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

0, 1, 4, 27, 1354, 401429925999155061

Offset: 0

Author

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

Keywords

• nonn
• hard

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

Examples

			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.
		

References

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

Links

• 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]
• S. S. Pillai, On an Arithmetic Function concerning Primes, Journal Of The Annamalai University, Vol-1 (1932), pp. 159-167 (pages 204-212 in pdf).

Crossrefs

Cf. A007924.

Programs

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

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).

Extensions

Edited by Charles R Greathouse IV, Oct 28 2009

Entry rewritten by Charles R Greathouse IV, Aug 28 2010