A257757 Quasi-Carmichael numbers to exactly seven bases.

777923, 1030189, 1060459, 4903309, 5493247, 5659637, 6431071, 6673087, 6778969, 9790577, 11390429, 11860969, 12053263, 12390319, 12602059, 21215011, 21842629, 22991989, 24005239, 39339667, 39929437, 40080661, 40761169, 42314449, 50979479, 51876007, 54345943

Offset

1,1

Comments

All known terms have only two prime factors, one slightly larger than the other.

a(435) = 7523021437 = 1597 * 1933 * 2437 is the first term which has more than two prime factors. - Hiroaki Yamanouchi, Sep 28 2015

a(5586) > 10^12. - Hiroaki Yamanouchi, Sep 28 2015

Links

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..5885

Example

a(1) = 777923 because this is the first squarefree composite number n such that exactly seven integers b except 0 exist such that for every prime factor p of n, p+b divides n+b (-879, -878, -875, -872, -867, -863, -839): 777923=881*883 and 2, 4 both divide 777044 and 3, 5 both divide 777045 and 6, 8 both divide 777048 and 9, 11 both divide 777051 and 14, 16 both divide 777056 and 18, 20 both divide 777060 and 42, 44 both divide 777084.

Prog

(PARI) for(n=2, 1000000, if(!isprime(n), if(issquarefree(n), f=factor(n); k=0; for(b=-(f[1, 1]-1), n, c=0; for(i=1, #f[, 1], if((n+b)%(f[i, 1]+b)>0, c++)); if(c==0, if(!b==0, k++))); if(k==7, print1(n, ", ")))))

Crossrefs

Cf. A257750 (every number of bases).

Cf. A257751, A257752, A257753, A257754, A257755, A257756, A258842 (1 to 6 and 8 bases).

Cf. A257758 (first occurrences).

Sequence in context: A244071 A157357 A146953 * A250690 A233948 A204312

Adjacent sequences: A257754 A257755 A257756 * A257758 A257759 A257760

Keyword

nonn

Author

Tim Johannes Ohrtmann, May 12 2015

Extensions

a(16)-a(27) from Hiroaki Yamanouchi, Sep 26 2015

Status

approved