A051664 -id:A051664 - cp's OEIS frontend

A087073 Mobius transform of A051664, the number of nonzero terms in the n-th cyclotomic polynomial.

2, 0, 1, 0, 3, 0, 5, 0, 0, 0, 9, 0, 11, 0, 1, 0, 15, 0, 17, 0, 1, 0, 21, 0, 0, 0, 0, 0, 27, 0, 29, 0, 3, 0, 7, 0, 35, 0, 3, 0, 39, 0, 41, 0, 0, 0, 45, 0, 0, 0, 5, 0, 51, 0, 3, 0, 5, 0, 57, 0, 59, 0, 0, 0, 15, 0, 65, 0, 7, 0, 69, 0, 71, 0, 0, 0, 15, 0, 77, 0, 0, 0, 81, 0, 21, 0, 9, 0, 87, 0, 5, 0, 9, 0, 9

Offset: 1

T. D. Noe, Aug 08 2003

nonn

Note that a(n) = 0 for even n and a(n) = n-2 for prime n. It appears that the following are true: a(1) is the only positive even term, all odd numbers eventually appear in the sequence, a(n) = 0 if n is squareful and a(n) > 0 if n is squarefree. Assuming the truth of the last statement, we can show that if distinct odd primes p and q divide n, then A051664(n) > p + q - 2.

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

Cf. A051664.

a(n) = Sum{d|n} mu(n/d) A051664(d)

Definition corrected: "inverse moebius transform" to "moebius transform" by Wouter Meeussen, Jan 17 2009

A070776 Numbers k such that number of terms in the k-th cyclotomic polynomial is equal to the largest prime factor of k.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 40, 41, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 56, 58, 59, 61, 62, 64, 67, 68, 71, 72, 73, 74, 76, 79, 80, 81, 82, 83, 86, 88, 89, 92, 94, 96, 97, 98, 100

Offset: 1

Labos Elemer, May 07 2002

Numbers k such that A051664(k) = A006530(k).
This is also numbers in the form of 2^i*p^j, i >= 0 and j >= 0, p is an odd prime number. - Lei Zhou, Feb 18 2012
From Zhou's formulation (where the exponents i and j should actually have been specified as i > 0 OR j > 0, to exclude 1) it follows that this is a subsequence of A324109. It also follows that A005940(a(n)) = A324106(a(n)) for all n >= 1. - Antti Karttunen, Feb 15 2019
Also from Zhou's formulation, the union (disjoint) of A000079\{1} and A336101. - Peter Munn, Jul 16 2020
Numbers k>=2 such that A078701(k) = A299766(k). - Juri-Stepan Gerasimov, Jun 02 2021

			n=10: Cyclotomic[10,x]=1-x+x^2-x^3+x^4 with 5 terms [including 1] which equals largest prime factor (5) of 10=n.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Positions of zeros in A070536.
Cf. A005940, A006530, A051664, A061345, A070537 (complement), A324106, A324111.
Subsequence of A324109.
Subsequences: A000079\{1}, A336101.

Select[Range[1000],(a=FactorInteger[#];b=Length[a];(b==1)||((b==2)&&(a[[1]][[1]]==2)))&] (* Lei Zhou, Feb 18 2012 *)

A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1); \\ From A006530.
A051664(n) = length(select(x->x!=0, Vec(polcyclo(n)))); \\ After program in A051664
A070536(n) = (A051664(n) - A006530(n));
isA070776(n) = (!A070536(n)); \\ Antti Karttunen, Feb 15 2019
k=0; n=0; while(k<10000, n++; if(isA070776(n), k++; write("b070776.txt", k, " ", n)));

A086780 Number of negative terms in n-th cyclotomic polynomial.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 3, 3, 0, 0, 1, 0, 2, 4, 5, 0, 1, 0, 6, 0, 3, 0, 3, 0, 0, 7, 8, 8, 1, 0, 9, 8, 2, 0, 4, 0, 5, 3, 11, 0, 1, 0, 2, 11, 6, 0, 1, 8, 3, 12, 14, 0, 3, 0, 15, 4, 0, 15, 7, 0, 8, 15, 8, 0, 1, 0, 18, 3, 9, 15, 8, 0, 2, 0, 20, 0, 4, 20, 21, 19, 5, 0, 3, 11, 11, 20, 23, 15, 1

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 03 2003

nonn

See A051664

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

Cf. A086765, A000961.

Cf. A051664 (number of nonzero terms in n-th cyclotomic polynomial).

