A033502 -id:A033502 - cp's OEIS frontend

A255441 Carmichael numbers of the form (60k+41)(90k+61)(150k+101), where 60k+41, 90k+61 and 150k+101 are all primes.

252601, 3828001, 82929001, 366652201, 8251854001, 12173703001, 25749237001, 67495942201, 82380774001, 285983187601, 802204143001, 1039708768201, 1068674250001, 1158788350201, 2997587784001, 3236207713201, 4467180843001, 7902515425201, 8470346587201

Offset: 1

Vincenzo Librandi, Feb 24 2015

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Umberto Cerruti, Pseudoprimi di Fermat e numeri di Carmichael (in Italian), p. 10.
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A033502.

Cf. A255512 (associated k).

[(60*n+41)*(90*n+61)*(150*n+101): n in [0..300]| IsPrime(60*n+41) and IsPrime(90*n+61) and IsPrime(150*n+101)];

f[k_] := {60*k + 41, 90*k + 61, 150*k + 101}; Times @@ f[#]& /@ Select[Range[0, 500], And @@ PrimeQ[f[#]] &] (* Amiram Eldar, Apr 24 2024 *)

lista(kmax) = for(k = 0, kmax, if(isprime(60*k + 41) && isprime(90*k + 61) && isprime(150*k + 101), print1((60*k+41)*(90*k+61)*(150*k+101), ", "))); \\ Amiram Eldar, Apr 24 2024

First term added from Bruno Berselli, Feb 24 2015

A290811 Numbers n such that (6n-1, 6n+1), (12n-1, 12n+1) and (18n-1, 18n+1) are 3 pairs of twin primes.

Original entry on oeis.org

1, 8925, 70070, 70385, 270725, 355040, 566650, 866635, 874335, 1091545, 1230740, 1295980, 1586095, 1594285, 1738380, 1974210, 2201325, 2427145, 2436665, 3124660, 3349990, 3599470, 3661350, 4059825, 4101790, 4486020, 4726540, 5139680, 5613370, 5898655, 6279035

Offset: 1

Amiram Eldar, Aug 11 2017

nonn

If n is in the sequence then (6n+1)*(12n+1)*(18n+1) is a Carmichael number (A002997) and (6n-1)*(12n-1)*(18n-1) is a Lucas-Carmichael number (A006972).
Intersection of A046025 and A290810.
The first 10 pairs of corresponding Lucas-Carmichael and Carmichael numbers ((6n-1)*(12n-1)*(18n-1), (6n+1)*(12n+1)*(18n+1)) are:
(935, 1729)
(921329139943799, 921392227198801)
(445860973748310119, 445864862313790921)
(451901165073782759, 451905088679976961)
(25715181770344848599, 25715239817629143601)
(58001133699332691839, 58001233533626759041)
(235803065459494289399, 235803319764534509401)
(843555229160685647759, 843555823997214441961)
(866240412591524160959, 866241018045184403161)
(1685504102154302331719, 1685505045798928055521)
(2416038446298343361039, 2416039645957333860241)

			1 is in the sequence since (6*1 - 1, 6*1 + 1) = (5, 7), (12*1 - 1, 12*1 + 1) = (11, 13) and (18*1 - 1, 18*1 + 1) = (17, 19) are all pairs of twin primes.

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..10000

Cf. A002997, A006972, A033502, A046025, A290810.

seq = {}; Do[ If[ AllTrue[{6 m - 1, 6 m + 1, 12 m - 1, 12 m + 1, 18 m - 1,
    18 m + 1}, PrimeQ ], AppendTo[seq, m]], {m, 1, 10^7} ]; seq
Select[Range[6280000],AllTrue[{6#+1,6#-1,12#+1,12#-1,18#+1,18#-1},PrimeQ]&] (* Harvey P. Dale, Jun 21 2024 *)

isok(n) = isprime(6*n-1) && isprime(6*n+1) && isprime(12*n-1) && isprime(12*n+1) && isprime(18*n-1) && isprime(18*n+1); \\ Michel Marcus, Aug 11 2017

A317126 Numbers of the form: (6m + 1) (12m + 1) Product_{i=1..k-2} (9 * 2^i * m + 1), where k >= 3, with the condition that each of the factors is prime and that m is divisible by 2^(k-4).

