cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A104017 Devaraj numbers (A104016) which are not Carmichael numbers.

Original entry on oeis.org

11305, 39865, 96985, 401401, 464185, 786961, 1106785, 1296505, 1719601, 1993537, 2242513, 2615977, 2649361, 2722681, 3165961, 3181465, 3755521, 4168801, 4229601, 4483297, 4698001, 5034601, 5381265, 5910121, 5977153, 7177105
Offset: 1

Views

Author

Max Alekseyev, Feb 25 2005

Keywords

Comments

Counterexamples to Devaraj's 2nd conjecture: A.K. Devaraj conjectured that these numbers are exactly Carmichael numbers. It was proved (see A104016 ) that every Carmichael number is indeed a Devaraj number, but the converse is not true. Devaraj numbers that are not Carmichael are listed here.
It is sufficient to scan only odd numbers (cf. A104016), which makes the computation of the list twice as fast. - M. F. Hasler, Apr 03 2009

Links

Crossrefs

Cf. A104016, A002997.

Programs

  • PARI
    DNC() = for(n=2,10^8, f=factorint(n); if(vecmax(f[,2])>1,next); f=f[,1]; r=length(f); if(r==1,next); Carmichael=1; d=f[1]-1; p=1; for(i=1,r, d=gcd(d,f[i]-1); p*=f[i]-1; if((n-1)%(f[i]-1),Carmichael=0)); if( ((n-1)^(r-2)*d^2)%p==0 && !Carmichael, print1(" ",n)) )
  • PARI
    forstep( n=3, 10^7, 2, vecmax((f=factor(n))[,2])>1 && next; #(f*=[1,-1]~)>1 || next; gcd(f)^2*(n-1)^(#f-2) % prod(i=1,#f,f[i]) && next; for( i=1,#f, (n-1)%f[i] && !print1(n",") && break)) \\ M. F. Hasler, Apr 03 2009
  • PARI
    Korselt(n,p)=for(i=1, #p, if((n-1)%(p[i]-1), return(0))); 1
Devaraj(n,p)=my(u=apply(q->q-1,p)); gcd(u)^2*(n-1)^(#p-2)%vecprod(u)==0
list(lim)=my(v=List()); forsquarefree(N=11305,lim\=1, my(p=N[2][,1],n=N[1]); if(p[1]>2 && #p>2 && Devaraj(n,p) && !Korselt(n,p), listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Mar 09 2023

A163956 Multiplicative order of 2 in Z/mZ with m = A002997(n).

Original entry on oeis.org

40, 24, 36, 56, 60, 660, 198, 252, 45, 180, 60, 144, 153, 1012, 36, 120, 300, 72, 36, 1160, 60, 36, 300, 56, 36, 660, 4284, 264, 420, 3060, 2268, 180, 540, 1680, 120, 4900, 1080, 396, 72, 72, 60, 60, 144, 2970, 612, 396, 324, 210, 180, 540, 504, 792, 198, 180
Offset: 1

Views

Author

A.K. Devaraj, Aug 28 2009

Keywords

Comments

Related sequence: A162990. - A.K. Devaraj, Aug 31 2009

References

  • A. K. Devaraj, "Minimum universal exponent generalisation of Fermat's theorem" (ISSN 1550-3747)

Links

Crossrefs

Cf. A002326, A002997, A007733, A104016, A162990, A165139.

Programs

  • Mathematica
    MultiplicativeOrder[2, #] & /@ Select[Range[1, 10^6, 2], CompositeQ[#] && Divisible[# - 1, CarmichaelLambda[#]] &] (* Amiram Eldar, Jul 30 2020 *)

Formula

a(n) = A002326((A002997(n)-1)/2) = A007733(A002997(n)). - Amiram Eldar, Jul 30 2020

Extensions

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
More terms from Amiram Eldar, Jul 30 2020

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)).

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

Views

Author

A.K. Devaraj, Aug 08 2010

Keywords

Comments

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.

Examples

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

Links

Crossrefs

Cf. A104016, A104017, A002997, A162290.

Programs

  • Magma
    [ (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
  • Mathematica
    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 *)

Extensions

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

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

Original entry on oeis.org

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

Views

Author

A.K. Devaraj, Sep 02 2009

Keywords

Comments

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

Examples

			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.

Crossrefs

Cf. A104016, A104017, A162290.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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