A087788 -id:A087788 - cp's OEIS frontend

A064238 Values of m such that N = (am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,3.

6, 36, 210, 270, 306, 330, 336, 600, 726, 1170, 1236, 1296, 1530, 1656, 2220, 2280, 2556, 3036, 3060, 3066, 4260, 4446, 4800, 4950, 5226, 5580, 5850, 6150, 6360, 6690, 6840, 6966, 7620, 7680, 7686, 7866, 8016, 8166, 8190, 8286, 8520, 8526, 8646, 8940, 9090

Offset: 1

N. J. A. Sloane, Sep 23 2001

nonn

am+1, bm+1, cm+1 are primes and am | (N-1), bm | (N-1), cm |(N-1).

All m's are multiples of 6 and m, 2m and 3m divide m(2m+1)(3m+1)-1 automatically.

Harvey Dubner (harvey(AT)dubner.com), personal communication, Jun 27 2001.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A046025, A087788.

q:= n-> andmap(isprime, [6*j*n+1$j=1..3]):
map(x-> 6*x, select(q, [$1..2000]))[];  # Alois P. Heinz, Jun 25 2023

CarmichaelNbrQ[n_] := ! PrimeQ@ n && Mod[n, CarmichaelLambda[n]] == 1; Select[ Range@ 9000, PrimeQ[# + 1] && PrimeQ[2# + 1] && PrimeQ[3# + 1] && CarmichaelNbrQ[(# + 1)(2 # + 1)(3 # + 1)] &] (* Robert G. Wilson v, Aug 23 2012 *)

a(n) = 6 * A046025(n).

Offset corrected by Amiram Eldar, Oct 16 2019

A064262 Values of m such that N=(am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,51.

Original entry on oeis.org

330, 1656, 1758, 6960, 11958, 14406, 27258, 30318, 30930, 31236, 37356, 40110, 43986, 44088, 50820, 53268, 55818, 63366, 65100, 67650, 71526, 74586, 81930, 90906, 93150, 94476, 98250, 99678, 109470, 119568, 121710, 129768, 135276, 141906, 146190, 147516, 155166

Offset: 1

N. J. A. Sloane, Sep 23 2001

nonn

am+1, bm+1, cm+1 are primes and am | (N-1), bm | (N-1), cm |(N-1).

Harvey Dubner (harvey(AT)dubner.com), personal communication, Jun 27 2001.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A087788.

CarmichaelNbrQ[n_] := ! PrimeQ@ n && Mod[n, CarmichaelLambda[n]] == 1; Select[ Range@ 9000, PrimeQ[# + 1] && PrimeQ[2# + 1] && PrimeQ[51# + 1] && CarmichaelNbrQ[(# + 1)(2# + 1)(51# + 1)] &] (* Robert G. Wilson v, Aug 23 2012 *)

Offset corrected and more terms added by Amiram Eldar, Oct 17 2019

A065703 Values of m such that N = (m+1)(2m+1)(71m+1) is a 3-Carmichael number (A087788).

Original entry on oeis.org

1170, 5430, 53568, 59106, 63366, 86370, 95316, 99576, 103836, 105966, 116190, 183498, 184776, 239730, 260178, 300648, 319818, 333450, 339840, 362418, 367530, 481698, 485958, 503850, 511518, 605238, 644856, 725370, 732186, 762006, 788418, 799920, 837408, 870210

Offset: 1

Harvey Dubner (harvey(AT)dubner.com), Nov 14 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A064238-A064262, A065695, A065696, A065697, A065698, A065699, A065700, A065701, A065702, A087788.

CarmichaelNbrQ[n_] := ! PrimeQ@ n && Mod[n, CarmichaelLambda@ n] == 1; Select[ Range@ 1000000, PrimeQ[# + 1] && PrimeQ[2# + 1] && PrimeQ[71# + 1] && CarmichaelNbrQ[(# + 1)(2# + 1)(71# + 1)] &] (* Robert G. Wilson v, Aug 23 2012 *)

for(m=1,1e6,is_A002997((m+1)*(2*m+1)*(71*m+1)) & print1(m","))  \\ - M. F. Hasler, Aug 23 2012

am+1, bm+1, cm+1 are primes and am|(N-1), bm|(N-1), cm|(N-1).

Definition simplified, missing terms inserted, and extended by M. F. Hasler, Aug 23 2012

More terms from Amiram Eldar, Oct 17 2019

A065695 Numbers m such that N = (am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,53.

