The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60th year, we have over 367,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A318555 "Strong impostors" not divisible by 4: Those numbers s !== 0 (mod 4) such that lambda(s) | 2(s-1), where lambda is the Carmichael function (A002322). 1
 6, 15, 66, 91, 435, 561, 703, 946, 1105, 1729, 1891, 2465, 2701, 2821, 2926, 3367, 5551, 6601, 8646, 8695, 8911, 10585, 11305, 12403, 13981, 15051, 15841, 16471, 18721, 23001, 26335, 29341, 30889, 38503, 39865, 41041, 46657, 49141, 52633, 53131, 62745, 63973, 68101, 75361, 76627, 76798, 79003, 88561, 88831, 91001 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Strong impostors not == 0 (mod 4) have the property that, even though they are composite, when paired with any odd prime r such that (s,r) = 1, they produce valid RSA key pairs. More specifically, if n=sr, all a in Z_n will be correctly encrypted and decrypted for any (e,d) key pair such that ed == 1 mod (s-1)(r-1). They include the Carmichael numbers and are squarefree. The set of their odd prime factors is always normal: If p_i and p_j are odd prime factors, no p_i == 1 mod p_j. LINKS Barry Fagin, Table of n, a(n) for n = 1..2773 (all terms up to 32 bits) D. Borwein, J. M. Borwein, P. B. Borwein and R. Girgensohn, Giuga's Conjecture on Primality, Amer. Math. Monthly 103, No. 1, 40-50 (1996). B. S. Fagin, Composite Numbers That Give Valid RSA Key Pairs For Any Coprime p, Information, 9, 216; doi:10.3390/info9090216. J. M. Grau and Antonio Oller-Marcén, Generalizing Giuga's conjecture, arXiv:1103.3483 [math.NT], 2011. MATHEMATICA Reap[For[s = 1, s < 10^5, s++, If[!Divisible[s, 4] && CompositeQ[s], If[ Divisible[2(s-1), CarmichaelLambda[s]], Print[s]; Sow[s]]]]][[2, 1]] (* Jean-François Alcover, Feb 18 2019 *) PROG (Python with numbthy library) for s in range(min_s, max_s): if numbthy.is_prime(s): continue elif s % 4 == 0: continue elif (2*(s-1) % numbthy.carmichael_lambda(s) == 0): print("s =", s) (PARI) isok(s) = s>1 && s%4>0 && !isprime(s) && (2*s-2)%lcm(znstar(s)[2])==0; \\ Jinyuan Wang, Mar 01 2020 CROSSREFS Cf. A002997 (Carmichael numbers), A005117 (squarefree numbers). Subsequence of A231569. Sequence in context: A233450 A298374 A324973 * A359923 A035077 A032164 Adjacent sequences: A318552 A318553 A318554 * A318556 A318557 A318558 KEYWORD nonn AUTHOR Barry Fagin, Aug 28 2018 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 4 10:17 EST 2023. Contains 367560 sequences. (Running on oeis4.)