Original entry on oeis.org

1729, 63973, 294409, 56052361, 118901521, 172947529, 216821881, 228842209, 1299963601, 2301745249, 9624742921, 11346205609, 13079177569, 21515221081, 27278026129, 65700513721, 71171308081, 100264053529, 168003672409, 172018713961, 173032371289, 192739365541

Offset: 1

Daniel Suteu, Jul 21 2018

nonn

Also known as extended Chernick Carmichael numbers.

Each term of this sequence is the product of 3 or more distinct prime factors.

Daniel Suteu, Table of n, a(n) for n = 1..10081
Jack Chernick, On Fermat's simple theorem, Bull. Amer. Math. Soc. 45:4 (1939), pp. 269-274.
Daniel Suteu, Perl program
Eric Weisstein's World of Mathematics, Carmichael Number
Index entries for sequences related to Carmichael numbers.

Cf. A033502.

A382809 a(n) = (6n + 1)(12n + 1)(18*n + 1).

Original entry on oeis.org

1, 1729, 12025, 38665, 89425, 172081, 294409, 464185, 689185, 977185, 1335961, 1773289, 2296945, 2914705, 3634345, 4463641, 5410369, 6482305, 7687225, 9032905, 10527121, 12177649, 13992265, 15978745, 18144865, 20498401, 23047129, 25798825, 28761265, 31942225, 35349481

Offset: 0

Stefano Spezia, Apr 05 2025

a(n) is a Carmichael number if all the three factors (6*n + 1), (12*n + 1), and (18*n + 1) are prime (see Chernick and Ribenboim).

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 101.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 146.

Jack Chernick, On Fermat's simple theorem, Bulletin of the American Mathematical Society, Vol. 45, No. 4 (1939), pp. 269-274.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002997, A033502, A046025.

Cf. A002476, A002997, A016921, A017533, A068228, A161705.

LinearRecurrence[{4,-6,4,-1},{1,1729,12025,38665},31]

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3.
G.f.: (1 + 1725*x + 5115*x^2 + 935*x^3)/(1 - x)^4.
E.g.f.: exp(x)*(1 + 1728*x + 4284*x^2 + 1296*x^3).
a(n) = A016921(n) * A017533(n) * A161705(n).
a(n) == 1 (mod 72).

A319008 Let k = A000396(n) be the n-th perfect number, a(n) is the least number m such that kdm + 1 is prime for all of the proper divisors d of k so their product is a Carmichael number.

Original entry on oeis.org

1, 2136, 13494274080, 216818853118725

Offset: 1

Amiram Eldar, Sep 07 2018

Chernick proved that (6m + 1)*(12m + 1)*(18m + 1) is a Carmichael number, if all the 3 factors are primes (A033502, A046025).
Lieuwens generalized it to Product_{i} (k*d(i)*m + 1), for k a perfect number.
a(1) corresponds to 6. It was found by Jack Chernick in 1939.
a(2) corresponds to 28. It was found by Dubner in 1996. Lieuwens evaluated that the least corresponding Carmichael number > 10^27.
a(3) corresponds to 496. It was found by Jim Fougeron in 2002 (Dubner found a larger value: 474382033125).
a(4) corresponds to 8128. It was found by Phil Carmody in 2002.
The corresponding Carmichael numbers are 1729, 599966117492747584686619009, 1.631... * 10^126, 4.559... * 10^260, ...

			28 = 1 + 2 + 4 + 7 + 14 is the second perfect number. 2136 is the least number m such that 28*1*333 + 1 = 59809, 28*2*2136 + 1 = 119617, 28*4*2136 + 1 = 239233, 28*7*2136 + 1 =  418657 and 28*14*2136 + 1 = 837313 are all primes, therefore 59809*119617*239233*418657*837313 = 599966117492747584686619009 is a Carmichael number.

