A006972 -id:A006972 - cp's OEIS frontend

A292572 Lucas-Carmichael numbers whose Dedekind psi value is a cube.

8855, 31535, 73535, 265895, 12676799, 30071327, 86450399, 561645839, 674628479, 741722399, 945066527, 1066699799, 1304305415, 2239506719, 2423951999, 2693338559, 3512071871, 4708417055, 4811496767, 8194093919, 9140299199, 9184665599, 9405512639, 11729537855

Offset: 1

Amiram Eldar, Sep 19 2017

nonn

			psi(8855) = 24^3.

Amiram Eldar, Table of n, a(n) for n = 1..89 (terms below 10^12)

Cf. A001615, A006972, A290793, A292571, A292573.

psi[n_] := If[n < 1, 0, n*Sum[MoebiusMu[d]^2/d, {d, Divisors@n}]]; s = Import["b006972.txt","Data"][[All,-1]];Select[s, IntegerQ@Power[psi@#, 1/3] &]

A349028 Lucas-Carmichael numbers with 9 prime factors.

Original entry on oeis.org

14563696180319, 16569718534655, 20203946790335, 22034564147519, 23315834862719, 23889526894079, 27074874805055, 28932092649215, 31534433588735, 34236981827279, 34249223161439, 45373136257295, 45593377151399, 50103079391519, 50415330959279, 50683388926247

Offset: 1

Tim Johannes Ohrtmann, Nov 06 2021

nonn

			14563696180319 = 11*13*17*23*29*41*47*59*79 and 12, 14, 18, 24, 30, 42, 48, 60, and 80 all divide 14563696180320.

Daniel Suteu, Table of n, a(n) for n = 1..14207 (all terms < 10^17).

Intersection of A006972 and A046312.
Cf. A216928 (least Lucas-Carmichael number with n prime factors).
Cf. A216925, A216926, A216927, A217002, A217003, A217091, A349029, A349030 (Lucas-Carmichael numbers with 3-8, 10 and 11 prime factors).

PARI
```
is(n)={omega(n)==9&&is_A006972(n)}
```

A349029 Lucas-Carmichael numbers with 10 prime factors.

Original entry on oeis.org

989565001538399, 1250312791224959, 1419432982021439, 1518134614712639, 2240225337903839, 2493922560242399, 2708548708646879, 2786001880066559, 2807577905060159, 2808521396058455, 3157015238986895, 3210972445532159, 3221015190555239, 3407706183722399, 3614740529402519

Offset: 1

Tim Johannes Ohrtmann, Nov 06 2021

nonn

			989565001538399 = 11*13*17*19*29*31*41*47*83*149 and 12, 14, 18, 20, 30, 32, 42, 48, 84, and 150 all divide 989565001538400.

Daniel Suteu, Table of n, a(n) for n = 1..5249 (all terms < 10^18)

Intersection of A006972 and A046314.
Cf. A216928 (least Lucas-Carmichael number with n prime factors).
Cf. A216925, A216926, A216927, A217002, A217003, A217091, A349028, A349030 (Lucas-Carmichael numbers with 3-9 and 11 prime factors).

PARI
```
is(n)={omega(n)==10&&is_A006972(n)}
```

A349030 Lucas-Carmichael numbers with 11 prime factors.

Original entry on oeis.org

20576473996736735, 42380075646230399, 75943207554554879, 83668951228080959, 96195222056687039, 116436396482735615, 132525862783734959, 134052021887096159, 162544912900261199, 175900784368936319, 186326804496197519, 190523141606006495, 196467189590024639

Offset: 1

Tim Johannes Ohrtmann, Nov 06 2021

nonn

			20576473996736735 = 5*7*11*17*23*31*47*53*71*107*233 and 6, 8, 12, 18, 24, 32, 48, 54, 72, 108, and 234 all divide 20576473996736736.

Daniel Suteu, Table of n, a(n) for n = 1..2124 (all terms < 2^64).

Intersection of A006972 and A069272.
Cf. A216928 (least Lucas-Carmichael number with n prime factors).
Cf. A216925, A216926, A216927, A217002, A217003, A217091, A349028, A349029 (Lucas-Carmichael numbers with 3-10 prime factors).

