cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A257750 Quasi-Carmichael numbers.

Original entry on oeis.org

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
Offset: 1

Views

Author

Tim Johannes Ohrtmann, May 07 2015

Keywords

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)

Examples

			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.

Links

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).

Programs

  • 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 *)
  • 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,", ")))))

Extensions

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

A259282 Quasi-Carmichael numbers to at least one negative base.

Original entry on oeis.org

35, 77, 143, 187, 209, 221, 247, 299, 323, 391, 437, 493, 527, 561, 589, 713, 899, 943, 989, 1073, 1105, 1147, 1189, 1247, 1271, 1295, 1333, 1517, 1537, 1591, 1595, 1705, 1729, 1739, 1763, 1829, 1927, 1961, 2021, 2093, 2257, 2279, 2419, 2465, 2479, 2501, 2623
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Jun 23 2015

Keywords

Examples

			a(1) = 35 because this is the first squarefree composite number n such that at least one negative integer b 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.

Links

Crossrefs

Cf. A257750, A257759, A259283.

Programs

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

A257759 Quasi-Carmichael numbers to at least one negative base and at least one positive base.

Original entry on oeis.org

1105, 1595, 2093, 2465, 2821, 7843, 10373, 17963, 19721, 29341, 31003, 33143, 46189, 46657, 62647, 66263, 70151, 70219, 88559, 101813, 106361, 115843, 193343, 200777, 206471, 209933, 230159, 234883, 252601, 285619, 294409, 308267, 343027, 369799, 423181, 467273
Offset: 1

Views

Author

Tim Johannes Ohrtmann, May 12 2015

Keywords

Comments

It is a open question whether any Carmichael number exists that is also a Lucas-Carmichael number.

Examples

			a(1) = 1105 because this is the first squarefree composite number n such that at least one negative integer and at least one positive integer except 0 exist such that for every prime factor p of n applies that p+b divides n+b (-1, 15): 1105=5*13*17 and 4, 12, 16 both divide 1104 and 20, 28, 32 both divide 1120.

Links

Crossrefs

Intersection of A259282 and A259283.

Programs

  • 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, if(k==0, k++), if(b>0, if(k==1, k++))))); if(k==2, print1(n,", ")))))
Showing 1-3 of 3 results.

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