A019320 -id:A019320 - cp's OEIS frontend

A063696 Positions of positive coefficients in cyclotomic polynomial Phi_n(x), converted from binary to decimal.

0, 2, 3, 7, 5, 31, 5, 127, 17, 73, 21, 2047, 17, 8191, 85, 297, 257, 131071, 65, 524287, 273, 4681, 1365, 8388607, 257, 1082401, 5461, 262657, 4369, 536870911, 387, 2147483647, 65537, 1198665, 87381, 17454241, 4097, 137438953471, 349525

Offset: 0

Antti Karttunen, Aug 03 2001

nonn

Maple procedures Phi_pos_terms and Phi_neg_terms are modeled after the formula given in Lam and Leung paper and they compute correct results for all integers x > 1 and for all n with at most two distinct odd prime factors (that is, up to n=104). Other procedures as in A063698 and A063694.

D. M. Bloom, On the Coefficients of the Cyclotomic Polynomials, Amer.Math.Monthly 75, 372-377, 1968.
T. Y. Lam and K. H. Leung, On the Cyclotomic Polynomial Phi_pq(X), Amer.Math.Monthly 103, 562-564, August-September 1996.
H. W. Lenstra, Vanishing sums of roots of unity, in Proc. Bicentennial Congress Wiskundig Genootschap (Vrije Univ. Amsterdam, 1978), Part II, pp. 249-268.
Index entries for cyclotomic polynomials, positions of coefficients

Cf. A013594, A063697 (binary version), A063698 (negative terms), A063670 (nonzero terms).

A019320(n) = a(n) - A063698(n) for up to n=104.

with(numtheory); [seq(Phi_pos_terms(j,2),j=0..104)];
inv_p_mod_q := (p,q) -> op(2,op(1,msolve(p*x=1,q))); # Find's p's inverse modulo q.
dilate := proc(nn,x,e) local n,i,s; n := nn; i := 0; s := 0; while(n > 0) do s := s + (((x^e)^i)*(n mod x)); n := floor(n/x); i := i+1; od; RETURN(s); end;
Phi_pos_terms := proc(n,x) local a,m,p,q,e,f,r,s; if(n < 2) then RETURN(x); fi; a := op(2, ifactors(n)); m := nops(a); p := a[1][1]; e := a[1][2]; if(1 = m) then RETURN(((x^(p^e))-1)/((x^(p^(e-1)))-1)); fi; if(2 = m) then q := a[2][1]; f := a[2][2]; r := inv_p_mod_q(p,q)-1; s := inv_p_mod_q(q,p)-1; RETURN( (`if`(0=s,1,(((x^((s+1)*((q^f)*(p^(e-1)))))-1)/((x^((q^f)*(p^(e-1))))-1)))) * (`if`(0=r,1,(((x^((r+1)*((p^e)*(q^(f-1)))))-1)/((x^((p^e)*(q^(f-1))))-1)))) ); fi; if((3 = m) and (2 = p)) then if(1 = e) then RETURN(every_other_pos(Phi_pos_terms(n/2,x),x,0)+every_other_pos(Phi_neg_terms(n/2,x),x,1)); else RETURN(dilate(Phi_pos_terms((n/(2^(e-1))),x),x,2^(e-1))); fi; else printf(`Cannot handle argument %a with three or more distinct odd prime factors!\n`,n); RETURN(0); fi; end;

a[n_] := 2^(Flatten[Position[CoefficientList[Cyclotomic[n, x], x], ?Positive]] - 1) // Total; a[0] = 0; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover, Mar 05 2016 *)

a(n)=local(p); if(n<1,0,p=polcyclo(n); sum(i=0,n,2^i*(polcoeff(p,i)>0)))

A019325 Cyclotomic polynomials at x=7.

