A126797 -id:A126797 - cp's OEIS frontend

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

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

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);

A372189 Number of terms of A372186 that do not exceed 10^n.

Original entry on oeis.org

0, 0, 2, 17, 87, 487, 2959, 18960, 126997, 878559, 6263608, 45854245

Offset: 1

Amiram Eldar, Apr 21 2024

Ken Nakamula, Hirofumi Tsumura, and Hiroaki Komai, New polynomials producing absolute pseudoprimes with any number of prime factors, arXiv:math/0702410 [math.NT], 2007. See Table 1, p. 8.

Cf. A002997, A036060, A126797, A372186, A372190.

A372190 Number of terms of A372188 that do not exceed 10^n.

Original entry on oeis.org

1, 2, 10, 33, 149, 824, 5116, 32077, 213075, 1463213, 10397977, 75903023

Offset: 1

Amiram Eldar, Apr 21 2024

Ken Nakamula, Hirofumi Tsumura, and Hiroaki Komai, New polynomials producing absolute pseudoprimes with any number of prime factors, arXiv:math/0702410 [math.NT], 2007. See Table 3, p. 9.

Cf. A002997, A036060, A126797, A372188, A372189.

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

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

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A372189 Number of terms of A372186 that do not exceed 10^n.

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A372190 Number of terms of A372188 that do not exceed 10^n.

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cp's OEIS Frontend

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

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Author

Keywords

Comments

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Mathematica

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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Author

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PARI

A372189 Number of terms of A372186 that do not exceed 10^n.

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Author

Keywords

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A372190 Number of terms of A372188 that do not exceed 10^n.

Views

Author

Keywords

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