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.

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

Views

Author

A.K. Devaraj, Sep 05 2009

Keywords

Crossrefs

Cf. A104017, A162290, A163956.

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

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.

A104016 Devaraj numbers: squarefree r-prime-factor (r>1) integers N=p1*...*pr such that phi(N)=(p1-1)*...*(pr-1) divides gcd(p1-1,...,pr-1)^2*(N-1)^(r-2).

Original entry on oeis.org

561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 11305, 15841, 29341, 39865, 41041, 46657, 52633, 62745, 63973, 75361, 96985, 101101, 115921, 126217, 162401, 172081, 188461, 252601, 278545, 294409, 314821, 334153, 340561, 399001, 401401, 410041
Offset: 1

Views

Author

Max Alekseyev, Feb 25 2005

Keywords

Comments

A.K. Devaraj conjectured that these numbers are exactly Carmichael numbers. It was proved (see Alekseyev link) that every Carmichael number is indeed a Devaraj number, but the converse is not true. Devaraj numbers that are not Carmichael are given by A104017.
These numbers can't be even, since phi(N) is always even (N>2) but p1=2 implies that gcd{pi-1}=1 and N-1 is odd. - M. F. Hasler, Apr 03 2009

Links

Crossrefs

Subsequence of A350352 and hence of A033942.
Cf. A104017, A002997.

Programs

  • PARI
    Devaraj() = for(n=2,10^8, f=factorint(n); if(vecmax(f[,2])>1,next); f=f[,1]; r=length(f); if(r==1,next); d=f[1]-1; p=f[1]-1; for(i=2,r,d=gcd(d,f[i]-1); p*=f[i]-1); if( ((n-1)^(r-2)*d^2)%p==0, print1(" ",n)) )
  • PARI
    isA104016(n)= local(f=factor(n)); vecmax(f[,2])==1 && #(f*=[1,-1]~)>1 && gcd(f)^2*(n-1)^(#f-2)%prod(i=1,#f,f[i])==0
/* To print the list: */ forstep( n=3, 10^6, 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]) || print1(n","))
/* The following version could be efficient for large omega(n) */
isA104016(n) = issquarefree(n) && !isprime(n) && Mod(n-1,prod(i=1,#n=factor(n)*[1,-1]~,n[i]))^(#n-2)*gcd(n)^2==0 \\ M. F. Hasler, Apr 03 2009

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
Showing 1-4 of 4 results.

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