^3,q=12*n-prime(n);if(isprime(q),print1(q,", "))) \\ Derek Orr, Jul 30 2014

a(n) is the n-th prime member of the sequence 12*i - prime(i). a(n) = prime(n) if (12*i - prime(i)) = prime(n).

A072916 Number of m such that floor(prime(m)/m) = n.

Original entry on oeis.org

3, 8, 19, 41, 117, 254, 616, 1642, 3766, 9461, 24183, 60252, 151368, 385600, 979844, 2507393, 6428977, 16513542, 42642649, 110283280, 285776799, 742428731, 1932223170, 5038580446, 13159683245, 34423463648, 90173540312

Offset: 1

Zak Seidov Aug 11 2002

nonn

			Only m = 2,3,4 give [p(m)/m] = 1, so a(1) = 3.
There are 8 values of m giving floor(prime(m)/m) = 2, namely m = 1,5,6,7,8,9,10,11, so a(2) = 8.

Cf. A062742, A102281.

a(n_) := Length[Cases[Table[Floor[Prime[m]/m], {m, 1, 1000000}], n]]

a(16)-a(27) from Farideh Firoozbakht, Sep 13 2005

Typo corrected by David W. Wilson, Oct 22 2005

A364635 a(n) is the largest prime p such that p/PrimePi(p) < n.

Original entry on oeis.org

7, 31, 113, 359, 1129, 3089, 8467, 24281, 64717, 175141, 481447, 1304713, 3524621, 9560081, 25874773, 70119967, 189969349, 514282961, 1394199299, 3779856617, 10246936393, 27788573801, 75370126379, 204475055189, 554805820519, 1505578026059, 4086199303001, 11091501632977

Offset: 2

Jon E. Schoenfield, Sep 09 2023

nonn

Sequence begins at a(2) because there exists no prime p such that p/PrimePi(p) < 1.

			a(4) = 113 because 113/PrimePi(113) = 113/30 = 3.766... but p/PrimePi(p) >= 4 for all primes > 113.

Cf. A000720, A038625, A062743, A102281.

a(n) = prime(A102281(n)). - Michel Marcus, Sep 10 2023

cp's OEIS Frontend

A038627 Number of solutions x to n * pi(x) = x, where pi(x) = number of primes <= x.

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A038626 Smallest positive integer m such that m = pi(n*m) = A000720(n*m).

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A087237 a(n) = (Max{x : n*pi(x) = x} - Min{x : n*pi(x) = x})/n = A087236(n)/n.

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A245071 a(n) = 12n - prime(n).

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A102279 Numbers n such that n = phi(d_1)!*phi(d_2)!* ... *phi(d_k)! where d_1 d_2 ... d_k is the decimal expansion of n and assume that phi(0)=0.

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A245650 Primes in the sequence 12*n - prime(n), (A245071).

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A072916 Number of m such that floor(prime(m)/m) = n.

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A364635 a(n) is the largest prime p such that p/PrimePi(p) < n.

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A038626 Smallest positive integer m such that m = pi(nm) = A000720(nm).

A087237 a(n) = (Max{x : npi(x) = x} - Min{x : npi(x) = x})/n = A087236(n)/n.

A102279 Numbers n such that n = phi(d_1)!phi(d_2)! ... *phi(d_k)! where d_1 d_2 ... d_k is the decimal expansion of n and assume that phi(0)=0.