Harold Davenport, The Higher Arithmetic, Cambridge University Press, 7th ed., 1999, exercise 8.4.
Harvey Dubner, Carmichael numbers and Egyptian fractions, Mathematica japonicae, Vol. 43, No. 2 (1996), pp. 411-419.

Jack Chernick, On Fermat's simple theorem, Bulletin of the American Mathematical Society, Vol. 45, No. 4 (1939), pp. 269-274.
Claude Goutier, De l'utilité d'une vieille curiosité grecque, Crux Mathematicorum, Vol. 46, No. 8 (2020), pp. 397-403.
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 59996...19009 (27-digits).
Erik Lieuwens, Fermat pseudo primes, Doctoral Thesis, Delft University of Technology, 1971, pp. 29-30.
Carlos Rivera, Puzzle 171. Perfect & Carmichael numbers, The Prime Puzzles & Problems Connection.

Cf. A000396, A002997, A033502, A046025, A067199.

ms = {2, 3, 5, 7, 13}; ns = Length[ms]; M[p_] := 2^(p - 1)*(2^p - 1); L[m_] := Module[{}, d = Most[Divisors[m]]*m; aQ[n_] := AllTrue[d*n + 1, PrimeQ]; n=1; While[!aQ[n], n++];n]; s={}; Do[m = M[ms[[k]]]; b = L[m]; AppendTo[s, b], {k, 1, ns}]; s

A343979 Composite numbers m such that lambda(m) = lambda(D_{m-1}), where lambda(n) is the Carmichael function of n (A002322) and D_k is the denominator (A027642) of Bernoulli number B_k.

Original entry on oeis.org

5615659951, 36901698733, 55723044637, 776733036121, 2752403727511, 7725145165297, 14475486778537, 15723055492417, 22824071195485, 29325910221631, 54669159894469, 62086332981241, 125685944708905, 180225455689481, 298620660945331, 335333122310629, 426814989321721

Offset: 1

Amiram Eldar and Thomas Ordowski, May 06 2021

nonn

Squarefree composites m such that LCM_{prime p|m} (p-1) = LCM_{prime p, p-1|m-1} (p-1).
Carmichael numbers m such that LCM_{prime p|m} (p-1) = LCM_{prime p, p-1|m-1} (p-1), i.e., with A173614(m) = A346467(m).
Carmichael numbers m such that their index (m-1)/lambda(m) = A346468(m), cf. A174590.
Carl Pomerance noted that, for k = 40826, Chernick's Carmichael number (6k+1)*(12k+1)*(18k+1) = 88189878776579929 satisfies this condition.
Theorem: lambda(m) | lambda(D_{m-1}) if and only if m | D_{m-1}.
Composites m such that lambda(m) | lambda(D_{m-1}) are all Carmichael numbers, defined as composites m such that lambda(m) | m-1, while lambda(D_{m-1}) | m-1 for every m.
Note that if p is prime, then lambda(p) = lambda(D_{p-1}) = p-1.

Cf. A002322, A002997, A027642, A033502, A173614, A174590, A346467, A346468.

c = Cases[Import["https://oeis.org/A002997/b002997.txt", "Table"], {, }][[;; , 2]]; q[d_] := If[PrimeQ[d + 1], d, 1]; Select[c, LCM @@ (FactorInteger[#][[;; , 1]] - 1) == LCM @@ (q /@ Divisors[# - 1]) &]

A002322(n) = lcm(znstar(n)[2]); \\ From A002322
A173614(n) = lcm(apply(p->p-1, factor(n)[, 1]));
A346467(n) = if(1==n,n,my(m=1); fordiv(n-1,d,if(isprime(1+d),m = lcm(m,d))); (m));
isA343979(n) = ((n>1) && !isprime(n) && (!((n-1)%A002322(n))) && A173614(n)==A346467(n)); \\ Antti Karttunen, Jul 22 2021

A379656 Carmichael numbers that are the sum of 2 positive cubes.

