A.K. Devaraj - cp's OEIS frontend

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)).

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

A180225 Primes of the form 3^m-2 where m are the terms of A123239.

Original entry on oeis.org

7, 450283905890997361, 36472996377170786401

Offset: 1

A.K. Devaraj, Aug 17 2010

nonn

Subsequence of A086214. - R. J. Mathar, Aug 18 2010

The next term has 259 digits and is too large to display.

Amiram Eldar, Table of n, a(n) for n = 1..4

Cf. A086214, A123239.

Erroneous 4th term deleted and offset corrected by R. J. Mathar, Aug 18 2010

Edited by N. J. A. Sloane, Aug 18 2010

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

A162290 Let A087788(n) = pqr, where p

Original entry on oeis.org

7, 23, 48, 22, 47, 45, 45, 21, 44, 163, 162, 43, 161, 280, 1684, 1363, 159, 351, 950, 1675, 1358, 949, 158, 345, 1829, 947, 1353, 510, 938, 1660, 2796, 1820, 820, 10208, 2779, 935, 1650, 817, 937, 1822

Offset: 1

A.K. Devaraj, Jul 01 2009

nonn

A.K. Devaraj conjectured that a(n) is always an integer, and this was proved by Carl Pomerance.
a(n) may be called the Pomerance index of the n-th 3-Carmichael number.
An application of Pomerance index: The index for the Carmichael number 561 is 7. This can be used to prove that 561 is the only 3-factor Carmichael number with 3 as one of its factors. Proof: Let N be a 3-factor composite number. Keep 3 fixed and increase the other two prime factors indefinitely. The relevant Pomerance index is a number less than 7 but greater than 6. As the other two prime factors are increased indefinitely the Pomerance index becomes asymptotic to 6. Hence 561 is the only 3-factor Carmichael number with 3 as a factor. - A.K. Devaraj, Jul 27 2010
Let p be a prime number. Then, along the lines indicated above, it can be proved that there are only a finite number of 3-Carmichael numbers divisible by p. - A.K. Devaraj, Aug 06 2010

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A002997, A087788, A162990.

do(lim)=my(v=List()); forprime(p=3, sqrtnint(lim\=1,3), forprime(q=p+1, sqrtint(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,(p*q*r-1)*(p-1)/(q-1)/(r-1)]))))); v=vecsort(v,1); vector(#v,i,v[i][2]) \\ Charles R Greathouse IV, Sep 07 2016

Edited by N. J. A. Sloane, Sep 14 2009, based on email messages from David Broadhurst and M. F. Hasler, Jul 10 2009

Spelling corrected by Jason G. Wurtzel, Aug 23 2010

A168235 1+5n+7n^2.

Original entry on oeis.org

13, 39, 79, 133, 201, 283, 379, 489, 613, 751, 903, 1069, 1249, 1443, 1651, 1873, 2109, 2359, 2623, 2901, 3193, 3499, 3819, 4153, 4501, 4863, 5239, 5629, 6033, 6451, 6883, 7329, 7789, 8263, 8751, 9253, 9769, 10299, 10843, 11401, 11973, 12559, 13159, 13773

Offset: 1

A.K. Devaraj, Nov 21 2009

Consider the quadratic cyclotomic polynomial f(x) = x^2+x+1 and the quotients defined by f(x + n*f(x))/f(x). a(n) is the quotient at x=2.

See A168240 for x=3 or A168244 for x= 1+sqrt(-5).

			When x = 2, f(x) = 7. Hence at n=1, f( x + f(x))/f(x) = 13 = a(1).

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A165806, A165808, A165809.

Table[1+5n+7n^2,{n,60}] (* or *) LinearRecurrence[{3,-3,1},{13,39,79},60] (* Harvey P. Dale, Feb 07 2015 *)

a(n)=1+5*n+7*n^2 \\ Charles R Greathouse IV, Jun 17 2017

a(1)=13, a(2)=39, a(3)=79, a(n)=3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Feb 07 2015
From G. C. Greubel, Apr 09 2016: (Start)
G.f.: (1 + 10*x + 3*x^2)/(1-x)^3.
E.g.f.: (1 + 12*x + 7*x^2)*exp(x). (End)

Edited, definition simplified, sequence extended beyond a(8) by R. J. Mathar, Nov 23 2009

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

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.

A173838 A sequence of coefficients of 3^n, when x = x_oj.

Original entry on oeis.org

1, 2, 1, 38, 43, 242, 67, 1238, 1289, 2278, 1657, 14662, 68321

Offset: 1

A.K. Devaraj, Feb 26 2010

Appendix to " A theorem a la Ramanujan " - Joint meeting of AMS - BENELUX, '96.

