A046025 -id:A046025 - cp's OEIS frontend

A033503 First differences of A046025.

5, 29, 10, 6, 4, 1, 44, 21, 74, 11, 10, 39, 21, 94, 10, 46, 80, 4, 1, 199, 31, 59, 25, 46, 59, 45, 50, 35, 55, 25, 21, 109, 10, 1, 30, 25, 25, 4, 16, 39, 1, 20, 49, 25, 181, 109, 70, 10, 46, 109, 25, 21, 105, 44, 11, 10, 15, 95, 59, 95, 45, 6, 85, 25, 30, 165, 84, 60, 55, 56

Offset: 1

Marc LeBrun

nonn

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

Cf. A046025, A002997.

Differences[Select[Range[3500],And@@PrimeQ[{6,12,18}#+1]&]] (* Harvey P. Dale, May 26 2014 *)

A033502 Carmichael numbers of the form (6k+1)(12k+1)(18k+1), where 6k+1, 12k+1 and 18k+1 are all primes.

Original entry on oeis.org

1729, 294409, 56052361, 118901521, 172947529, 216821881, 228842209, 1299963601, 2301745249, 9624742921, 11346205609, 13079177569, 21515221081, 27278026129, 65700513721, 71171308081, 100264053529, 168003672409, 172018713961, 173032371289, 464052305161

Offset: 1

Marc LeBrun

nonn

Also called Chernick's Carmichael numbers. The polynomial (6*k+1)*(12*k+1)*(18*k+1) is the simplest Chernick polynomial. [Named after the American physicist and mathematician Jack Chernick (1911-1971). - Amiram Eldar, Jun 15 2021]
The first term, 1729, is the Hardy-Ramanujan number and the smallest primary Carmichael number (A324316).
Dickson's conjecture implies that this sequence is infinite, as pointed out by Chernick.
All terms of this sequence are primary Carmichael numbers (A324316) having the following remarkable property. Let m be a term of A033502. For each prime divisor p of m, the sum of the base-p digits of m equals p. This property also holds for "almost all" 3-term Carmichael numbers (A087788), since they can be represented by certain Chernick polynomials, whose values obey a strict s-decomposition (A324460) besides certain exceptions, see Kellner 2019. - Bernd C. Kellner, Aug 03 2022

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A13, pp. 50-53.

Donovan Johnson, Table of n, a(n) for n = 1..10000
Jack Chernick, On Fermat's simple theorem, Bull. Amer. Math. Soc., Vol. 45, No. 4 (1939), pp. 269-274.
Shyam Sunder Gupta, On Some Special Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 22, 527-565.
Douglas E. Iannucci, When the small divisors of a natural number are in arithmetic progression, INTEGERS, Electronic Journal of Combinatorial Number Theory, Vol. 18 (2018), #77. See p. 9.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.
G. Tarry, I. Franel, A. Korselt, and G. Vacca. Problème chinois. L'intermédiaire des mathématiciens, Vol. 6 (1899), pp. 142-144.
Eric Weisstein's World of Mathematics, Carmichael Number.
Index entries for sequences related to Carmichael numbers.

Values of k are given by A046025. Subsequence of A002997, A087788, and A324316.

Cf. A242980, A242981.

[n : k in [1..710] | IsPrime(a) and IsPrime(b) and IsPrime(c) and IsOne(n mod CarmichaelLambda(n)) where n is a*b*c where a is 6*k+1 where b is 12*k+1 where c is 18*k+1]; // Arkadiusz Wesolowski, Oct 29 2013

CarmichaelNbrQ[n_] := ! PrimeQ@ n && Mod[n, CarmichaelLambda@ n] == 1; (6# + 1)(12# + 1)(18# + 1) & /@
Select[ Range@ 1000, PrimeQ[6# + 1] && PrimeQ[12# + 1] && PrimeQ[18# + 1] && CarmichaelNbrQ[(6# + 1)(12# + 1)(18# + 1)] &]

Definition corrected (thanks to Umberto Cerruti) by Bruno Berselli, Jan 18 2013

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.

Original entry on oeis.org

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

A206024 Numbers k such that 6k+1, 12k+1, 18k+1 and 36k+1 are all primes.

Original entry on oeis.org

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

Offset: 1

José María Grau Ribas, Feb 03 2012

nonn

(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

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Muniru A Asiru)
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A046025, A206349, A257035.

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

[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

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

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 *)

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

is(m,c=36)=!until(bittest(c\=2,0)&&9>c+=3, isprime(m*c+1)||return) \\ M. F. Hasler, Apr 15 2015

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);

A067199 Integers k such that k28c + 1 is prime for c = 1, 2, 4, 7 and 14.

Original entry on oeis.org

2136, 2211, 4071, 5106, 5430, 9000, 10656, 17655, 18315, 20220, 20805, 21381, 22356, 22920, 23025, 29616, 37050, 39261, 45795, 49920, 55686, 60435, 62205, 64380, 79356, 81345, 91455, 94800, 95910, 96285, 105336, 108585, 111885, 118626

Offset: 1

Frank Ellermann, Feb 19 2002

The product of the 5 primes is a Carmichael number. 28=1+2+4+7+14.