Original entry on oeis.org

7, 6, 8, 57, 50, 2801, 43, 137257, 2402, 117993, 2101, 329554457, 2353, 16148168401, 102943, 4956001, 5764802, 38771752331201, 117307, 1899815864228857, 5649505, 11898664849, 247165843, 4561457890013486057, 5762401, 79797014141614001, 12111126301, 1628413638264057

Offset: 0

Simon Plouffe

Sequence has a(0) = x; see comments in A020501.

Robert Israel, Table of n, a(n) for n = 0..1186
Index entries for cyclotomic polynomials, values at X

Cf. A019320, A019321, A019322, A019323, A019324.

with(numtheory,cyclotomic); f := n->subs(x=7,cyclotomic(n,x)); seq(f(i),i=0..64);

Join[{7}, Cyclotomic[Range[50], 7]] (* Paolo Xausa, Feb 26 2024 *)

a(n) = if(n==0, 7, polcyclo(n, 7)); \\ Michel Marcus, Dec 16 2017

More terms from Michel Marcus, Dec 17 2017

A063703 Cyclotomic polynomials Phi_n at x=phi, floored down (where phi = tau = (sqrt(5)+1)/2).

Original entry on oeis.org

1, 0, 2, 5, 3, 16, 2, 45, 7, 23, 4, 320, 5, 841, 11, 25, 47, 5776, 14, 15125, 34, 166, 76, 103680, 41, 16626, 199, 5855, 233, 1860496, 56, 4870845, 2207, 7601, 1364, 45080, 305, 87403801, 3571, 51940, 1926, 599074576, 407, 1568397605, 10946, 80320

Offset: 0

Antti Karttunen, Aug 03 2001

nonn

Cf. A019320, A063705, A063707, A063704.

with(numtheory); Phi_at_x := (n,y) -> subs(x=y,cyclotomic(n,x)); [seq(floor(evalf(simplify(Phi_at_x(j,(sqrt(5)+1)/2)))),j=0..120)];

Floor[Simplify[Cyclotomic[Range[0, 50],GoldenRatio]]] (* Paolo Xausa, Feb 27 2024 *)

a(43) corrected by Sean A. Irvine, May 08 2023

A063705 Cyclotomic polynomials Phi_n at x=phi, rounded to nearest integer (where phi = tau = (sqrt(5)+1)/2).

Original entry on oeis.org

2, 1, 3, 5, 4, 16, 2, 45, 8, 23, 5, 320, 5, 841, 11, 26, 48, 5776, 15, 15125, 34, 167, 76, 103680, 41, 16626, 199, 5855, 233, 1860496, 56, 4870845, 2208, 7602, 1364, 45081, 305, 87403801, 3571, 51941, 1926, 599074576, 407, 1568397605, 10946, 80321

Offset: 0

Antti Karttunen, Aug 03 2001

nonn

Robert Israel, Table of n, a(n) for n = 0..2500
Index entries for cyclotomic polynomials, values at phi

Cf. A019320, A063703, A063707, A063706.

with(numtheory); Phi_at_x := (n,y) -> subs(x=y,cyclotomic(n,x)); [seq(round(evalf(simplify(Phi_at_x(j,(sqrt(5)+1)/2)))),j=0..120)];

Join[{2}, Round[Simplify[Cyclotomic[Range[50], GoldenRatio]]]] (* Paolo Xausa, Feb 27 2024 *)

a(43) corrected by Sean A. Irvine, May 08 2023

A063707 Cyclotomic polynomials Phi_n at x=phi, ceiled up (where phi = tau = (sqrt(5)+1)/2).