Table[Count[CoefficientList[Cyclotomic[n, x], x], _?(#<0&)], {n, 0, 100}]

a(n) = #select(x->(x<0), Vec(polcyclo(n))); \\ Michel Marcus, Apr 18 2018

a(n) = 0 iff n is a prime power. - T. D. Noe, Aug 08 2003

a(n) = (A051664(n)-1)/2 if n is not a prime power and has at most two distinct odd prime divisors. So 105 is the smallest n>1 where neither formula applies. - Aaron Meyerowitz, Apr 18 2018

More terms from T. D. Noe, Aug 08 2003

A070537 Numbers k such that the k-th cyclotomic polynomial has more terms than the largest prime factor of k.

Original entry on oeis.org

1, 15, 21, 30, 33, 35, 39, 42, 45, 51, 55, 57, 60, 63, 65, 66, 69, 70, 75, 77, 78, 84, 85, 87, 90, 91, 93, 95, 99, 102, 105, 110, 111, 114, 115, 117, 119, 120, 123, 126, 129, 130, 132, 133, 135, 138, 140, 141, 143, 145, 147, 150, 153, 154, 155, 156, 159, 161, 165

Offset: 1

Labos Elemer, May 03 2002

nonn

When (as at k=105) coefficients are not equal to 1 or -1, terms in C[k,x] are counted with multiplicity. This comment was left by the original author, but please see my comment in A070536. - Antti Karttunen, Feb 15 2019
Union of A324110 and A324111. - Antti Karttunen, Feb 15 2019
It appears that except for the initial 1, the terms are products of two or more distinct odd primes. - Enrique Navarrete, Oct 16 2022

			k=21: Cyclotomic[21,x] = 1 - x + x^3 - x^4 + x^6 - x^8 + x^9 - x^11 + x^12 has 9 terms while the largest prime factor of 21 is 7; 9 > 7, so 21 is in the sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A006530, A051664, A070536, A070776 (complement), A324110, A324111.

A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1); \\ From A006530.
A051664(n) = length(select(x->x!=0, Vec(polcyclo(n)))); \\ After program in A051664
isA070537(n) = (A051664(n) > A006530(n)); \\ Antti Karttunen, Feb 15 2019
for(n=1,165,if(isA070537(n), print1(n,", ")))

Numbers n satisfying A070536(n) = A051664(n) - A006530(n) > 0.

Edited by N. J. A. Sloane, Nov 30 2022

A086765 Number of positive coefficients in n-th cyclotomic polynomial.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 3, 3, 11, 2, 13, 4, 4, 2, 17, 2, 19, 3, 5, 6, 23, 2, 5, 7, 3, 4, 29, 4, 31, 2, 8, 9, 9, 2, 37, 10, 9, 3, 41, 5, 43, 6, 4, 12, 47, 2, 7, 3, 12, 7, 53, 2, 9, 4, 13, 15, 59, 4, 61, 16, 5, 2, 16, 8, 67, 9, 16, 9, 71, 2, 73, 19, 4, 10, 16, 9, 79, 3, 3, 21, 83, 5, 21, 22, 20, 6

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 02 2003

nonn

			The 1st cyclotomic polynomial is -1+1*x, which has 1 positive coefficient.
The 2nd cyclotomic polynomial is 1+1*x, which has 2 positive coefficients.
The 4th cyclotomic polynomial s 1+1*x^2, which has 2 positive coefficients.

See A051664

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

Cf. A007947.

Cf. A051664 (number of nonzero terms in n-th cyclotomic polynomial).

Table[Count[CoefficientList[Cyclotomic[n, x], x], _?(#>0&)], {n, 0, 100}]

If n = p^m is a prime power then a(n) = p.

More terms from T. D. Noe, Aug 08 2003

A070536 Number of terms in n-th cyclotomic polynomial minus largest prime factor of n; a(1)=1 by convention.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 4, 0, 10, 0, 0, 0, 4, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0, 6, 0, 6, 0, 0, 2, 0, 0, 2, 0, 18, 4, 0, 0, 8, 10, 0, 0, 0, 0, 2, 0, 20, 4, 0, 0, 0, 0, 0, 2, 24, 0, 10, 0, 0, 2, 10, 0, 10, 0, 12, 0, 0, 0, 4, 0, 0, 6, 0, 0, 26

Offset: 1

Labos Elemer, May 03 2002

nonn

When (as at n=105) coefficients are not equal 1 or -1 then terms in C[n,x] are counted with multiplicity. - This is the comment by the original author. However, the claim contradicts the given formula, as A051664 counts each nonzero coefficient just once, regardless of its value. For the version summing the absolute values of the coefficients (thus "with multiplicity"), see A318886. - Antti Karttunen, Sep 10 2018

			n=21: Cyclotomic[21,x]=1-x+x^3-x^4+x^6-x^8+x^9-x^11+x^12 has 9 terms while largest prime factor of 21 is 7

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A006530, A051664, A070537, A070776.

