A083737 Pseudoprimes to bases 2, 3 and 5.

1729, 2821, 6601, 8911, 15841, 29341, 41041, 46657, 52633, 63973, 75361, 101101, 115921, 126217, 162401, 172081, 188461, 252601, 294409, 314821, 334153, 340561, 399001, 410041, 488881, 512461, 530881, 552721, 658801, 670033, 721801, 748657

Offset: 1

Author

Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), May 05 2003

Keywords

• easy
• nonn

Comments

a(n) = n-th positive integer k(>1) such that 2^(k-1) == 1 (mod k), 3^(k-1) == 1 (mod k) and 5^(k-1) == 1 (mod k)

See A153580 for numbers k > 1 such that 2^k-2, 3^k-3 and 5^k-5 are all divisible by k but k is not a Carmichael number (A002997).

Note that a(1)=1729 is the Hardy-Ramanujan number. - Omar E. Pol, Jan 18 2009

Examples

			a(1)=1729 since it is the first number such that 2^(k-1) == 1 (mod k), 3^(k-1) == 1 (mod k) and 5^(k-1) == 1 (mod k).
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 102 from R. J. Mathar)
• J. Bernheiden, Pseudoprimes (Text in German)
• F. Richman, Primality testing with Fermat's little theorem
• Index entries for sequences related to pseudoprimes

Crossrefs

Proper subset of A052155. Superset of A230722. Cf. A153580, A002997, A001235, A011541.

Programs

Mathematica

Select[ Range[838200], !PrimeQ[ # ] && PowerMod[2, # - 1, # ] == 1 && PowerMod[3, 1 - 1, # ] == 1 && PowerMod[5, # - 1, # ] == 1 & ]

PARI

is(n)=!isprime(n)&&Mod(2,n)^(n-1)==1&&Mod(3,n)^(n-1)==1&&Mod(5,n)^(n-1)==1 \\ Charles R Greathouse IV, Apr 12 2012

Extensions

Edited by Robert G. Wilson v, May 06 2003

Edited by N. J. A. Sloane, Jan 14 2009