Cf. A173252.

A173252 A sequence of coefficients of 3^n when x_n = x_oi.

Original entry on oeis.org

2, 1, 2, 19, 2, 289, 118, 41, 182578, 239803, 495074, 3866257, 1158454, 2629057, 56207062, 82084427, 4638842098, 5389722857, 30867186934, 8585039713, 5319558074, 2, 193589999521, 616960854422, 5663407855939, 5264528838038

Offset: 1

A.K. Devaraj, Feb 14 2010

The paper is not yet published - it can be furnished on request.
Sequence of x_n: 0, 3, 12, 39, 66, 795, 1524, 8085, 539526, 1070967, 2665290, ...
The x_n are given by recurrence x_(n+1) = x_n + 3^(s_n - 1), where s_n is the exponent of the highest power of 3 in v_n = x_n^2 + 18, and the a(n) are equal to v_n / 3^s_n.

A. K. Devaraj, A theorem a la Ramanujan, Joint Meeting of AMS-BENELUX, '96.

lista(nn) = {x = 0; for (i=1, nn, y = x^2 + 18; s = valuation(y, 3); f = z^2 + 18; fx = subst(f, z, x); p3 = valuation (fx, 3); print1(fx/3^p3, ", "); x += 3^(s-1););} \\ Michel Marcus, Aug 08 2013

More terms from Michel Marcus, Aug 08 2013

A168325 Terms of A123239 which are also primes in Z(i).

Original entry on oeis.org

3, 11, 59, 67, 83, 103, 107, 131, 151, 179, 227, 251, 271, 307, 347, 367, 419, 443, 467, 491, 523, 547, 563, 587, 619, 659, 683, 787, 827, 883, 947, 967, 971, 991, 1019, 1051, 1091, 1163, 1171, 1187, 1223, 1259, 1283, 1303, 1307, 1427, 1451, 1471

Offset: 1

A.K. Devaraj, Nov 23 2009

nonn

Cf. A002145, A123239.

MangammalQ[p_] := Block[{k = 3}, While[k > 2, k = Mod[3 k, p]]; k != 2];
A168325 = Select[Prime[Range[235]], MangammalQ[#] && Mod[#, 4] == 3 &] (* Ray Chandler, Jul 21 2011 *)

Extended by Ray Chandler, Jul 21 2011

A168367 Terms of A123239 which are 2 mod 3.

Original entry on oeis.org

2, 11, 41, 59, 83, 107, 131, 179, 227, 251, 347, 419, 443, 467, 491, 563, 587, 659, 683, 761, 827, 947, 971, 1019, 1091, 1163, 1181, 1187, 1223, 1259, 1283, 1289, 1307, 1427, 1451, 1499, 1511, 1523, 1571, 1583, 1619, 1667, 1787, 1811, 1871, 1877

Offset: 1

A.K. Devaraj, Nov 23 2009, Nov 25 2009

nonn

Terms of A123239 which are also primes in Z(w) where w = (-1+sqrt(-3))/2 is a cube root of unity.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A003627, A123239, A168325.

MangammalQ[p_] := Block[{k = 3}, While[k > 2, k = Mod[3 k, p]]; k != 2];
A168367 = Select[Prime[Range[290]], MangammalQ[#] && Mod[#, 3] == 2 &] (* Ray Chandler, Jul 21 2011 *)

is(n)=if(n<9, return(n==2)); if(n%6!=5 || !isprime(n) || znorder(Mod(3, n)) == n-1, return(0)); my(m=Mod(3, n)); while(m!=1, m*=3; if(m==2, return(0))); 1 \\ Charles R Greathouse IV, May 21 2017

Extended by Ray Chandler, Jul 21 2011

Definition simplified by Charles R Greathouse IV, May 21 2017

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User: A.K. Devaraj

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)).

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A180225 Primes of the form 3^m-2 where m are the terms of A123239.

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

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A162290 Let A087788(n) = p*q*r, where p

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A168235 1+5*n+7*n^2.

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

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A173838 A sequence of coefficients of 3^n, when x = x_oj.

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A173252 A sequence of coefficients of 3^n when x_n = x_oi.

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Author

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Programs

PARI

Extensions

A168325 Terms of A123239 which are also primes in Z(i).

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Author

Keywords

Crossrefs

Programs

Mathematica

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)).

A162290 Let A087788(n) = pqr, where p

A168235 1+5n+7n^2.