PARI
```
is(n)={omega(n)==11&&is_A006972(n)}
```

A110885 Lucas-Carmichael numbers that are not congruent to 11 (mod 12).

Original entry on oeis.org

399, 5719, 20705, 80189, 120581, 162687, 482143, 663679, 1162349, 7274249, 8734109, 9486399, 10260809, 10397407, 14658349, 14970499, 25603599, 29010079, 32869759, 49412285, 77801359, 90393029, 95972799, 99467679, 105818129, 110066669, 125532329, 126325399

Offset: 1

Walter Kehowski, Sep 19 2005

nonn

There are 9967 Lucas-Carmichael numbers less than 10^12, and all but 332 are congruent to 11 (mod 12).

			5719=7*19*43=7 mod 12.

Robert G. Wilson v, Table of n, a(n) for n = 1..333

Cf. A006972.

with(numtheory); LC:=[]: for z from 1 to 1 do for m from 1 to 2000000 do n:=2*m+1; if not(isprime(n)) and issqrfree(n) then PF:=factorset(n); lcb:=true; for x in PF do if (n+1) mod (x+1) > 0 then lcb:=false fi od; if lcb then LC:=[op(LC),n]; fi; fi; #not od; #m od; #z select(proc(z) not(z mod 12 = 11) end, LC);

a(10) onward from Robert G. Wilson v, Feb 12 2015

A216929 Number of Lucas-Carmichael numbers less than 10^n.

Original entry on oeis.org

0, 0, 2, 8, 26, 60, 135, 323, 791, 1840, 4216, 9967, 23070, 54356, 129125

Offset: 1

Tim Johannes Ohrtmann, Sep 20 2012

Cf. A006972 (Lucas-Carmichael numbers).

lucas_carmichael(A, B, k) = A=max(A, vecprod(primes(k+1))\2); (f(m, l, lo, k) = my(list=List()); my(hi=min(sqrtint(B+1)-1, sqrtnint(B\m, k))); if(lo > hi, return(list)); if(k==1, lo=max(lo, ceil(A/m)); my(t=lift(-1/Mod(m,l))); while(t < lo, t += l); forstep(p=t, hi, l, if(isprime(p), my(n=m*p); if((n+1)%(p+1) == 0, listput(list, n)))), forprime(p=lo, hi, if(gcd(m, p+1) == 1, list=concat(list, f(m*p, lcm(l, p+1), p+1, k-1))))); list); f(1, 1, 3, k);
a(n) = my(N=10^n); my(count=0); for(k=3, oo, if(vecprod(primes(k+1))\2 > N, break); count += #lucas_carmichael(1, N, k)); count; \\ Daniel Suteu, Dec 01 2023

a(9)-a(11) from Donovan Johnson, Sep 22 2012
a(12) from Donovan Johnson, Sep 26 2012
a(13)-a(15) from Daniel Suteu, Dec 01 2023

A290810 Numbers k such that 6k-1, 12k-1 and 18k-1 are all primes.

Original entry on oeis.org

1, 4, 5, 14, 15, 29, 39, 40, 49, 70, 110, 159, 169, 204, 235, 260, 264, 315, 334, 355, 390, 425, 449, 490, 560, 565, 599, 634, 725, 729, 735, 820, 824, 889, 1019, 1029, 1349, 1379, 1419, 1510, 1580, 1590, 1694, 1719, 1765, 1925, 1930, 1950, 1985, 2044, 2150

Offset: 1

Amiram Eldar, Aug 11 2017

nonn

If k is in the sequence then (6k-1)(12k-1)(18k-1) = 36k * (36k^2 - 11k + 1) - 1 is a Lucas-Carmichael number (A006972).

Analogous to A046025 as A006972 (Lucas-Carmichael numbers) is analogous to A002997 (Carmichael numbers).

			1 is in the sequence since 6*1 - 1 = 5, 12*1 - 1 = 11 and 18*1 - 1 = 17 are all primes, and 5*11*17 = 935 is a Lucas-Carmichael number.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A002997, A006972, A046025, A067256, A087788, A216925.

