A318646 -id:A318646 - cp's OEIS frontend

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

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);

A382835 Array read by ascending antidiagonals: A(n,k) = (6n + 1)(12n + 1)Product_{i=0..k-2} (92^in + 1) with k >= 2.

Original entry on oeis.org

1, 91, 1, 325, 1729, 1, 703, 12025, 63973, 1, 1225, 38665, 877825, 4670029, 1, 1891, 89425, 4214485, 127284625, 677154205, 1, 2701, 172081, 12966625, 914543245, 36785256625, 195697565245, 1, 3655, 294409, 31146661, 3747354625, 395997225085, 21225093072625, 112917495146365, 1

Offset: 0

Stefano Spezia, Apr 06 2025

A(n,k) is a Carmichael number with k prime factors if n is such all k factors are prime numbers and 2*k-4 divides n (see Ribenboim).

			The array begins as:
     1,      1,        1,           1,              1, ...
    91,   1729,    63973,     4670029,      677154205, ...
   325,  12025,   877825,   127284625,    36785256625, ...
   703,  38665,  4214485,   914543245,   395997225085, ...
  1225,  89425, 12966625,  3747354625,  2162223618625, ...
  1891, 172081, 31146661, 11243944621,  8106884071741, ...
  2701, 294409, 63886753, 27662964049, 23928463902385, ...
  ...

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 101.

Cf. A000012 (n=0), A002997, A318646, A382809 (k=3), A382836 (antidiagonal sums).

A[n_,k_]:=(6n+1)(12n+1)Product[9*2^i*n+1,{i,k-2}];Table[A[n-k,k],{n,0,9},{k,2,n}]//Flatten

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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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A382835 Array read by ascending antidiagonals: A(n,k) = (6*n + 1)*(12*n + 1)*Product_{i=0..k-2} (9*2^i*n + 1) with k >= 2.

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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.

A382835 Array read by ascending antidiagonals: A(n,k) = (6n + 1)(12n + 1)Product_{i=0..k-2} (92^in + 1) with k >= 2.