A029591 -id:A029591 - cp's OEIS frontend

A257750 Quasi-Carmichael numbers.

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

Tim Johannes Ohrtmann, May 07 2015

nonn

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)

			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.

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

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).

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 *)

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,", ")))))

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

A202157 a(n) = smallest k having at least two prime divisors d such that (d + n) | ( k + n).

Original entry on oeis.org

63, 18, 45, 50, 75, 66, 63, 102, 75, 50, 165, 198, 147, 258, 165, 110, 663, 182, 399, 442, 147, 242, 705, 678, 455, 786, 483, 182, 1015, 950, 1023, 988, 363, 506, 637, 1446, 1083, 322, 885, 590, 1155, 1443, 1935, 2118, 627, 770, 3243, 2502, 1407, 2706, 845

Offset: 1

Michel Lagneau, Dec 13 2011

nonn

The sequence of numbers k composite and squarefree, prime p | k ==> p+n | k+n is given by A029591 (least quasi-Carmichael number of order -n).
If k is squarefree, for n = 1, we obtain Lucas-Carmichael numbers: A006972.
In this sequence, the majority of terms are not squarefree: 63, 18, 45, ...

			a(8) = 102 because the prime divisors of 102 are 2, 3 and 17;
(2 + 8) | (102 + 8) = 110 = 10*11;
(3 + 8) | 110 = 11*10.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 399, p. 89, Ellipses, Paris 2008.

Amiram Eldar, Table of n, a(n) for n = 1..2500

Cf. A006972, A029591, A202158, A202159.

with(numtheory):for n from 1 to 52 do:i:=0:for k from 1 to 5000 while(i=0) do:x:=factorset(k):n1:=nops(x):y:=k+n: j:=0:for m from 1 to n1 do:if  n1>=2 and irem(y,x[m]+n)=0 then j:=j+1:else fi:od:if j>=2 then i:=1:printf(`%d, `,k):else fi:od:od:

numd[n_, k_] := Module[{p=FactorInteger[k][[;;,1]], c=0}, Do[If[Divisible[n+k, n+p[[i]]], c++], {i,1,Length[p]}]; c]; a[n_]:=Module[{k=1}, While[numd[n, k] <= 1, k++]; k]; Array[a, 50] (* Amiram Eldar, Sep 09 2019 *)

a(n) >= n^2 + 4n + 6. [Charles R Greathouse IV, Dec 13 2011]

A202158 a(n) = smallest k having at least three prime divisors d such that (d + n) | (k + n).

Original entry on oeis.org

399, 598, 165, 1886, 715, 2370, 273, 532, 231, 935, 3445, 828, 1547, 2821, 1105, 3710, 12903, 4182, 6669, 4732, 2475, 4466, 2737, 2706, 1595, 5658, 10413, 3542, 7315, 24225, 23769, 22578, 3927, 12818, 1885, 64119, 11063, 20482, 10881, 4370, 52275, 7878, 14645

Offset: 1

Michel Lagneau, Dec 13 2011

nonn

The sequence of numbers k composite and squarefree, prime p | k ==> p+n | k+n is given by A029591 (least quasi-Carmichael number of order -n).
If k is squarefree, for n = 1, we obtain Lucas-Carmichael numbers: A006972.
In this sequence, the majority of terms are not squarefree.

			a(3) = 165 because the prime divisors of 165 are 3, 5, 11 =>
(3 + 3) | (165 + 3) = 168 = 6*28;
(5 + 3) | 168 = 8*21;
(11 + 3) | 168 = 14*12.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 399, p. 89, Ellipses, Paris 2008.

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A006972, A029591, A202157, A202159.

with(numtheory):for n from 1 to 45 do:i:=0:for k from 1 to 100000 while(i=0) do:x:=factorset(k):n1:=nops(x):y:=k+n: j:=0:for m from 1 to n1 do:if  n1>=2 and irem(y,x[m]+n)=0 then j:=j+1:else fi:od:if j>2 then i:=1:printf(`%d, `,k):else fi:od:od:

numd[n_, k_] := Module[{p=FactorInteger[k][[;;,1]], c=0}, Do[If[Divisible[n+k, n+p[[i]]], c++], {i,1,Length[p]}]; c]; a[n_]:=Module[{k=1}, While[numd[n, k] <= 2, k++]; k]; Array[a, 40] (* Amiram Eldar, Sep 09 2019 *)

A202159 a(n) = smallest k having at least four prime divisors d such that (d + n) | (k + n).