Intersection of A024898, A138620 and A138918.

seq = {}; Do[ If[ AllTrue[{6 m - 1, 12 m - 1, 18 m - 1}, PrimeQ ], AppendTo[seq, m] ], {m, 1, 10^5} ]; seq

isok(n) = isprime(6*n-1) && isprime(12*n-1) && isprime(18*n-1); \\ Michel Marcus, Aug 11 2017

6*a(n) - 1 = A067256(n+1).

A290811 Numbers n such that (6n-1, 6n+1), (12n-1, 12n+1) and (18n-1, 18n+1) are 3 pairs of twin primes.

Original entry on oeis.org

1, 8925, 70070, 70385, 270725, 355040, 566650, 866635, 874335, 1091545, 1230740, 1295980, 1586095, 1594285, 1738380, 1974210, 2201325, 2427145, 2436665, 3124660, 3349990, 3599470, 3661350, 4059825, 4101790, 4486020, 4726540, 5139680, 5613370, 5898655, 6279035

Offset: 1

Amiram Eldar, Aug 11 2017

nonn

If n is in the sequence then (6n+1)*(12n+1)*(18n+1) is a Carmichael number (A002997) and (6n-1)*(12n-1)*(18n-1) is a Lucas-Carmichael number (A006972).
Intersection of A046025 and A290810.
The first 10 pairs of corresponding Lucas-Carmichael and Carmichael numbers ((6n-1)*(12n-1)*(18n-1), (6n+1)*(12n+1)*(18n+1)) are:
(935, 1729)
(921329139943799, 921392227198801)
(445860973748310119, 445864862313790921)
(451901165073782759, 451905088679976961)
(25715181770344848599, 25715239817629143601)
(58001133699332691839, 58001233533626759041)
(235803065459494289399, 235803319764534509401)
(843555229160685647759, 843555823997214441961)
(866240412591524160959, 866241018045184403161)
(1685504102154302331719, 1685505045798928055521)
(2416038446298343361039, 2416039645957333860241)

			1 is in the sequence since (6*1 - 1, 6*1 + 1) = (5, 7), (12*1 - 1, 12*1 + 1) = (11, 13) and (18*1 - 1, 18*1 + 1) = (17, 19) are all pairs of twin primes.

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..10000

Cf. A002997, A006972, A033502, A046025, A290810.

seq = {}; Do[ If[ AllTrue[{6 m - 1, 6 m + 1, 12 m - 1, 12 m + 1, 18 m - 1,
    18 m + 1}, PrimeQ ], AppendTo[seq, m]], {m, 1, 10^7} ]; seq
Select[Range[6280000],AllTrue[{6#+1,6#-1,12#+1,12#-1,18#+1,18#-1},PrimeQ]&] (* Harvey P. Dale, Jun 21 2024 *)

isok(n) = isprime(6*n-1) && isprime(6*n+1) && isprime(12*n-1) && isprime(12*n+1) && isprime(18*n-1) && isprime(18*n+1); \\ Michel Marcus, Aug 11 2017

A292352 Numbers that generate Lucas-Carmichael numbers using an adjusted version of Erdős's method.

Original entry on oeis.org

24, 36, 40, 48, 60, 72, 80, 84, 96, 108, 120, 144, 168, 180, 192, 200, 216, 240, 252, 270, 300, 324, 336, 360, 384, 400, 420, 432, 440, 468, 480, 504, 528, 540, 576, 588, 600, 624, 648, 660, 672, 714, 720, 744, 756, 768, 792, 810, 840, 864, 900, 912, 936, 960

Offset: 1

Amiram Eldar, Sep 14 2017

nonn

Erdős showed in 1956 how to construct Carmichael numbers from a given number n (see A287840). With appropriate sign changes the method can be used to generate Lucas-Carmichael numbers. Given a number n, let P be the set of primes p such that (p+1)|n but p is not a factor of n. Let c be a product of a subset of P with at least 3 elements. If c == -1 (mod n) then c is a Lucas-Carmichael number.

Numbers with only one generated Lucas-Carmichael number: 24, 36, 40, 48, 60, 80, 84, 96, 108, 200, 252, 270, 300, 324, 336, 400, 440, 468, ...

			The set of primes for n = 24 is P={2, 3, 5, 7, 11, 23}. One subset, {5, 7, 11, 23} have c == -1 (mod n): c = 5*7*11*23 = 8855. 24 is the least number that generates Lucas-Carmichael numbers thus a(1)=24.