Original entry on oeis.org

2, 1, 3, 6, 4, 17, 2, 46, 8, 24, 5, 321, 6, 842, 12, 26, 48, 5777, 15, 15126, 35, 167, 77, 103681, 42, 16627, 200, 5856, 234, 1860497, 57, 4870846, 2208, 7602, 1365, 45081, 306, 87403802, 3572, 51941, 1927, 599074577, 408, 1568397606, 10947, 80321

Offset: 0

Antti Karttunen, Aug 03 2001

nonn

Cf. A019320, A063703, A063705, A063708.

with(numtheory); Phi_at_x := (n,y) -> subs(x=y,cyclotomic(n,x)); [seq(ceil(evalf(simplify(Phi_at_x(j,(sqrt(5)+1)/2)))),j=0..120)];

Join[{2}, Ceiling[Simplify[Cyclotomic[Range[50], GoldenRatio]]]] (* Paolo Xausa, Feb 27 2024 *)

A066845 a(n) = (lcm_{k=0..n} (2^k + 1))/(lcm_{k=0..n-1} (2^k + 1)).

Original entry on oeis.org

3, 5, 3, 17, 11, 13, 43, 257, 57, 205, 683, 241, 2731, 3277, 331, 65537, 43691, 4033, 174763, 61681, 5419, 838861, 2796203, 65281, 1016801, 13421773, 261633, 15790321, 178956971, 80581, 715827883, 4294967297, 1397419, 3435973837

Offset: 1

Vladeta Jovovic, Jan 20 2002

The primitive part of 2^n + 1. Bisection of A019320. - T. D. Noe, Jul 24 2008

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A051844, A034268, A019320.

Cf. A019320.

Cyclotomic[2*Range[40],2] (* Harvey P. Dale, Apr 19 2019 *)

a(n) = polcyclo(2*n, 2); \\ Michel Marcus, Mar 06 2015

a(n) = cyclotomic(2*n, 2). - Vladeta Jovovic, Apr 05 2004

A140797 Numbers of the form (2^p^N-1)/(2^p^(N-1)-1), where N>0, p is prime.

Original entry on oeis.org

3, 5, 7, 17, 31, 73, 127, 257, 2047, 8191, 65537, 131071, 262657, 524287, 1082401, 8388607, 536870911, 2147483647, 4294967297, 137438953471, 2199023255551, 4432676798593, 8796093022207, 140737488355327, 9007199254740991, 18014398643699713, 576460752303423487

Offset: 1

Vladimir Shevelev, Jul 15 2008

nonn

Contains Fermat numbers A000215 (p=2) and Mersenne numbers A001348 (N=1). The terms of the sequence are either primes A000040 or overpseudoprimes A141232.

The values of A019320(n) for prime power n, sorted. This sequence is a subsequence of A064896, which means that all terms are sturdy numbers (A125121). It appears that the largest prime factor of each of these numbers is a sturdy prime (A143027). - T. D. Noe, Jul 21 2008

T. D. Noe, Table of n, a(n) for n=1..199
Vladimir Shevelev, Process of "primoverization" of numbers of the form a^n-1, arXiv:0807.2332 [math.NT], 2008.

Cf. A000040, A000215, A001348, A141232.

nmax[p_] := Which[p == 2, 6, p == 3, 4, True, 2];
Reap[Do[If[IntegerQ[k = (2^p^n-1)/(2^p^(n-1)-1)] && k<10^18, Print[{p, n, k}]; Sow[k]], {p, Prime[Range[17]]}, {n, 1, nmax[p]}]][[2, 1]] // Union (* Jean-François Alcover, Dec 10 2018 *)

Definition corrected by and more terms from T. D. Noe, Jul 21 2008

A250197 Numbers k such that the left Aurifeuillian primitive part of 2^k+1 is prime.

Original entry on oeis.org

10, 14, 18, 22, 26, 30, 42, 54, 58, 66, 70, 86, 94, 98, 106, 110, 126, 130, 138, 146, 158, 174, 186, 210, 222, 226, 258, 302, 334, 434, 462, 478, 482, 522, 566, 602, 638, 706, 734, 750, 770, 782, 914, 1062, 1086, 1114, 1126, 1226, 1266, 1358, 1382, 1434, 1742, 1926

Offset: 1

Eric Chen, Jan 18 2015

nonn

All terms are congruent to 2 modulo 4.
Phi_n(x) is the n-th cyclotomic polynomial.
Numbers n such that Phi_{2nL(n)}(2) is prime.
Let J(n) = 2^n+1, J*(n) = the primitive part of 2^n+1, this is Phi_{2n}(2).
Let L(n) = the Aurifeuillian L-part of 2^n+1, L(n) = 2^(n/2) - 2^((n+2)/4) + 1 for n congruent to 2 (mod 4).
Let L*(n) = GCD(L(n), J*(n)).
This sequence lists all n such that L*(n) is prime.