			2136 results in Carmichael number 599966117492747584686619009.

H. Davenport, The Higher Arithmetic. Cambridge Univ. Press, 7th ed., 1999, exercise 8.4.

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

Cf. A002997, A046025 (based on 6 instead of 28, exercise 8.3 in Davenport), A112428.

aQ[n_] := AllTrue[{1, 2, 4, 7, 14}, PrimeQ[28 * n * # + 1] &]; Select[Range[10^5], aQ] (* Amiram Eldar, Sep 19 2019 *)

Offset corrected by Amiram Eldar, Sep 19 2019

A101186 Values of k for which 7m+1, 8m+1 and 11m+1 are prime, with m = 1848k + 942.

Original entry on oeis.org

13, 123, 218, 223, 278, 411, 513, 551, 588, 733, 743, 796, 856, 928, 1168, 1226, 1263, 1401, 1533, 1976, 1981, 2013, 2096, 2138, 2241, 2376, 2556, 2676, 2703, 3626, 3703, 3718, 3971, 4008, 4121, 4138, 4163, 4188, 4211, 4313, 4423, 4653, 4656, 4901, 5018

Offset: 1

Gerard P. Michon, Dec 03 2004

nonn

The number (7m+1)(8m+1)(11m+1) is a 3-factor Carmichael number if and only if m is equal to 1848k+942 with k in this sequence. The sequence includes the value k = 10^329 - 4624879 which yields a 1000-digit Carmichael number with three prime factors of 334 digits each. Other Carmichael numbers of the same form would necessarily have 4 prime factors or more; the smallest such example is 3664585=127*(7*29)*199, for m=18.

			a(1)=13 because k=13 corresponds to m=24966, which yields a product of three primes (7m+1)(8m+1)(11m+1) equal to the Carmichael number 9585921133193329. (Among all Carmichael numbers with 16 or fewer digits, as first listed by Richard G. E. Pinch, this one features the largest "least prime factor".)

Robert Israel, Table of n, a(n) for n = 1..10000
Gérard P. Michon, Generic Carmichael Numbers.

Cf. A002997 (Carmichael numbers), A046025.

[k:k in [1..5100]| forall{s:s in [7,8,11]|IsPrime(m*s+1) where m is 1848*k+942}]; // Marius A. Burtea, Nov 01 2019

filter:= proc(n) local m;
  m:= 1848*n+942;
  andmap(isprime,[7*m+1,8*m+1,11*m+1])
end proc:
select(filter, [$1..10000]); # Robert Israel, May 14 2019

q[k_] := Module[{m = 1848*k + 942}, PrimeQ[7*m + 1] && PrimeQ[8*m + 1] && PrimeQ[11*m + 1]]; Select[Range[6000], q] (* Amiram Eldar, Apr 27 2024 *)

is(k) = {my(m = 1848*k + 942); isprime(7*m + 1) && isprime(8*m + 1) && isprime(11*m + 1);} \\ Amiram Eldar, Apr 27 2024

A101187 Values of m for which (6m+1)(12m+1)(18m+1) is a Carmichael number.

Original entry on oeis.org

1, 5, 6, 11, 15, 22, 33, 35, 45, 51, 55, 56, 61, 85, 96, 100, 103, 105, 115, 121, 195, 206, 216, 225, 242, 255, 276, 370, 380, 426, 455, 470, 506, 510, 511, 550, 561, 588, 609, 628, 661, 700, 710, 741, 800, 805, 825, 871, 920, 930, 975, 1025, 1060, 1115, 1140

Offset: 1

Gerard P. Michon, Dec 03 2004

nonn

A046025 is a subsequence giving the values of m for which the three factors are prime, which is a sufficient condition for the product (6m+1)(12m+1)(18m+1) to be a Carmichael number.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Bruno Berselli)
Harvey Dubner, Carmichael Numbers of the form (6m+1)(12m+1)(18m+1), Journal of Integer Sequences, Vol. 5 (2002) Article 02.2.1.
Gérard P. Michon, Generic Carmichael Numbers.

Cf. A002997 (Carmichael numbers), A046025 (subsequence), A101186.

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A033503 First differences of A046025.

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A033502 Carmichael numbers of the form (6*k+1)*(12*k+1)*(18*k+1), where 6*k+1, 12*k+1 and 18*k+1 are all primes.

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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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A206024 Numbers k such that 6k+1, 12k+1, 18k+1 and 36k+1 are all primes.

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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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A033502 Carmichael numbers of the form (6k+1)(12k+1)(18k+1), where 6k+1, 12k+1 and 18k+1 are all primes.

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.

A067199 Integers k such that k28c + 1 is prime for c = 1, 2, 4, 7 and 14.