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Carmichael numbers equal to the product of 5 primes.
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13
825265, 1050985, 9890881, 10877581, 12945745, 13992265, 16778881, 18162001, 27336673, 28787185, 31146661, 36121345, 37167361, 40280065, 41298985, 41341321, 41471521, 47006785, 67371265, 67994641, 69331969, 74165065
COMMENTS
A subsequence is given by (6n+1)*(12n+1)*(18n+1)*(36n+1)*(72n+1) with n in A206349. - M. F. Hasler, Apr 14 2015
Numbers k such that 6k+1, 12k+1, 18k+1 and 36k+1 are all primes.
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6
1, 45, 56, 121, 206, 255, 380, 506, 511, 710, 871, 1025, 1421, 1515, 1696, 2191, 2571, 2656, 2681, 3341, 3566, 3741, 3796, 3916, 3976, 4235, 4340, 4426, 5645, 5875, 6006, 7066, 7616, 7826, 7976, 8900, 8925, 8976, 9025, 9186, 9600, 9761, 10920, 11301, 11385
COMMENTS
(6k+1)*(12k+1)*(18k+1)*(36k+1) is a Carmichael number for all k in this sequence. - José María Grau Ribas, Feb 06 2012
MAPLE
select(n->isprime(6*n+1) and isprime(12*n+1) and isprime(18*n+1) and isprime(36*n+1), [$1..12000]); # Muniru A Asiru, May 27 2018
MATHEMATICA
Select[Range[20000], PrimeQ[6 # + 1] && PrimeQ[12 # + 1] && PrimeQ[18 # + 1] && PrimeQ[36 # + 1] &]
Select[Range[12000], And@@PrimeQ[{6, 12, 18, 36}#+1]&] (* Harvey P. Dale, Mar 25 2013 *)
PROG
(PARI) forprime(p=2, 1e5, if(p%6!=1, next); if(isprime(2*p-1)&&isprime(3*p-2)&&isprime(6*p-5), print1(p\6", "))) \\ Charles R Greathouse IV, Feb 06 2012 (PARI) is(m, c=36)=!until(bittest(c\=2, 0)&&9>c+=3, isprime(m*c+1)||return) \\ M. F. Hasler, Apr 15 2015 (Magma) [n: n in [0..2*10^4] | IsPrime(6*n+1) and IsPrime(12*n+1) and IsPrime(18*n+1) and IsPrime(36*n+1)]; // Vincenzo Librandi, Apr 15 2015 (GAP) Filtered([1..12000], n->IsPrime(6*n+1) and IsPrime(12*n+1) and IsPrime(18*n+1) and IsPrime(36*n+1)); # Muniru A Asiru, May 27 2018
Numbers m such that 6m+1, 12m+1, 18m+1, 36m+1 and 72m+1 are all prime.
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5
1, 121, 380, 506, 511, 3796, 5875, 6006, 8976, 9025, 9186, 10920, 12245, 12896, 14476, 14800, 15386, 22451, 23471, 32326, 35175, 38460, 39536, 40420, 41456, 43430, 44415, 59901, 60076, 61341, 74676, 76615, 76986, 82530, 87390, 99486, 101101, 107926, 112315, 112840, 115101
COMMENTS
A subsequence of A206024, which contains A206349 as a subsequence, see there for motivations.
MAPLE
f:=isprime: select(m->f(6*m+1) and f(12*m+1) and f(18*m+1) and f(36*m+1) and f(72*m+1), [$1..120000] ); # Muniru A Asiru, Jun 06 2018
MATHEMATICA
Select[Range[120000], PrimeQ[6 # + 1] && PrimeQ[12 # + 1] && PrimeQ[18 # + 1] && PrimeQ[36 # + 1] && PrimeQ[72 # + 1] &] (* Vincenzo Librandi, Apr 15 2015 *) Select[Range[120000], AllTrue[{6, 12, 18, 36, 72}#+1, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 23 2016 *)
PROG
(PARI) is(m, c=72)=!until(bittest(c\=2, 0)&&9>c+=3, isprime(m*c+1)||return)
(Magma) [n: n in [0..2*10^5] | IsPrime(6*n+1) and IsPrime(12*n+1) and IsPrime(18*n+1) and IsPrime(36*n+1)and IsPrime(72*n+1)]; // Vincenzo Librandi, Apr 15 2015 (GAP) Filtered([1..120000], m->IsPrime(6*m+1) and IsPrime(12*m+1) and IsPrime(18*m+1) and IsPrime(36*m+1) and IsPrime(72*m+1)); # Muniru A Asiru, Jun 06 2018
Numbers m such that 18*m + 1, 36*m + 1, 108*m + 1, and 162*m + 1 are all primes.
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5
1, 71, 155, 176, 241, 346, 420, 540, 690, 801, 1145, 1421, 1506, 2026, 2066, 3080, 3235, 3371, 3445, 3511, 3640, 4746, 4925, 5681, 5901, 6055, 6520, 7931, 8365, 8970, 9006, 9556, 9685, 10186, 11396, 11750, 11935, 12055, 12666, 13205, 13266, 13825, 13881, 14606
COMMENTS
If m is a term, then (18*m + 1) * (36*m + 1) * (108*m + 1) * (162*m + 1) is a Carmichael number ( A002997). These are the Carmichael numbers of the form W_4(3*m) in Nakamula et al. (2007). The corresponding Carmichael numbers are 12490201, 288503529142321, 6548129556412321, ...
