A066352 Pillai sequence: a(n) is the smallest term in A007924 requiring n primes.

0, 1, 4, 27, 1354, 401429925999155061

Offset

0,3

Comments

a(5) computed independently in 2007 by R. J. Mathar and Luca & Thangadurai, both using Thomas Nicely's tables.

On Cramer's conjecture, the number of primes required is O(log* n), where log* is the iterated logarithm, so the rate of growth of a(n) is tetrational in n. - Charles R Greathouse IV, Aug 28 2010

The next term likely has hundreds of millions of digits. - Charles R Greathouse IV, Jun 29 2015

References

S. S. Pillai, "An arithmetical function concerning primes", Annamalai University Journal (1930), pp. 159-167.

Links

Table of n, a(n) for n=0..5.

Florian Luca and Ravindranathan Thangadurai, On an arithmetic function considered by Pillai, Journal de théorie des nombres de Bordeaux 21:3 (2009), pp. 695-701.

M. T. Marcos, Smarandache Prime Base representation, prime puzzle 141.

Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]

Formula

a(n) = 2*p(m) - p(m-1) with minimal m = pi(a(n)) so that p(m) = a(n-1) + p(m-1), where p(n) is A008578(n).

Example

The greatest prime <= 27 is 23; the greatest prime <= 27-23 is 3; 27-23-3 = 1, so the Pillai representation of 27 is 23+3+1, which uses more terms than all preceding numbers.

Prog

(PARI) A072491(n)=if(n<4, n>0, 1+A072491(n-precprime(n)))
print1(r=0); for(n=1, 1e7, if(A072491(n)>r, r=a(n); print1(", "n)))
\\ Charles R Greathouse IV, Feb 04 2013

Crossrefs

Cf. A007924.

Sequence in context: A271385 A347146 A110763 * A357561 A249105 A249110

Adjacent sequences: A066349 A066350 A066351 * A066353 A066354 A066355

Keyword

nonn,hard

Author

Copied from www.primepuzzles.net by Frank Ellermann, Dec 19 2001

Extensions

Edited by Charles R Greathouse IV, Oct 28 2009

Entry rewritten by Charles R Greathouse IV, Aug 28 2010

Status

approved