A206024 -id:A206024 - cp's OEIS frontend

A257035 Numbers m such that 6m+1, 12m+1, 18m+1, 36m+1 and 72m+1 are all prime.

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

Offset: 1

M. F. Hasler, Apr 14 2015

A subsequence of A206024, which contains A206349 as a subsequence, see there for motivations.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A002997, A112428, A174613, A206024, A206349.

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

[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

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

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 *)

is(m,c=72)=!until(bittest(c\=2,0)&&9>c+=3,isprime(m*c+1)||return)

A318646 The least Chernick's "universal form" Carmichael number with n prime factors.

Original entry on oeis.org

1729, 63973, 26641259752490421121, 1457836374916028334162241, 24541683183872873851606952966798288052977151461406721, 53487697914261966820654105730041031613370337776541835775672321, 58571442634534443082821160508299574798027946748324125518533225605795841

Offset: 3

Amiram Eldar, Aug 31 2018

nonn

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, ...

			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.

Amiram Eldar, Table of n, a(n) for n = 3..11
Jack Chernick, On Fermat's simple theorem, Bulletin of the American Mathematical Society, Vol. 45, No. 4 (1939), pp. 269-274.
Daniel Suteu, C++ program
Samuel S. Wagstaff, Jr., Large Carmichael numbers, Mathematical Journal of Okayama University, Vol. 22, (1980), pp. 33-41.

Cf. A002997, A033502 (3 prime factors), A206024 (4 prime factors), A206349 (5 prime factors), A126797.

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

A372188 Numbers m such that 18m + 1, 36m + 1, 108m + 1, and 162m + 1 are all primes.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Apr 21 2024

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, ...

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ken Nakamula, Hirofumi Tsumura, and Hiroaki Komai, New polynomials producing absolute pseudoprimes with any number of prime factors, arXiv:math/0702410 [math.NT], 2007.

Cf. A002997, A318646, A372190.

Similar sequences: A046025, A257035, A206024, A206349, A372186, A372187.

q[n_] := AllTrue[{18, 36, 108, 162}, PrimeQ[#*n + 1] &]; Select[Range[15000], q]

is(n) = isprime(18*n + 1) && isprime(36*n + 1) && isprime(108*n + 1) && isprime(162*n + 1);

A372186 Numbers m such that 20m + 1, 80m + 1, 100m + 1, and 200m + 1 are all primes.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Apr 21 2024

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, ...

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ken Nakamula, Hirofumi Tsumura, and Hiroaki Komai, New polynomials producing absolute pseudoprimes with any number of prime factors, arXiv:math/0702410 [math.NT], 2007.

Cf. A002997, A318646, A372189.

Similar sequences: A046025, A257035, A206024, A206349, A372187, A372188.

q[n_] := AllTrue[{20, 80, 100, 200}, PrimeQ[# * n + 1] &]; Select[Range[40000], q]

is(n) = isprime(20*n + 1) && isprime(80*n + 1) && isprime(100*n + 1) && isprime(200*n + 1);

A372187 Numbers m such that 72m + 1, 576m + 1, 648m + 1, 1296m + 1, and 2592*m + 1 are all primes.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Apr 21 2024

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, ...

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ken Nakamula, Hirofumi Tsumura, and Hiroaki Komai, New polynomials producing absolute pseudoprimes with any number of prime factors, arXiv:math/0702410 [math.NT], 2007.

Cf. A002997, A126797, A318646.

Similar sequences: A046025, A257035, A206024, A206349, A372186, A372188.

q[n_] := AllTrue[{72, 576, 648, 1296, 2592}, PrimeQ[#*n + 1] &]; Select[Range[200000], q]

is(n) = isprime(72*n + 1) && isprime(576*n + 1) && isprime(648*n + 1) && isprime(1296*n + 1) && isprime(2592*n + 1);

cp's OEIS Frontend

A257035 Numbers m such that 6m+1, 12m+1, 18m+1, 36m+1 and 72m+1 are all prime.

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A318646 The least Chernick's "universal form" Carmichael number with n prime factors.

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A372188 Numbers m such that 18*m + 1, 36*m + 1, 108*m + 1, and 162*m + 1 are all primes.

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A372186 Numbers m such that 20*m + 1, 80*m + 1, 100*m + 1, and 200*m + 1 are all primes.

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A372187 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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A372188 Numbers m such that 18m + 1, 36m + 1, 108m + 1, and 162m + 1 are all primes.

A372186 Numbers m such that 20m + 1, 80m + 1, 100m + 1, and 200m + 1 are all primes.

A372187 Numbers m such that 72m + 1, 576m + 1, 648m + 1, 1296m + 1, and 2592*m + 1 are all primes.