A162290 -id:A162290 - cp's OEIS frontend

A164946 The value A104017(n) rescaled with its decremented prime factors as described in A162290.

295788, 1003244, 2419212, 20140245, 10178892, 35839470, 24413481, 32157228, 295702416, 95828168, 107785924, 353180006543727, 320682950, 457591752, 909143104, 78888524, 735661013336064, 193098816, 26308112, 215405768, 1125114110156250, 3418986808281250, 236301822947449, 269517889, 287152344, 157098832

Offset: 1

A.K. Devaraj, Sep 02 2009

nonn

The entry N = A104017(n) is written as the product N = p*q*r*... of its k distinct prime factors p < q < r < ... . The (k-2)nd power multiplied by the decremented p-1 of the smallest prime factor and divided by the product of the decremented other prime factors defines a(n): a(n) = (N-1)^(k-2)*(p-1)/( (q-1)*(r-1)*...). - R. J. Mathar, Dec 16 2010

			The 4 factors of the first member of A104017 (11305) are 5, 7, 17 and 19. Hence the first term of the present sequence is (4*11304^2)/(6*16*18) = 295788.

Cf. A104016, A104017, A162290.

A087788 3-Carmichael numbers: Carmichael numbers equal to the product of 3 primes: k = pqr, where p < q < r are primes such that a^(k-1) == 1 (mod k) if a is prime to k.

Original entry on oeis.org

561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, 46657, 52633, 115921, 162401, 252601, 294409, 314821, 334153, 399001, 410041, 488881, 512461, 530881, 1024651, 1152271, 1193221, 1461241, 1615681, 1857241, 1909001, 2508013

Offset: 1

Miklos Kristof, Oct 07 2003

It is interesting that most of the numbers have the last digit 1. For example 530881, 3581761, 7207201, etc.
Granville & Pomerance conjecture that there are ~ c x^(1/3)/(log x)^3 terms of this sequence up to x. Heath-Brown proves that, for any e > 0, there are O(x^(7/20 + e)) terms of this sequence up to x. - Charles R Greathouse IV, Nov 19 2012
All 3-term Carmichael numbers can be represented by certain Chernick polynomials, whose values obey a strict s-decomposition (A324460) besides certain exceptions. Under Dickson's conjecture, "almost all" 3-term Carmichael numbers are primary Carmichael numbers (A324316) in the sense that C'3(x)/C_3(x) -> 1 as x -> infinity, where C_3(x) counts the 3-term Carmichael numbers and C'_3(x) counts the 3-term primary Carmichael numbers up to x. A primary Carmichael number m has the unique property (*) that for each prime divisor p of m, the sum of the base-p digits of m equals p. As a nontrivial result, all 3-term Carmichael numbers m have at least the property that (*) holds for the greatest prime divisor p of m, see Kellner 2019. - _Bernd C. Kellner, Aug 03 2022

			a(6)=6601=7*23*41: 7-1|6601-1, 23-1|6601-1, 41-1|6601-1, i.e., 6|6600, 22|6600, 40|6600.

O. Ore, Number Theory and Its History, McGraw-Hill, 1948, Reprinted by Dover Publications, 1988, Chapter 14.

R. J. Mathar and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 3284 terms from Mathar)
F. Arnault, Constructing Carmichael numbers which are strong pseudoprimes to several bases, Journal of Symbolic Computation, vol. 20, no 2, Aug. 1995, pp. 151-161.
Jack Chernick, On Fermat's simple theorem, Bull. Amer. Math. Soc., Vol. 45, No. 4 (1939), pp. 269-274.
Harvey Dubner, Carmichael Numbers of the form (6m+1)(12m+1)(18m+1), Journal of Integer Sequences, Vol. 5 (2002) Article 02.2.1,
A. Granville and C. Pomerance, Two contradictory conjectures concerning Carmichael numbers, Math. Comp. 71 (2002), pp. 883-90.
D. R. Heath-Brown, Carmichael numbers with three prime factors, Hardy-Ramanujan Journal 30 (2007), pp. 6-12.
G. Jaeschke, The Carmichael numbers to 10^12, Math. Comp., 55 (1990), 383-389.
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.
Math Reference Project, Carmichael Numbers
R. G. E. Pinch, The Carmichael numbers up to 10^18, 2008.
Rosetta Code, Programs for finding 3-Carmichael numbers

Intersection of A002997 and A007304.

Cf. A162290.

list(lim)=my(v=List());forprime(p=3,(lim)^(1/3), forprime(q=p+1, sqrt(lim\p),forprime(r=q+1,lim\(p*q),if((q*r-1)%(p-1)||(p*r-1)%(q-1)||(p*q-1)%(r-1),,listput(v,p*q*r)))));vecsort(Vec(v)) \\ Charles R Greathouse IV, Nov 19 2012

k is composite and squarefree and for p prime, p|k => p-1|k-1. A composite odd number k is a Carmichael number if and only if k is squarefree and p-1 divides k-1 for every prime p dividing k (Korselt, 1899) k = p*q*r, p-1|k-1, q-1|k-1, r-1|k-1.

Minor edit to definition by N. J. A. Sloane, Sep 14 2009

A165139 Multiplicative order of 2 in Z/mZ with m=A104017(n).

