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A257750 Quasi-Carmichael numbers. 15
35, 77, 143, 165, 187, 209, 221, 231, 247, 273, 299, 323, 357, 391, 399, 437, 493, 527, 561, 589, 598, 713, 715, 899, 935, 943, 989, 1015, 1073, 1105, 1147, 1189, 1247, 1271, 1295, 1333, 1517, 1537, 1547, 1591, 1595, 1705, 1729, 1739, 1763, 1829, 1885, 1886, 1927 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

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.

Are 1885 and 1886 the only two consecutive integers such that both numbers are Quasi-Carmichael numbers?

From Robert G. Wilson v, Dec 05 2015: (Start)

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.

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.

Less than 0.5% are even (A262252). Of course they are == 2 (mod 4).

Least k-almost prime quasi-Carmichael number with k>1: 35, 165, 6545, 179998, 7509579, ..., .

(End)

LINKS

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..16869

EXAMPLE

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.

MATHEMATICA

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 *)

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>0, print1(n, ", ")))))

CROSSREFS

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).

Sequence in context: A043985 A254443 A300160 * A257751 A259282 A201068

Adjacent sequences:  A257747 A257748 A257749 * A257751 A257752 A257753

KEYWORD

nonn

AUTHOR

Tim Johannes Ohrtmann, May 07 2015

EXTENSIONS

All terms less than 1000000 checked by Robert G. Wilson v, Dec 13 2015

STATUS

approved

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Last modified March 20 05:34 EDT 2019. Contains 321344 sequences. (Running on oeis4.)