			14 is in this sequence because the left Aurifeuillian primitive part of 2^14+1 is 113, which is prime.
34 is not in this sequence because the left Aurifeuillian primitive part of 2^34+1 is 130561, which equals 137 * 953 and is not prime.

Eric Chen, Gord Palameta, Factorization of Phi_n(2) for n up to 1280
Samuel Wagstaff, The Cunningham project
Eric W. Weisstein's World of Mathematics, Aurifeuillean Factorization.

Cf. A250198, A153443, A019320, A072226, A161508, A092440, A229767, A061442.

Select[Range[2000], Mod[#, 4] == 2 && PrimeQ[GCD[2^(#/2) - 2^((#+2)/4) + 1, Cyclotomic[2*#, 2]]] &]

isok(n) = isprime(gcd(2^(n/2) - 2^((n+2)/4) + 1, polcyclo(2*n, 2))); \\ Michel Marcus, Jan 27 2015

A253240 Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.

Original entry on oeis.org

1, 1, -1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 3, 3, 1, 1, 3, 4, 7, 2, 1, 1, 4, 5, 13, 5, 5, 1, 1, 5, 6, 21, 10, 31, 1, 1, 1, 6, 7, 31, 17, 121, 3, 7, 1, 1, 7, 8, 43, 26, 341, 7, 127, 2, 1, 1, 8, 9, 57, 37, 781, 13, 1093, 17, 3, 1, 1, 9, 10, 73, 50, 1555, 21, 5461, 82, 73, 1, 1, 1, 10, 11, 91, 65, 2801, 31, 19531, 257, 757, 11, 11, 1, 1, 11, 12, 111, 82, 4681, 43, 55987, 626, 4161, 61, 2047, 1, 1

Offset: 0

Eric Chen, Apr 22 2015

Outside of rows 0, 1, 2 and columns 0, 1, only terms of A206942 occur.
Conjecture: There are infinitely many primes in every row (except row 0) and every column (except column 0), the indices of the first prime in n-th row and n-th column are listed in A117544 and A117545. (See A206864 for all the primes apart from row 0, 1, 2 and column 0, 1.)
Another conjecture: Except row 0, 1, 2 and column 0, 1, the only perfect powers in this table are 121 (=Phi_5(3)) and 343 (=Phi_3(18)=Phi_6(19)).

			Read by antidiagonals:
m\n  0   1   2   3   4   5   6   7   8   9  10  11  12
------------------------------------------------------
0    1   1   1   1   1   1   1   1   1   1   1   1   1
1   -1   0   1   2   3   4   5   6   7   8   9  10  11
2    1   2   3   4   5   6   7   8   9  10  11  12  13
3    1   3   7  13  21  31  43  57  73  91 111 133 157
4    1   2   5  10  17  26  37  50  65  82 101 122 145
5    1   5  31 121 341 781 ... ... ... ... ... ... ...
6    1   1   3   7  13  21  31  43  57  73  91 111 133
etc.
The cyclotomic polynomials are:
n        n-th cyclotomic polynomial
0        1
1        x-1
2        x+1
3        x^2+x+1
4        x^2+1
5        x^4+x^3+x^2+x+1
6        x^2-x+1
...