Original entry on oeis.org

6120, 11526, 104700, 108516, 115830, 122826, 297726, 298680, 320940, 338430, 339066, 367686, 374046, 387720, 448140, 531456, 534636, 538770, 587106, 618270, 709536, 746106, 762006, 857406, 863766, 897156, 963300, 1115940, 1150920

Offset: 1

Harvey Dubner (harvey(AT)dubner.com), Nov 14 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A064238-A064262, A065696-A065703, A087788.

CarmichaelNbrQ[n_] := ! PrimeQ@ n && Mod[n, CarmichaelLambda@ n] == 1; Select[ Range@ 1000000, PrimeQ[# + 1] && PrimeQ[2# + 1] && PrimeQ[53# + 1] && CarmichaelNbrQ[(# + 1)(2# + 1)(53# + 1)] &] (* Robert G. Wilson v, Aug 23 2012 *)

am+1, bm+1, cm+1 are primes and am|(N-1), bm|(N-1), cm|(N-1).

More terms from Robert G. Wilson v, Aug 23 2012

A162290 Let A087788(n) = pqr, where p

Original entry on oeis.org

7, 23, 48, 22, 47, 45, 45, 21, 44, 163, 162, 43, 161, 280, 1684, 1363, 159, 351, 950, 1675, 1358, 949, 158, 345, 1829, 947, 1353, 510, 938, 1660, 2796, 1820, 820, 10208, 2779, 935, 1650, 817, 937, 1822

Offset: 1

A.K. Devaraj, Jul 01 2009

nonn

A.K. Devaraj conjectured that a(n) is always an integer, and this was proved by Carl Pomerance.
a(n) may be called the Pomerance index of the n-th 3-Carmichael number.
An application of Pomerance index: The index for the Carmichael number 561 is 7. This can be used to prove that 561 is the only 3-factor Carmichael number with 3 as one of its factors. Proof: Let N be a 3-factor composite number. Keep 3 fixed and increase the other two prime factors indefinitely. The relevant Pomerance index is a number less than 7 but greater than 6. As the other two prime factors are increased indefinitely the Pomerance index becomes asymptotic to 6. Hence 561 is the only 3-factor Carmichael number with 3 as a factor. - A.K. Devaraj, Jul 27 2010
Let p be a prime number. Then, along the lines indicated above, it can be proved that there are only a finite number of 3-Carmichael numbers divisible by p. - A.K. Devaraj, Aug 06 2010

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A002997, A087788, A162990.

do(lim)=my(v=List()); forprime(p=3, sqrtnint(lim\=1,3), forprime(q=p+1, sqrtint(lim\p), forprime(r=q+1, lim\(p*q), if((q*r-1)%(p-1)||(p*r-1)%(q-1)||(p*q-1)%(r-1), , listput(v, [p*q*r,(p*q*r-1)*(p-1)/(q-1)/(r-1)]))))); v=vecsort(v,1); vector(#v,i,v[i][2]) \\ Charles R Greathouse IV, Sep 07 2016

Edited by N. J. A. Sloane, Sep 14 2009, based on email messages from David Broadhurst and M. F. Hasler, Jul 10 2009

Spelling corrected by Jason G. Wurtzel, Aug 23 2010

A065696 Numbers m such that N = (am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,55.

Original entry on oeis.org

3876, 7506, 8166, 16746, 20706, 23676, 24336, 40506, 42156, 68226, 69876, 79776, 95286, 123996, 139176, 149076, 166236, 177786, 183066, 187686, 203856, 210126, 213096, 214086, 216396, 221676, 232566, 265566, 307146, 310116, 321006, 326946

Offset: 1

Harvey Dubner (harvey(AT)dubner.com), Nov 14 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A064238-A064262, A065695, A065696, A065697, A065698, A065699, A065700, A065701, A065703, A087788.

CarmichaelNbrQ[n_] := ! PrimeQ@ n && Mod[n, CarmichaelLambda@ n] == 1; Select[ Range@ 350000, PrimeQ[# + 1] && PrimeQ[2# + 1] && PrimeQ[55# + 1] && CarmichaelNbrQ[(# + 1)(2# + 1)(55# + 1)] &] (* Robert G. Wilson v, Aug 23 2012 *)

am+1, bm+1, cm+1 are primes and am|(N-1), bm|(N-1), cm|(N-1).

A065697 Values of m such that N = (am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,57.

Original entry on oeis.org

198, 996, 2706, 9090, 13536, 16728, 25620, 33486, 34056, 35310, 41010, 53550, 58566, 60960, 61986, 63240, 72816, 72930, 74526, 75780, 77490, 80340, 83760, 96756, 97326, 100746, 103140, 111918, 125028, 125370, 128676, 129360, 136656, 164700, 174048, 175758, 176898

Offset: 1

Harvey Dubner (harvey(AT)dubner.com), Nov 14 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A064238-A064262, A065695-A065703, A087788.