Cf. A006972, A287840.

a = {}; Do[p = Select[Divisors[n] - 1, PrimeQ]; pr = Times @@ p; pr = pr/GCD[n, pr]; ps = Divisors[pr]; c = 0; Do[p1 = FactorInteger[ps[[j]]][[;; , 1]]; If[Length[p1] < 3, Continue[]]; c1 = Times @@ p1; If[Mod[c1, n] == 1, c++], {j, 1, Length[ps]}]; If[c > 0, AppendTo[a, n]], {n, 1, 1000}]; a

A292538 Lucas-Carmichael numbers of the form k^2 - 1.

Original entry on oeis.org

399, 2915, 7055, 63503, 147455, 1587599, 1710863, 2249999, 2924099, 6656399, 9486399, 14288399, 19289663, 25603599, 26936099, 28451555, 31270463, 32148899, 45158399, 49280399, 71368703, 91011599, 105884099, 111513599, 144288143, 146894399, 150405695, 152028899, 175827599

Offset: 1

Amiram Eldar, Sep 18 2017

nonn

Intersection of A005563 and A006972.
The numbers k such that k^2 - 1 is a Lucas-Carmichael number are 20, 54, 84, 252, 384, 1260, 1308, 1500, 1710, 2580, 3080, 3780, 4392, ...
From David A. Corneth, Aug 26 2023: (Start)
As k^2 - 1 = (k - 1)*(k + 1) and k is even we have k-1 and k+1 are coprime. So we can factor k-1 and k+1 separately when checking if k^2 - 1 is a term.
Possible other ideas are factoring an odd number only once, keeping it for the factorization of k^2 - 1 and (k + 2)^2 - 1. Alternatively dodging k = 18m +- 8, 18m +- 10 or 50m +- 24, 50m +- 26 to not get numbers that are multiples of odd primes squared. (End)
Wagstaff (2024) found that among the first 10^4 Lucas-Carmichael numbers there are 164 that are also Cunningham numbers (A080262) and that all of them are in this sequence. Below 10^15 there are 682 Lucas-Carmichael numbers that are also Cunningham numbers, and all of them are in this sequence (checked using the list of Lucas-Carmichael numbers by Daniel Suteu at A006972). - Amiram Eldar, Dec 29 2024

David A. Corneth, Table of n, a(n) for n = 1..2391 (terms <= 4*10^17; first 164 terms from Amiram Eldar)
Samuel S. Wagstaff, Ramanujan's taxicab number and its ilk, The Ramanujan Journal, Vol. 64, No. 3 (2024), pp. 761-764; ResearchGate link, author's copy.

Cf. A005563, A006972, A080262.

filter:= t ->
  andmap(f -> f[2]=1 and (t+1) mod (f[1]+1) = 0, ifactors(t)[2]):
select(filter, [seq(k^2-1, k=3..10^5)]); # Robert Israel, Sep 24 2017

lcQ[n_] := !PrimeQ[n] && Union[Transpose[FactorInteger[n]][[2]]] == {1} && Union[Mod[n + 1, Transpose[FactorInteger[n]][[1]] + 1]] == {0}; Select[Range[2, 10^4]^2 - 1, lcQ]

More terms from David A. Corneth, Aug 26 2023

cp's OEIS Frontend

A292572 Lucas-Carmichael numbers whose Dedekind psi value is a cube.

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A349028 Lucas-Carmichael numbers with 9 prime factors.

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A349029 Lucas-Carmichael numbers with 10 prime factors.

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A349030 Lucas-Carmichael numbers with 11 prime factors.

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A110885 Lucas-Carmichael numbers that are not congruent to 11 (mod 12).

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Maple

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A216929 Number of Lucas-Carmichael numbers less than 10^n.

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Extensions

A290810 Numbers k such that 6k-1, 12k-1 and 18k-1 are all primes.

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A290811 Numbers n such that (6n-1, 6n+1), (12n-1, 12n+1) and (18n-1, 18n+1) are 3 pairs of twin primes.

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A292352 Numbers that generate Lucas-Carmichael numbers using an adjusted version of Erdős's method.