Eric Chen, Table of n, a(n) for n = 0..5049 (first 100 antidiagonals)
Eric Weisstein's World of Mathematics, Cyclotomic polynomial
Wikipedia, Cyclotomic polynomial
Index entries for Cyclotomic polynomials, values at X

Rows 0-16 are A000012, A023443, A000027, A002061, A002522, A053699, A002061, A053716, A002523, A060883, A060884, A060885, A060886, A060887, A060888, A060889, A060890.
Columns 0-13 are A158388, A020500, A019320, A019321, A019322, A019323, A019324, A019325, A019326, A019327, A019328, A019329, A019330, A019331.
Main diagonal is A070518.
Indices of primes in n-th row for n = 1-20 are A008864, A006093, A002384, A005574, A049409, A055494, A100330, A000068, A153439, A246392, A162862, A246397, A217070, A250174, A250175, A006314, A217071, A164989, A217072, A250176.
Indices of primes in n-th column for n = 1-10 are A246655, A072226, A138933, A138934, A138935, A138936, A138937, A138938, A138939, A138940.
Indices of primes in main diagonal is A070519.
Cf. A117544 (indices of first prime in n-th row), A085398 (indices of first prime in n-th row apart from column 1), A117545 (indices of first prime in n-th column).
Cf. A206942 (all terms (sorted) for rows>2 and columns>1).
Cf. A206864 (all primes (sorted) for rows>2 and columns>1).

Table[Cyclotomic[m, k-m], {k, 0, 49}, {m, 0, k}]

t1(n)=n-binomial(floor(1/2+sqrt(2+2*n)), 2)
t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)
T(m, n) = if(m==0, 1, polcyclo(m, n))
a(n) = T(t1(n), t2(n))

T(m, n) = Phi_m(n)

A051844 a(n) = LCM_{k=0..n} (2^k + 1).

Original entry on oeis.org

2, 6, 30, 90, 1530, 16830, 218790, 9407970, 2417848290, 137817352530, 28252557268650, 19296496614487950, 4650455684091595950, 12700394473254148539450, 41619192688853844763777650, 13775952780010622616810402150, 902834617343556174437903325704550

Offset: 0

Jeffrey Shallit, Apr 20 2000

nonn

			a(3) = lcm(2, 3, 5) = 30.

Cf. A034268.

Cf. A019320.

Module[{nn=20,c},c=Table[2^n+1,{n,0,nn}];Table[LCM@@Take[c,n],{n,nn}]] (* Harvey P. Dale, Aug 04 2017 *)

a(n) = {ret = 1; for (k=0, n, ret = lcm(ret, 2^k+1)); return(ret);} \\ Michel Marcus, May 24 2013

from math import lcm
from itertools import accumulate
def aupton(nn): return list(accumulate((2**k+1 for k in range(nn+1)), lcm))
print(aupton(16)) # Michael S. Branicky, Jul 04 2022

a(n) = lcm(2, 3, 5, ..., 2^n + 1).

Product_{k=1..n} cyclotomic(2*k-2, 2). - Vladeta Jovovic, Apr 05 2004

More terms from Harvey P. Dale, Aug 04 2017

cp's OEIS Frontend

A063696 Positions of positive coefficients in cyclotomic polynomial Phi_n(x), converted from binary to decimal.

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A019325 Cyclotomic polynomials at x=7.

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A063703 Cyclotomic polynomials Phi_n at x=phi, floored down (where phi = tau = (sqrt(5)+1)/2).

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A063705 Cyclotomic polynomials Phi_n at x=phi, rounded to nearest integer (where phi = tau = (sqrt(5)+1)/2).

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A063707 Cyclotomic polynomials Phi_n at x=phi, ceiled up (where phi = tau = (sqrt(5)+1)/2).

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A066845 a(n) = (lcm_{k=0..n} (2^k + 1))/(lcm_{k=0..n-1} (2^k + 1)).

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A140797 Numbers of the form (2^p^N-1)/(2^p^(N-1)-1), where N>0, p is prime.

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A250197 Numbers k such that the left Aurifeuillian primitive part of 2^k+1 is prime.

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