Original entry on oeis.org

1729, 15841, 46657, 126217, 188461, 1082809, 1773289, 2628073, 3146221, 5049001, 6868261, 14469841, 19683001, 31146661, 40917241, 78091201, 92625121, 144218341, 252141121, 1836304561, 2616662881, 3035837161, 4354716961, 4828075561, 10779325921, 13200275881, 14235803713

Offset: 1

Amiram Eldar, Dec 29 2024

nonn

Below 10^22 there are only 2 Carmichael numbers that are the sum of two positive cubes in two or more different ways (i.e., in A001235): 1729 = 1^3 + 12^3 = 9^3 + 10^3 and 23226658794001 = 9001^3 + 28230^3 = 19108^3 + 25329^3.

Chernick's Carmichael numbers (A033502) are 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 (k is a term of A046025). There are no Chernick's Carmichael numbers other than 1729 that are the sum of two positive cubes in two or more different ways (Lagarias, 2018). In the solution to Lagarias's problem it is noted that John P. Robertson showed that if there are Chernick's Carmichael numbers other than 1729 (corresponding to k = 1) that are the sum of two positive cubes (i.e., terms of this sequence), then they have k > 10^5000.

Amiram Eldar, Table of n, a(n) for n = 1..831 (terms below 10^22)
Jeffrey C. Lagarias, Problem 12048, Problems and Solutions, The American Mathematical Monthly, Vol. 125, No. 6 (2018), p. 562; JSTOR link; Carmichael in a Taxicab, Solution to Problem 12048 by Albert Stadler, ibid., Vol. 127, No. 1 (2020), p. 93; JSTOR link.
Samuel S. Wagstaff, Ramanujan's taxicab number and its ilk, The Ramanujan Journal, Vol. 64, No. 3 (2024), pp. 761-764; ResearchGate link, author's copy.

Intersection of A002997 and A003325.
A265628 is a subsequence.
Cf. A001235, A033502, A046025.

carmQ[n_] := CompositeQ[n] && Divisible[n-1, CarmichaelLambda[n]]; Select[Range[200000], carmQ[#] && Length[PowersRepresentations[#, 2, 3]] > 0 &]

isA003325(n) = #select(v->min(v[1], v[2])>0, thue(thueinit('z^3+1);, n)) > 0; \\ Charles R Greathouse IV at A003325
is(n) = (n > 1) && !isprime(n) && !((n-1) % lcm(znstar(n)[2])) && isA003325(n);

A319011 Let k = A064771(n) be the n-th pseudoperfect number such that {d(i)} is a unique subset of its proper divisors that sums to k, a(n) is the least number m such that kd(i)m + 1 is prime for all d(i) in this subset so their product is a Carmichael number.

Original entry on oeis.org

1, 333, 2136, 14, 72765, 49, 9765, 5, 154, 490, 276, 55, 86, 104, 228195, 5, 25597845, 264, 220, 181, 24403740, 70, 226, 234, 199250835, 215, 358293, 13494274080, 49, 70, 14753835, 685, 35, 154, 60, 7307904366, 1, 570, 21792528, 154, 216, 145, 770, 228, 236

Offset: 1

Amiram Eldar, Sep 07 2018

nonn

Product_{i} (k*d(i)*m + 1) is a Carmichael number.
Chernick proved that (6m + 1)*(12m + 1)*(18m + 1) is a Carmichael number, if all the 3 factors are prime (A033502, A046025).
Lieuwens generalized it to Product_{i} (k*d(i)*m + 1), for k a perfect number (A000396), e.g., A067199 for k = 28.
Rotkiewicz generalized it to any number k with a subset of its proper divisors that sums to k.
The corresponding generated Carmichael numbers are 1729, 393575432565765601, 599966117492747584686619009, 17167430884969, 11744090279809908081796578516491199598397832961, 3680409480386689, 617027029751094776871101828064081267143041, 7622722964881, 700705956080852569, 90694625332467786841, 24182595473200959889, 553229304821570521, 3915654940974324169, 9215447790472998049, 4890416189580986381506017143209122707839833885365268481, 1746281192537521, ...
Supersequence of A319008.

