A216926 -id:A216926 - cp's OEIS frontend

A006972 Lucas-Carmichael numbers: squarefree composite numbers k such that p | k => p+1 | k+1.

399, 935, 2015, 2915, 4991, 5719, 7055, 8855, 12719, 18095, 20705, 20999, 22847, 29315, 31535, 46079, 51359, 60059, 63503, 67199, 73535, 76751, 80189, 81719, 88559, 90287, 104663, 117215, 120581, 147455, 152279, 155819, 162687, 191807, 194327, 196559, 214199

Offset: 1

Richard Pinch and Jeffrey Shallit

nonn

Wright proves that this sequence is infinite (Main Theorem 2). - Charles R Greathouse IV, Nov 03 2015
Conjecture: if k = p*q*r, p = a*d - 1, q = b*d - 1, r = c*d - 1 are distinct odd primes, with d = gcd(p + 1, q + 1, r + 1) and a*b*c*d divides k + 1, then k is a Lucas-Carmichael number. - Davide Rotondo, Dec 23 2020
A composite k is a Lucas-Carmichael number if and only if k | A322702(k+1). - Thomas Ordowski, May 06 2021

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 399, p. 89, Ellipses, Paris 2008.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 550 terms from Paolo P. Lava)
Ed Copeland and Brady Haran, Something special about 399, Numberphile video (2015).
Sridhar Tamilvanan and Subramani Muthukrishnan, On Lucas-Carmichael Integer, arXiv:2311.08012 [math.NT], 2023.
Daniel Suteu, Table of n, a(n) for terms a(n) < 10^15.
Samuel S. Wagstaff, Jr., Ramanujan's Taxicab Number and its Ilk, Purdue Univ. (2024). See p. 2.
Wikipedia, Lucas-Carmichael number
Thomas Wright, There are infinitely many elliptic Carmichael numbers
Thomas Wright, There are infinitely many elliptic Carmichael numbers, arXiv:1609.00231 [math.NT], 2016.
Index entries for sequences related to Carmichael numbers.

Intersection of A024556 and A056729.
Cf. A216925, A216926, A216927, A217002, A217003, A217091 (terms having 3, 4, 5, 6, 7 and 8 factors).
Cf. A216929, A322702.

with(numtheory):
a:= proc(n) option remember; local k; for k from 1+
     `if`(n=1, 3, a(n-1)) while isprime(k) or not issqrfree(k)
       or add(irem(k+1,i+1), i=factorset(k))>0 do od; k
    end:
seq(a(n), n=1..15);  # Alois P. Heinz, Apr 05 2018

Select[ Range[ 2, 10^6 ], !PrimeQ[ # ] && Union[ Transpose[ FactorInteger[ # ] ][ [ 2 ] ] ] == {1} && Union[ Mod[ # + 1, Transpose[ FactorInteger[ # ] ][ [ 1 ] ] + 1 ] ] == {0} & ]

is(n)=my(f=factor(n));for(i=1,#f[,1],if((n+1)%(f[i,1]+1) || f[i,2]>1, return(0)));#f[,1]>1 \\ Charles R Greathouse IV, Sep 23 2012

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);
upto(n) = my(list=List()); for(k=3, oo, if(vecprod(primes(k+1))\2 > n, break); list=concat(list, lucas_carmichael(1, n, k))); vecsort(Vec(list)); \\ Daniel Suteu, Dec 01 2023

A216925 Lucas-Carmichael numbers with 3 prime factors.

Original entry on oeis.org

399, 935, 2015, 2915, 4991, 5719, 7055, 12719, 20705, 20999, 22847, 46079, 60059, 63503, 67199, 76751, 80189, 88559, 90287, 104663, 120581, 152279, 155819, 196559, 214199, 230159, 388079, 482143, 663679, 676799, 709019, 741311, 761039, 776567, 880319, 966239

Offset: 1

Tim Johannes Ohrtmann, Sep 20 2012

nonn

			a(1) = 399 = 3*7*19

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A006972 (Lucas-Carmichael numbers), A216926, A216927, A217002, A217003, A217091.

A216927 Lucas-Carmichael numbers with 5 prime factors.

Original entry on oeis.org

588455, 1010735, 2276351, 2756159, 4107455, 4874639, 5669279, 6539819, 8421335, 13670855, 16184663, 16868159, 21408695, 23176439, 24685199, 25111295, 26636687, 30071327, 34347599, 34541639, 36149399, 36485015, 38999519, 39715319, 42624911, 43134959, 49412285

Offset: 1

Tim Johannes Ohrtmann, Sep 20 2012

nonn

			a(1) = 588455 = 5*7*17*23*43

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A006972 (Lucas-Carmichael numbers), A216925, A216926, A217002, A217003, A217091.

A217002 Lucas-Carmichael numbers with 6 prime factors.

Original entry on oeis.org

139501439, 196377335, 206238815, 239875559, 287432495, 336545495, 353107799, 381626399, 394426655, 406335215, 461829599, 464972255, 577901519, 592557119, 649351295, 653067359, 674628479, 761212655, 775931519, 777724415, 929892095, 993625919, 1073352959

Offset: 1

Donovan Johnson, Sep 22 2012

nonn

			A006972(385) = 139501439 = 7*11*17*19*71*79.