Original entry on oeis.org

8855, 11590, 27885, 122360, 16555, 10290, 6545, 61642, 71799, 65195, 14245, 142788, 63635, 580930, 39585, 21098, 69003, 258482, 59885, 378952, 8715, 266090, 133285, 690501, 27335, 704790, 1017423, 299222, 187891, 771650, 293405, 1638598, 282315, 553610, 227205

Offset: 1

Michel Lagneau, Dec 13 2011

nonn

The sequence of numbers k composite and squarefree, prime p | k ==> p+n | k+n is given by A029591 (least quasi-Carmichael number of order -n).
If k is squarefree, for n = 1, we obtain Lucas-Carmichael numbers: A006972.
In this sequence, the majority of terms are not squarefree.

			a(3) = 27885 because the prime divisors of 27885 are 3, 5, 11, 13  =>
(3 + 3)| (27885 + 3) = 27888 = 6*4648;
(5 + 3) | 27888 = 8*3486;
(11 + 3) | 27888 = 14*1992;
(13 + 3) | 27888 = 16*1743.

Amiram Eldar, Table of n, a(n) for n = 1..450

Cf. A006972, A029591, A202157, A202158.

with(numtheory):for n from 1 to 33 do:i:=0:for k from 1 to 5000000  while(i=0) do:x:=factorset(k):n1:=nops(x):y:=k+n: j:=0:for m from 1 to n1 do:if  n1>=2 and irem(y,x[m]+n)=0 then j:=j+1:else fi:od:if j>3 then i:=1:printf(`%d, `,k):else fi:od:od:

numd[n_, k_] := Module[{p=FactorInteger[k][[;;,1]], c=0}, Do[If[Divisible[n+k, n+p[[i]]], c++], {i,1,Length[p]}]; c]; a[n_]:=Module[{k=1}, While[numd[n, k] <= 3, k++]; k]; Array[a, 35] (* Amiram Eldar, Sep 09 2019 *)

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 *)

A202160 a(n) = smallest k having at least five prime divisors d such that (d + n) | (k + n).

Original entry on oeis.org

588455, 179998, 460317, 6265805, 1236235, 287274, 949025, 1436932, 794871, 2013650, 3797365, 1169688, 3739827, 1587586, 6872565, 7706270, 1529983, 7351242, 2528045, 5247970, 487179, 10920965, 1316497, 121894476, 1404455, 5814874, 12223653, 2260412, 8022531

Offset: 1

Michel Lagneau, Dec 13 2011

nonn

The sequence of numbers k composite and squarefree, prime p | k ==> p+n | k+n is given by A029591 (least quasi-Carmichael number of order -n).
If k is squarefree, for n = 1, we obtain Lucas-Carmichael numbers: A006972.
In this sequence, the majority of terms are not squarefree.

			a(3) = 460317 because the prime divisors of 460317 are 3, 11, 13, 29, 37  =>
(3 + 3) | (460317 + 3) = 460320 = 6*76720;
(11 + 3) | 460320 = 14*32880;
(13 + 3) | 460320 = 16*28770;
(29+3)  |  460320 = 32*14385;
(37+3) | 460320 = 40*11508.

Amiram Eldar, Table of n, a(n) for n = 1..125

Cf. A006972, A029591, A202157, A202158, A202159.

with(numtheory):for n from 1 to 23 do:i:=0:for k from 1 to 10^8 while(i=0) do:x:=factorset(k):n1:=nops(x):y:=k+n: j:=0:for m from 1 to n1 do:if  n1>=2 and irem(y,x[m]+n)=0 then j:=j+1:else fi:od:if j>4 then i:=1: printf ( "%d %d \n",n,k):else fi:od:od:

numd[n_, k_] := Module[{p=FactorInteger[k][[;;,1]], c=0}, Do[If[Divisible[n+k, n+p[[i]]], c++], {i,1,Length[p]}]; c]; a[n_]:=Module[{k=1}, While[numd[n, k] <= 4, k++]; k]; Array[a, 30] (* Amiram Eldar, Sep 09 2019 *)

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A257750 Quasi-Carmichael numbers.

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A202157 a(n) = smallest k having at least two prime divisors d such that (d + n) | ( k + n).

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A202158 a(n) = smallest k having at least three prime divisors d such that (d + n) | (k + n).

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A202159 a(n) = smallest k having at least four prime divisors d such that (d + n) | (k + n).

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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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A202160 a(n) = smallest k having at least five prime divisors d such that (d + n) | (k + n).

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