			20 = 1 + 4 + 5 + 10 is the sum of a single subset of the proper divisors of 20. 333 is the least number such that 20*1*333 + 1 = 6661, 20*4*333 + 1 = 26641, 20*5*333 + 1 = 33301, and 20*10*333 + 1 = 66601 are all primes, therefore 6661*26641*33301*66601 = 393575432565765601 is a Carmichael number.

Jack Chernick, On Fermat's simple theorem, Bulletin of the American Mathematical Society, Vol. 45, No. 4 (1939), pp. 269-274.
Bill Daly, Perfect numbers and Carmichael numbers - a hidden relation, transcription of a network conversation at David Eppstein's Egyptian Fractions web site.
Erik Lieuwens, Fermat pseudo primes, Doctoral Thesis, Delft University of Technology, 1971, pp. 29-30.
Andrzej Rotkiewicz, On some problems of W. Sierpinski, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.

Cf. A000396, A002997, A033502, A046025, A064771, A067199, A319008.

okQ[n_] := Module[{d = Most[Divisors[n]]}, SeriesCoefficient[Series[Product[1 + x^i, {i, d}], {x, 0, n}], n] == 1]; s = Select[Range[300], okQ]; divSubset[n_] := Module[{d = Most[Divisors[n]]}, divSets = Subsets[d]; ns = Length[divSets];
  Do[divs = divSets[[k]]; If[Total[divs] == n, Break[]], {k, 1, ns}]; divs]; leastMultiplier[n_] := Module[{divs = divSubset[n]}, m = 1;
  While[! AllTrue[n*m*divs + 1, PrimeQ], m++]; m]; seq = {}; Do[s1 = s[[k]]; m = leastMultiplier[s1]; AppendTo[seq, m], {k, 1, Length[s]}]; seq (* after Harvey P. Dale at A064771 *)

cp's OEIS Frontend

A255441 Carmichael numbers of the form (60k+41)*(90k+61)*(150k+101), where 60k+41, 90k+61 and 150k+101 are all primes.

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A290811 Numbers n such that (6n-1, 6n+1), (12n-1, 12n+1) and (18n-1, 18n+1) are 3 pairs of twin primes.

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A317126 Numbers of the form: (6*m + 1) * (12*m + 1) * Product_{i=1..k-2} (9 * 2^i * m + 1), where k >= 3, with the condition that each of the factors is prime and that m is divisible by 2^(k-4).

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A382809 a(n) = (6*n + 1)*(12*n + 1)*(18*n + 1).

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A319008 Let k = A000396(n) be the n-th perfect number, a(n) is the least number m such that k*d*m + 1 is prime for all of the proper divisors d of k so their product is a Carmichael number.

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A343979 Composite numbers m such that lambda(m) = lambda(D_{m-1}), where lambda(n) is the Carmichael function of n (A002322) and D_k is the denominator (A027642) of Bernoulli number B_k.

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A379656 Carmichael numbers that are the sum of 2 positive cubes.

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A319011 Let k = A064771(n) be the n-th pseudoperfect number such that {d(i)} is a unique subset of its proper divisors that sums to k, a(n) is the least number m such that k*d(i)*m + 1 is prime for all d(i) in this subset so their product is a Carmichael number.

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A255441 Carmichael numbers of the form (60k+41)(90k+61)(150k+101), where 60k+41, 90k+61 and 150k+101 are all primes.

A317126 Numbers of the form: (6m + 1) (12m + 1) Product_{i=1..k-2} (9 * 2^i * m + 1), where k >= 3, with the condition that each of the factors is prime and that m is divisible by 2^(k-4).

A382809 a(n) = (6n + 1)(12n + 1)(18*n + 1).

A319008 Let k = A000396(n) be the n-th perfect number, a(n) is the least number m such that kdm + 1 is prime for all of the proper divisors d of k so their product is a Carmichael number.

A319011 Let k = A064771(n) be the n-th pseudoperfect number such that {d(i)} is a unique subset of its proper divisors that sums to k, a(n) is the least number m such that kd(i)m + 1 is prime for all d(i) in this subset so their product is a Carmichael number.