Original entry on oeis.org

72, 264, 648, 600, 56, 720, 224, 2904, 600, 288, 648, 792, 4560, 180, 840, 792, 360, 1500, 1700, 324, 720, 360, 5152, 900, 576, 828, 4356, 26460, 19800, 144, 972, 700, 21780, 1152, 78408, 7128, 960, 540, 8064, 7968, 139080, 1620, 1296, 71148, 1960, 6624, 2280, 8820, 4680, 144, 495, 19800, 2016, 2592, 4356, 468, 1320, 3204, 2880

Offset: 1

A.K. Devaraj, Sep 05 2009

nonn

Cf. A104017, A162290, A163956.

Corrected and extended by M. F. Hasler, Sep 23 2009

Edited by N. J. A. Sloane, Sep 23 2009, following suggestions from M. F. Hasler

A180044 Let the n-th Carmichael number A002997(n) = p1p2...pr, where p1 < p2 < ... < pr are primes. Then a(n) = (p1-1) (p1p2...pr - 1)^(r-2) / ((p2-1)...*(pr-1)).

Original entry on oeis.org

7, 23, 48, 22, 47, 45, 45, 21, 44, 163, 2105352, 162, 43, 486266, 3157729, 9859600, 5110605, 161, 6146018, 280, 8225424, 9135075, 1684, 6185169, 1363, 159, 351, 59907600, 950, 1675, 9879408, 1358, 949, 158, 95468562, 4399220, 83722500

Offset: 1

A.K. Devaraj, Aug 08 2010

nonn

a(n) is always an integer as proved at the Alekseyev link.
The conjecture referred to in A162290 was generalized as follows: Let k be an r-factor Carmichael number (p_1 < p_2 < ... < p_r). Then (p_1-1)*(k-1)^(r-2)/((p_2-1)*(p_3-1)*...*(p_r-1)) is an integer. This was proved by Max Alekseyev (see link).
Contains A162290 as a subsequence.

			Since A002997(11) = 41041 = 7*11*13*41, we have a(11) = (6*41040^2) / (10*12*40) = 2105352.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, Pomerance's proof.

Cf. A104016, A104017, A002997, A162290.

[ (d[1]-1)*(n-1)^(r-2) / &*[ d[i]-1: i in [2..r] ]: n in [3..700000 by 2] | not IsPrime(n) and IsSquarefree(n) and forall(t){x: x in d | (n-1) mod (x-1) eq 0} where r is #d where d is PrimeDivisors(n)]; // Klaus Brockhaus, Aug 10 2010

lim = 1000001; CarmichaelQ[n_] := Divisible[n - 1, CarmichaelLambda[n]] && ! PrimeQ[n]; cc = Select[Table[k, {k, 561, lim, 2}], CarmichaelQ]; lg = Length[cc]; a[n_] := (c = cc[[n]]; pp = FactorInteger[c][[All, 1]]; r = Length[pp]; (pp[[1]] - 1)*((Times @@ pp - 1)^(r - 2)/ Times @@ (Drop[pp, 1] - 1))); Table[a[n], {n, 1, lg}] (* Jean-François Alcover, Sep 28 2011 *)

Edited and extended by Max Alekseyev and Klaus Brockhaus, Aug 10 2010

cp's OEIS Frontend

A164946 The value A104017(n) rescaled with its decremented prime factors as described in A162290.

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A087788 3-Carmichael numbers: Carmichael numbers equal to the product of 3 primes: k = pqr, where p < q < r are primes such that a^(k-1) == 1 (mod k) if a is prime to k.

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A165139 Multiplicative order of 2 in Z/mZ with m=A104017(n).

Views

Author

Keywords

Crossrefs

Extensions

A180044 Let the n-th Carmichael number A002997(n) = p1p2...pr, where p1 < p2 < ... < pr are primes. Then a(n) = (p1-1) (p1p2...pr - 1)^(r-2) / ((p2-1)...*(pr-1)).

Views

Author

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Programs

Magma

Mathematica

Extensions

cp's OEIS Frontend

A164946 The value A104017(n) rescaled with its decremented prime factors as described in A162290.

Views

Author

Keywords

Comments

Examples

Crossrefs

A087788 3-Carmichael numbers: Carmichael numbers equal to the product of 3 primes: k = p*q*r, where p < q < r are primes such that a^(k-1) == 1 (mod k) if a is prime to k.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

Formula

Extensions

A165139 Multiplicative order of 2 in Z/mZ with m=A104017(n).

Views

Author

Keywords

Crossrefs

Extensions

A180044 Let the n-th Carmichael number A002997(n) = p1*p2*...*pr, where p1 < p2 < ... < pr are primes. Then a(n) = (p1-1) * (p1*p2*...*pr - 1)^(r-2) / ((p2-1)*...*(pr-1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Extensions

A087788 3-Carmichael numbers: Carmichael numbers equal to the product of 3 primes: k = pqr, where p < q < r are primes such that a^(k-1) == 1 (mod k) if a is prime to k.

A180044 Let the n-th Carmichael number A002997(n) = p1p2...pr, where p1 < p2 < ... < pr are primes. Then a(n) = (p1-1) (p1p2...pr - 1)^(r-2) / ((p2-1)...*(pr-1)).