cp's OEIS Frontend

A257750 Quasi-Carmichael numbers.

%I A257750 #29 Jan 09 2016 13:52:33
%S A257750 35,77,143,165,187,209,221,231,247,273,299,323,357,391,399,437,493,
%T A257750 527,561,589,598,713,715,899,935,943,989,1015,1073,1105,1147,1189,
%U A257750 1247,1271,1295,1333,1517,1537,1547,1591,1595,1705,1729,1739,1763,1829,1885,1886,1927
%N A257750 Quasi-Carmichael numbers.
%C A257750 Quasi-Carmichael numbers are squarefree composites n with the property that for every prime factor p of n, p+b divides n+b positively with b being any integer besides 0.
%C A257750 If b is negative, then it is always larger than 0 minus the square root of the corresponding Quasi-Carmichael number. But if b is positive, how large can it be in relation to its corresponding Quasi-Carmichael number? Conjecture: It is always smaller than the square root of the corresponding Quasi-Carmichael number.
%C A257750 Are 1885 and 1886 the only two consecutive integers such that both numbers are Quasi-Carmichael numbers?
%C A257750 From _Robert G. Wilson v_, Dec 05 2015: (Start)
%C A257750 The conjecture that b < sqrt(n) is false. Look at n = 87061 = 13*37*181, 87365 = 5*101*173, and 96473 = 13*41*181. Their b values are 299, 331, and 351, while the corresponding sqrt(n) values are 295, 295, and 310, respectively.
%C A257750 For b to result in (n+b)/(p+b) > 0 with n = P_1*p_2*...*p_i and P_1 < p_2 < ... < p_i, -p_1 < b < |(n-p_i^2)/p_i|. (n+b)/(p+b) >= b+1. Solve for b.
%C A257750 Less than 0.5% are even (A262252). Of course they are == 2 (mod 4).
%C A257750 Least k-almost prime quasi-Carmichael number with k>1: 35, 165, 6545, 179998, 7509579, ..., .
%C A257750 (End)
%H A257750 Tim Johannes Ohrtmann, <a href="/A257750/b257750.txt">Table of n, a(n) for n = 1..16869</a>
%e A257750 a(1) = 35 because this is the first squarefree composite number n such that at least one integer b except 0 exists such that for every prime factor p of n applies that p+b divides n+b (-3): 35 = 5*7 and 2, 4 both divide 32.
%t A257750 fQ[n_] := Block[{c = -1, fi = FactorInteger@ n, k, lmt, p}, If[Times @@ (Last@# & /@ fi) == 1 < Plus @@ (Last@# & /@ fi), p = First@# & /@ fi; k = -fi[[1, 1]] + 1; lmt = Abs[(n - fi[[-1, 1]]^2)/fi[[-1, 1]]]; While[k < lmt, If[ Union[ IntegerQ@# & /@ ((n + k)/(p + k))] == {True}, c++; If[c > 0, Goto [fini]]]; k++]]; Label[fini]; c > 0]; Select[ Range@ 2000, fQ] (* _Robert G. Wilson v_, Dec 05 2015 *)
%o A257750 (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>0, print1(n,", ")))))
%Y A257750 Subsequences: A002997 (Carmichael numbers), A006972 (Lucas-Carmichael numbers), A029553 (-10), A029554 (-9), A029555 (-8), A029556 (-7), A029557 (-6), A029558 (-5), A029559 (-4), A029560 (-3), A029561 (-2), A029562 (+2), A029563 (+3), A029564 (+4), A029565 (+5), A029566 (+6), A029567 (+7), A029568 (+8), A029569 (+9), A029570 (+10), A029590 (Least quasi-Carmichael number of order n), A029591 (Least quasi-Carmichael number of order -n), A257751 (1 base), A257752 (2 bases), A257753 (3 bases), A257754 (4 bases), A257755 (5 bases), A257756 (6 bases), A257757 (7 bases), A258842 (8 bases), A257758 (first occurrences), A259282 (at least one negative base), A259283 (at least one positive base), A257759 (at least one negative base and at least one positive base).
%K A257750 nonn
%O A257750 1,1
%A A257750 _Tim Johannes Ohrtmann_, May 07 2015
%E A257750 All terms less than 1000000 checked by _Robert G. Wilson v_, Dec 13 2015