CarmichaelNbrQ[n_] := ! PrimeQ@n && Mod[n, CarmichaelLambda@n] == 1; Select[ Range@140000, PrimeQ[# +1] && PrimeQ[2# +1] && PrimeQ[57# +1] && CarmichaelNbrQ[(# +1) (2# +1) (57# +1)] &] (* Robert G. Wilson v, Jul 31 2017 *)

am+1, bm+1, cm+1 are primes and am|(N-1), bm|(N-1), cm|(N-1).

More terms from Amiram Eldar, Oct 17 2019

A065698 Numbers m such that N = (am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,61.

Original entry on oeis.org

5580, 19488, 22050, 86466, 140268, 173208, 177966, 227010, 233598, 265806, 273126, 355110, 395736, 402690, 432336, 476988, 486138, 550188, 578370, 588618, 754416, 788088, 844086, 1044288, 1092600, 1204596, 1217406, 1386498, 1415778, 1446888, 1463358, 1563276, 1566936, 1599876

Offset: 1

Harvey Dubner (harvey(AT)dubner.com), Nov 14 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A064238-A064262, A065695-A065703, A087788.

CarmichaelNbrQ[n_] := ! PrimeQ@n && Mod[n, CarmichaelLambda@ n] == 1; Select[ Range@ 1600000, PrimeQ[# +1] && PrimeQ[2# +1] && PrimeQ[61# +1] &&  CarmichaelNbrQ[(# +1) (2# +1) (61# +1)] &] (* Robert G. Wilson v, Jul 31 2017 *)

am+1, bm+1, cm+1 are primes and am|(N-1), bm|(N-1), cm|(N-1).

a(8) onward from Robert G. Wilson v, Jul 31 2017

A065699 Values of m such that N = (am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,63.

Original entry on oeis.org

156, 2550, 3180, 19686, 29640, 40350, 41610, 43626, 46020, 51060, 65550, 72480, 79536, 80670, 85836, 97176, 133716, 150096, 159420, 170760, 184116, 191550, 214986, 229980, 255180, 262110, 278490, 279120, 293106, 294996, 301926, 337080, 350940, 369210, 370596

Offset: 1

Harvey Dubner (harvey(AT)dubner.com), Nov 14 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A064238-A064262, A065695-A065703, A087788.

carmQ[n_] := CompositeQ[n] && Divisible[n - 1, CarmichaelLambda[n]]; Select[Range[10^5], AllTrue[(v = {1, 2, 63}*# + 1), PrimeQ] && carmQ[Times @@ v] &] (* Amiram Eldar, Oct 17 2019 *)

am+1, bm+1, cm+1 are primes and am|(N-1), bm|(N-1), cm|(N-1).

More terms from Amiram Eldar, Oct 17 2019

A065700 Values of m such that N = (am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,65.

Original entry on oeis.org

876, 1656, 7506, 9066, 12966, 33636, 67956, 74586, 83556, 89796, 111636, 126456, 129186, 143616, 150246, 154926, 166626, 184566, 222786, 241116, 252036, 252816, 261786, 271926, 288306, 303906, 304686, 319116, 340956, 344856, 351096, 357726, 362406, 363966, 365526

Offset: 1

Harvey Dubner (harvey(AT)dubner.com), Nov 14 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A064238-A064262, A065695-A065703, A087788.

carmQ[n_] := CompositeQ[n] && Divisible[n - 1, CarmichaelLambda[n]]; Select[Range[10^5], AllTrue[(v = {1, 2, 65}*# + 1), PrimeQ] && carmQ[Times @@ v] &] (* Amiram Eldar, Oct 17 2019 *)

am+1, bm+1, cm+1 are primes and am|(N-1), bm|(N-1), cm|(N-1).

More terms from Amiram Eldar, Oct 17 2019

cp's OEIS Frontend

A064238 Values of m such that N = (am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,3.

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A064262 Values of m such that N=(am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,51.

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A065703 Values of m such that N = (m+1)(2m+1)(71m+1) is a 3-Carmichael number (A087788).

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A065695 Numbers m such that N = (am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,53.

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A162290 Let A087788(n) = p*q*r, where p

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A065696 Numbers m such that N = (am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,55.

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A065697 Values of m such that N = (am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,57.

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A065698 Numbers m such that N = (am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,61.

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A162290 Let A087788(n) = pqr, where p