A257750 -id:A257750 - cp's OEIS frontend

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

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

Tim Johannes Ohrtmann, Jun 23 2015

nonn

			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.

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..100000

Cf. A257750, A257759, A259283.

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, ", "))))))

A259283 Quasi-Carmichael numbers to at least one positive base.

Original entry on oeis.org

165, 231, 273, 357, 399, 598, 715, 935, 1015, 1105, 1547, 1595, 1885, 1886, 2015, 2093, 2387, 2397, 2451, 2465, 2585, 2679, 2737, 2821, 2915, 3059, 3445, 3913, 3965, 4123, 4991, 5015, 5467, 5719, 6097, 6545, 7055, 7189, 7285, 7553, 7843, 8555, 8569, 8715, 8855

Offset: 1

Tim Johannes Ohrtmann, Jun 23 2015

nonn

			a(1) = 165 because this is the first squarefree composite number n such that at least one positive integer b except 0 exists such that for every prime factor p of n applies that p+b divides n+b (3): 165=3*5*11 and 6, 8, 14 all divide 168.

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..2963

Cf. A257750, A257759, A259282.

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

A029590 For n>0, a(n) is the least quasi-Carmichael number to base n; a(0) = least composite squarefree integer.

Original entry on oeis.org

6, 561, 1595, 35, 1705, 77, 13481, 187, 143, 209, 4807, 221, 14807, 493, 20723, 323, 7429, 437, 36593, 943, 713, 989, 1147, 1073, 899, 1537, 1271, 899, 1333, 1517, 104355281, 1591, 1517, 2993, 1591, 1517, 621193, 3397, 1763, 1763, 2623, 2021

Offset: 0

David W. Wilson

nonn

a(n) is the least squarefree composite integer for which prime p | a(n) ==> p-n | a(n)-n.

			For n=6 the minimum is a(n)=13481. Prime factors of 13481 are 13, 17 and 61. We have 13481 - 6 = 13475, 13 - 6 = 7 and 13475 / 7 = 1925, 17 - 6 = 11 and 13475 / 11 = 1225, 61 - 6 = 55 and 13475 / 55 = 245. - _Elijah Beregovsky_, Feb 15 2020

Donovan Johnson, Table of n, a(n) for n = 0..500
Index entries for sequences related to Carmichael numbers.

Cf. A029591 (base -n), A257750 (quasi-Carmichael numbers).

qcQ[n_,k_] := Module[{f = FactorInteger[n]}, p = f[[;; , 1]]; e = f[[;; , 2]];om=Length[e]; om>=2 && Max[e] == 1 && Min[p]>k && Length@Select[p, Divisible[n-k, #-k]&] == om]; seq[k_]:=SelectFirst[Range[1,50000], qcQ[#,k]&]; Print[seq/@Range[0,29]]; (* Elijah Beregovsky, Feb 15 2020 *)

A262252 Even Quasi-Carmichael numbers.

Original entry on oeis.org

598, 1886, 11590, 21098, 24734, 32578, 91078, 95170, 107606, 134930, 143318, 179998, 253598, 258482, 259010, 287274, 361730, 374402, 568514, 706142, 751394, 831290, 920782, 1074026, 1105646, 1327562, 1514602, 1548318, 1579394, 1742830, 1794854, 1808678, 1952222

Offset: 1

Tim Johannes Ohrtmann, Sep 16 2015

nonn

			598 is even, composite and squarefree and at least one nonzero integer b exists such that for every prime factor p of n, p+b divides n+b (2): 598 = 2*13*23 and 4, 15, 25 all divide 600.

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..84

Cf. A029562, A029564, A029566, A029568, A029570, A257750.

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

A263930 Number of quasi-Carmichael numbers less than 10^n.

Original entry on oeis.org

0, 2, 27, 165, 734, 3109, 11568, 40820, 137850, 457191

Offset: 1

Tim Johannes Ohrtmann, Oct 30 2015

For quasi-Carmichael numbers see A257750.

			a(1) = 0 because there are no quasi-Carmichael numbers below 10^1.
a(2) = 2 because there are two quasi-Carmichael numbers below 10^2, namely, 35 and 77.

Cf. A055553, A216929, A257750.

use ntheory ":all"; my($s,$e)=(0,1); forcomposites { say $e++," $s" if $ >= 10**$e; $s++ if is_quasi_carmichael($) } 1e7; # Dana Jacobsen, Apr 27 2017

a(8)-a(10) from Dana Jacobsen, Apr 27 2017

A270860 Least Quasi-Carmichael number with n prime factors.

Original entry on oeis.org

35, 165, 6545, 179998, 7509579, 850253030

Offset: 2

Tim Johannes Ohrtmann, Mar 24 2016

			6545 = 5*7*11*17 and 12, 14, 18, 24 all divide 6552.

Cf. A257750 (Quasi-Carmichael numbers).

use ntheory ":all"; my $f=2; forcomposites { say $f++," $" if scalar(factor($))==$f && is_quasi_carmichael($); } 1e9; # _Dana Jacobsen, Apr 27 2017

a(7) from Dana Jacobsen, Apr 04 2016

cp's OEIS Frontend

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

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PARI

A259283 Quasi-Carmichael numbers to at least one positive base.

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PARI

A029590 For n>0, a(n) is the least quasi-Carmichael number to base n; a(0) = least composite squarefree integer.

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Mathematica

A262252 Even Quasi-Carmichael numbers.

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PARI

A263930 Number of quasi-Carmichael numbers less than 10^n.

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Perl

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A270860 Least Quasi-Carmichael number with n prime factors.

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Perl

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