Daniel Suteu, Table of n, a(n) for n = 1..9382 (first 1000 terms from Donovan Johnson)

Cf. A006972 (Lucas-Carmichael numbers), A216925, A216926, A216927, A217003, A217091.

upto(n, k=6) = my(A=vecprod(primes(k+1))\2, B=n); (f(m, l, p, k, u=0, v=0) = my(list=List()); if(k==1, forprime(p=u, v, my(t=m*p); if((t+1)%l == 0 && (t+1)%(p+1) == 0, listput(list, t))), forprime(q = p, sqrtnint(B\m, k), my(t = m*q); my(L=lcm(l, q+1)); if(gcd(L, t) == 1, my(u=ceil(A/t), v=B\t); if(u <= v, my(r=nextprime(q+1)); if(k==2 && r>u, u=r); list=concat(list, f(t, L, r, k-1, u, v)))))); list); vecsort(Vec(f(1, 1, 3, k))); \\ Daniel Suteu, Sep 03 2022

A217003 Lucas-Carmichael numbers with 7 prime factors.

Original entry on oeis.org

3512071871, 10470856319, 11956093919, 12283814015, 13150303199, 15128703359, 15966728855, 18063158399, 21887083295, 22572006479, 23388059519, 23836221695, 23940514367, 25231063319, 25638464159, 27742047839, 28160966735, 30070781279, 31251542399, 35160944399

Offset: 1

Donovan Johnson, Sep 22 2012

nonn

			A006972(1249) = 3512071871 = 7*11*17*23*31*53*71.

Daniel Suteu, Table of n, a(n) for n = 1..7464 (first 1000 terms from Donovan Johnson)

Cf. A006972 (Lucas-Carmichael numbers), A216925, A216926, A216927, A217002, A217091.

upto(n, k=7) = my(A=vecprod(primes(k+1))\2, B=n); (f(m, l, p, k, u=0, v=0) = my(list=List()); if(k==1, forprime(p=u, v, my(t=m*p); if((t+1)%l == 0 && (t+1)%(p+1) == 0, listput(list, t))), forprime(q = p, sqrtnint(B\m, k), my(t = m*q); my(L=lcm(l, q+1)); if(gcd(L, t) == 1, my(u=ceil(A/t), v=B\t); if(u <= v, my(r=nextprime(q+1)); if(k==2 && r>u, u=r); list=concat(list, f(t, L, r, k-1, u, v)))))); list); vecsort(Vec(f(1, 1, 3, k))); \\ Daniel Suteu, Aug 30 2022

A217091 Lucas-Carmichael numbers with 8 prime factors.

Original entry on oeis.org

199195047359, 220323712895, 259305479279, 325451502935, 472765412735, 491091874559, 498357905759, 517270926095, 609349053599, 769658803199, 832015353455, 853833772799, 898951575599, 962940227039, 1087044101759, 1122857491679, 1249765950719, 1297923596255

Offset: 1

Donovan Johnson, Sep 26 2012

nonn

			A006972(5453) = 199195047359 = 7*11*17*19*23*31*47*239.

Daniel Suteu and Donovan Johnson, Table of n, a(n) for n = 1..5323 (terms 1..1000 from Donovan Johnson)

Cf. A006972 (Lucas-Carmichael numbers), A216925, A216926, A216927, A217002, A217003.

upto(n, k=8) = my(A = vecprod(primes(k)), B=n); (f(m,l,p,k,u=0,v=0) = my(list=List()); if(k==1, forprime(p=u, v, my(t=m*p); if((t+1)%l == 0 && (t+1)%(p+1) == 0, listput(list, t))), my(s = sqrtnint(B\m, k)); forprime(q = p, s, my(t = m*q); my(L=lcm(l,q+1)); if(gcd(L,t) == 1, my(u=ceil(A/t), v=B\t); if(u <= v, my(r=nextprime(q+1)); if(k==2 && r>u, u=r); list=concat(list, f(t, L, r, k-1, u, v)))))); list); vecsort(Vec(f(1, 1, 3, k))); \\ Daniel Suteu, Aug 29 2022

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)}
```

A300959 Number of prime factors of the n-th Lucas-Carmichael number.

Original entry on oeis.org

3, 3, 3, 3, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 4, 3, 4, 3, 3, 3, 4, 3, 3, 4, 3, 3, 3, 4, 3, 4, 3, 3, 4, 4, 4, 3, 3, 4, 3, 4, 4, 3, 4, 3, 5, 4, 3, 3, 3, 3, 4, 3, 3, 4, 3, 4, 4, 3, 3, 3, 5, 4, 4, 3, 3, 4, 3, 4, 3, 3, 4, 4, 4, 3, 3, 4, 4, 3, 4, 4, 4, 4, 3, 5, 3, 4, 3

Offset: 1

Tim Johannes Ohrtmann, Mar 17 2018

nonn

The number of prime factors is always >= 3.

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

Cf. A006972 (Lucas-Carmichael numbers).
Cf. A216925, A216926, A216927, A217002, A217003, A217091 (Lucas-Carmichael numbers with 3 to 8 prime factors).
Cf. A216928 (Least Lucas-Carmichael number with n prime factors).

islc(n)=my(f=factor(n)); for(i=1, #f[, 1], if((n+1)%(f[i, 1]+1) || f[i, 2]>1, return(0))); #f[, 1]>1; \\ from A006972
lista(nn) = for (n=1, nn, if (islc(n), print1(omega(n), ", "))); \\ Michel Marcus, Mar 17 2018

a(n) = A001221(A006972(n)).

cp's OEIS Frontend

A006972 Lucas-Carmichael numbers: squarefree composite numbers k such that p | k => p+1 | k+1.

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A216925 Lucas-Carmichael numbers with 3 prime factors.

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A216927 Lucas-Carmichael numbers with 5 prime factors.

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A217002 Lucas-Carmichael numbers with 6 prime factors.

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A217003 Lucas-Carmichael numbers with 7 prime factors.

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A217091 Lucas-Carmichael numbers with 8 prime factors.

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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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