A071973 -id:A071973

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A061274

Number of primes between Pi^(n-1) and Pi^n.

+10
4

2, 2, 7, 14, 37, 100, 271, 742, 2054, 5813, 16507, 47501, 137306, 399685, 1169789, 3438915, 10153559, 30083443, 89426830, 266598341, 796863253, 2387439540, 7168302094, 21565117858, 64993345011, 196202749033, 593205591259, 1796057856430, 5445108651668, 16528162238845

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..52 (calculated from the b-file at A071973)

FORMULA

a(n) = A071973(n) - A071973(n-1). - Amiram Eldar, Jun 14 2024

EXAMPLE

a(1) = 2 as 2 and 3 are the two primes between 1 and Pi. a(4) = 14 as there are 14 primes between Pi^3 = 31.006... and Pi^4 = 97.409091...

MAPLE

See A061273.

MATHEMATICA

p = q = 0; Do[m = n; q = PrimePi[Pi^n]; Print[q - p]; p = q, {n, 1, 28}]

Differences[PrimePi[Pi^Range[0, 27]]] (* Harvey P. Dale, Aug 17 2016 *)

CROSSREFS

Cf. A061273, A071973.

KEYWORD

nonn

AUTHOR

Amarnath Murthy, Apr 25 2001

EXTENSIONS

More terms from Winston C. Yang (winston(AT)cs.wisc.edu), May 17 2001

Edited and extended by Robert G. Wilson v, Jun 18 2002

a(28)-a(30) from Amiram Eldar, Jun 14 2024

STATUS

approved

A137382

Number of primes between Pi^(prime(n)) and Pi^(prime(n+1)).

+10
0

7, 51, 371, 25116, 184807, 15161948, 119510273, 10619203228, 8117133311259, 66755371425339, 59943578764218680, 5202903726035784628, 44211655386352360485, 4351545315671679793432

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..14.

FORMULA

a(n) = A071973(prime(n+1)) - A071973(prime(n)). - Amiram Eldar, Jun 14 2024

EXAMPLE

7 primes between Pi^2 (9.8...) and Pi^3 (31.0...).

51 primes between Pi^3 (31.0...) and Pi^5 (306.0...).

371 primes between Pi^5 (306.0...) and Pi^7 (3020.29...).

MATHEMATICA

Table[PrimePi[Pi^(Prime[n + 1])] - PrimePi[Pi^(Prime[n])], {n, 1, 8}]

CROSSREFS

Cf. A000796, A061274, A071973.

KEYWORD

nonn,more

AUTHOR

Vladimir Joseph Stephan Orlovsky, Apr 09 2008

EXTENSIONS

a(9)-a(14) calculated from the data at A071973 and added by Amiram Eldar, Jun 14 2024

STATUS

approved

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