EXAMPLE
1 is a term since 18*1 + 1 = 19, 36*1 + 1 = 37, 108*1 + 1 = 109, and 162*1 + 1 = 163 are all primes.
71 is a term since 18*71 + 1 = 1279, 36*71 + 1 = 2557, 108*71 + 1 = 7669, and 162*71 + 1 = 11503 are all primes.
MATHEMATICA
q[n_] := AllTrue[{18, 36, 108, 162}, PrimeQ[#*n + 1] &]; Select[Range[15000], q]
PROG
(PARI) is(n) = isprime(18*n + 1) && isprime(36*n + 1) && isprime(108*n + 1) && isprime(162*n + 1);
The least Chernick's "universal form" Carmichael number with n prime factors.
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4
1729, 63973, 26641259752490421121, 1457836374916028334162241, 24541683183872873851606952966798288052977151461406721, 53487697914261966820654105730041031613370337776541835775672321, 58571442634534443082821160508299574798027946748324125518533225605795841
COMMENTS
Chernick proved that U(k, m) = (6m + 1)*(12m + 1)*Product_{i = 1..k-2} (9*(2^i)m + 1), for k >= 3 and m >= 1 is a Carmichael number, if all the factors are primes and, for k >= 4, 2^(k-4) divides m. He called U(k, m) "universal forms". This sequence gives a(k) = U(k, m) with the least value of m. The least values of m for k = 3, 4, ... are 1, 1, 380, 380, 780320, 950560, 950560, 3208386195840, 31023586121600, ...
LINKS
Samuel S. Wagstaff, Jr., Large Carmichael numbers, Mathematical Journal of Okayama University, Vol. 22, (1980), pp. 33-41.
EXAMPLE
For k=3, m = 1, a(3) = U(3, 1) = (6*1 + 1)*(12*1 + 1)*(18*1 + 1) = 1729.
For k=4, m = 1, a(4) = U(4, 1) = (6*1 + 1)*(12*1 + 1)*(18*1 + 1)*(36*1 + 1) = 63973.
For k=5, m = 380, a(5) = U(5, 1) = (6*380 + 1)*(12*380 + 1)*(18*380 + 1)*(36*380 + 1)*(72*380 + 1) = 26641259752490421121.
MATHEMATICA
fc[k_] := If[k < 4, 1, 2^(k - 4)]; a={}; Do[v = Join[{6, 12}, 2^Range[k-2]*9];
w = fc[k]; x = v*w; m = 1; While[! AllTrue[x*m + 1, PrimeQ], m++]; c=Times @@ (x*m + 1); AppendTo[a, c], {k, 3, 9}]; a
Numbers m such that 20*m + 1, 80*m + 1, 100*m + 1, and 200*m + 1 are all primes.
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4
333, 741, 1659, 1749, 2505, 2706, 2730, 4221, 4437, 4851, 5625, 6447, 7791, 7977, 8229, 8250, 9216, 10833, 12471, 13950, 14028, 15147, 16002, 17667, 18207, 18246, 19152, 20517, 23400, 23421, 23961, 25689, 26247, 28587, 28608, 30363, 31584, 34167, 36330, 36378
COMMENTS
If m is a term, then (20*m + 1) * (80*m + 1) * (100*m + 1) * (200*m + 1) is a Carmichael number ( A002997). These are the Carmichael numbers of the form U_{4,4}(m) in Nakamula et al. (2007). The corresponding Carmichael numbers are 393575432565765601, 9648687289456956001, 242412946401534283201, ...
EXAMPLE
333 is a term since 20*333 + 1 = 6661, 80*333 + 1 = 26641, 100*333 + 1 = 33301, and 200*333 + 1 = 66601 are all primes.
MATHEMATICA
q[n_] := AllTrue[{20, 80, 100, 200}, PrimeQ[# * n + 1] &]; Select[Range[40000], q]
PROG
(PARI) is(n) = isprime(20*n + 1) && isprime(80*n + 1) && isprime(100*n + 1) && isprime(200*n + 1);
Numbers m such that 72*m + 1, 576*m + 1, 648*m + 1, 1296*m + 1, and 2592*m + 1 are all primes.
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4
95, 890, 3635, 8150, 9850, 12740, 13805, 18715, 22590, 23591, 32526, 36395, 38571, 49016, 49456, 57551, 58296, 61275, 80756, 81050, 84980, 99940, 104346, 115361, 116761, 121055, 122550, 129320, 140331, 142625, 149431, 153505, 159306, 159730, 169625, 173485, 181661
COMMENTS
If m is a term, then (72*m + 1) * (576*m + 1) * (648*m + 1) * (1296*m + 1) * (2592*m + 1) is a Carmichael number ( A002997). These are the Carmichael numbers of the form U_{5,5}(m) in Nakamula et al. (2007). The corresponding Carmichael numbers are 698669495582067436250881, 50411423376758357271937215361, 57292035175893741987253427965441, ...
EXAMPLE
95 is a term since 72*95 + 1 = 6841, 576*95 + 1 = 54721, 648*95 + 1 = 61561, 1296*95 + 1 = 123121, and 2592*95 + 1 = 246241 are all primes.
MATHEMATICA
q[n_] := AllTrue[{72, 576, 648, 1296, 2592}, PrimeQ[#*n + 1] &]; Select[Range[200000], q]
PROG
(PARI) is(n) = isprime(72*n + 1) && isprime(576*n + 1) && isprime(648*n + 1) && isprime(1296*n + 1) && isprime(2592*n + 1);
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