A083739 Pseudoprimes to bases 2, 3, 5 and 7.

29341, 46657, 75361, 115921, 162401, 252601, 294409, 314821, 334153, 340561, 399001, 410041, 488881, 512461, 530881, 552721, 658801, 721801, 852841, 1024651, 1152271, 1193221, 1461241, 1569457, 1615681, 1857241, 1909001, 2100901

Offset: 1

Author

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

Keywords

• nonn

Examples

			a(1)=29341 since it is the first number such that 2^(k-1) = 1 (mod k), 3^(k-1) = 1 (mod k), 5^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..7469 (terms below 10^12; terms 1..114 from R. J. Mathar)
• Jens Bernheiden, Pseudoprimes (in German).
• Fred Richman, Primality testing with Fermat's little theorem.
• Index entries for sequences related to pseudoprimes.

Crossrefs

Cf. A001567, A005935, A005936, A005938.

Proper subset of A083737.

Programs

Maple

a001567 := [] : f := fopen("b001567.txt",READ) : bfil := readline(f) : while StringTools[WordCount](bfil) > 0 do if StringTools[FirstFromLeft]("#",bfil ) <> 0 then ; else bfil := sscanf(bfil,"%d %d") ; a001567 := [op(a001567), op(2,bfil) ] ; fi ; bfil := readline(f) ; od: fclose(f) : isPsp := proc(n,b) if n>3 and not isprime(n) and b^(n-1) mod n = 1 then true; else false; fi; end: isA001567 := proc(n) isPsp(n,2) ; end: isA005935 := proc(n) isPsp(n,3) ; end: isA005936 := proc(n) isPsp(n,5) ; end: isA005938 := proc(n) isPsp(n,7) ; end: isA083739 := proc(n) if isA001567(n) and isA005935(n) and isA005936(n) and isA005938(n) then true ; else false ; fi ; end: n := 1: for psp2 from 1 do i := op(psp2,a001567) ; if isA083739(i) then printf("%d %d ",n,i) ; n :=n+1 ; fi ; od: # R. J. Mathar, Feb 07 2008

Mathematica

Select[ Range[2113920], !PrimeQ[ # ] && PowerMod[2, # - 1, # ] == 1 && PowerMod[3, 1 - 1, # ] == 1 && PowerMod[5, # - 1, # ] == 1 && PowerMod[7, 1 - 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&&Mod(7,n)^(n-1)==1 \\ Charles R Greathouse IV, Apr 12 2012

Formula

a(n) = n-th positive integer k(>1) such that 2^(k-1) = 1 (mod k), 3^(k-1) = 1 (mod k), 5^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).

A005938 INTERSECT A083737. - R. J. Mathar, Feb 07 2008

Extensions

Edited by Robert G. Wilson v, May 06 2003