Differs from A318886 for the first time at n=105, where a(105) = 26, while A318886(105) = 28.

Array[Length@ Cyclotomic[#, x] - FactorInteger[#][[-1, 1]] &, 105] (* Michael De Vlieger, Sep 10 2018 *)

A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1); \\ From A006530.
A051664(n) = length(select(x->x!=0, Vec(polcyclo(n)))); \\ From A051664
A070536(n) = (A051664(n) - A006530(n)); \\ Antti Karttunen, Sep 10 2018

a(n) = A051664(n) - A006530(n).

Data section extended to 105 terms by Antti Karttunen, Sep 10 2018

A086761 Numbers k such that k-th cyclotomic polynomial has exactly 5 nonzero terms.

Original entry on oeis.org

5, 10, 20, 25, 40, 50, 80, 100, 125, 160, 200, 250, 320, 400, 500, 625, 640, 800, 1000, 1250, 1280, 1600, 2000, 2500, 2560, 3125, 3200, 4000, 5000, 5120, 6250, 6400, 8000, 10000, 10240, 12500, 12800, 15625, 16000, 20000, 20480, 25000, 25600, 31250, 32000

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 02 2003

nonn

A206787(a(n)) = 6. - Reinhard Zumkeller, Feb 12 2012
All terms have the form 2^a 5^b with a >= 0 and b > 0. - T. D. Noe, Feb 13 2012
If the above holds for all terms then this sequence is 5 * A003592. - David A. Corneth, Jul 04 2018

Cf. A003592, A051664, A065119.

Select[Range[1000], Count[CoefficientList[Cyclotomic[#, x], x], 0] == EulerPhi[#] - 4 &] (* T. D. Noe, Feb 13 2012 *)

is(n) = v = Vec(polcyclo(n)); sum(i=1,#v,v[i]!=0) == 5 \\ David A. Corneth, Jul 04 2018

More terms from T. D. Noe, Feb 13 2012

A086798 Number of coefficients equal to zero in n-th cyclotomic polynomial.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 3, 4, 0, 0, 2, 0, 0, 2, 7, 0, 4, 0, 4, 4, 0, 0, 6, 16, 0, 16, 6, 0, 2, 0, 15, 6, 0, 8, 10, 0, 0, 8, 12, 0, 4, 0, 10, 18, 0, 0, 14, 36, 16, 10, 12, 0, 16, 24, 18, 12, 0, 0, 10, 0, 0, 28, 31, 18, 6, 0, 16, 14, 8, 0, 22, 0, 0, 34, 18, 30, 8, 0, 28, 52, 0, 0, 16, 24, 0, 18, 30, 0

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 05 2003

nonn

See A051664.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (terms 1..1000 from T. D. Noe)

Cf. A086765, A086780.

Cf. A051664 (number of nonzero terms in n-th cyclotomic polynomial).

Table[Count[CoefficientList[Cyclotomic[n, x], x], _?(#==0&)], {n, 0, 100}]

a(n)=sum(k=0,eulerphi(n),if(polcoeff(polcyclo(n),k),0,1))

A086798(n) = (1 + eulerphi(n) - length(select(x->x!=0, Vec(polcyclo(n))))); \\ Antti Karttunen, Sep 21 2018

From Benoit Cloitre, Aug 06 2003: (Start)
a(4n+2) = a(2n+1); a(4n) = a(2n) + phi(2n).
When p is an odd prime and m integer >= 1: a(p^m) = a(2*p^m) = p^m - p^(m-1) - p + 1. In particular a(p) = a(2p) = 0. (End)
a(n) = 1 + phi(n) - A051664(n). - T. D. Noe, Aug 08 2003

More terms from Benoit Cloitre and T. D. Noe, Aug 06 2003

A318884 a(n) is the sum of absolute values of the coefficients in the n-th cyclotomic polynomial.

Original entry on oeis.org

2, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 7, 2, 17, 3, 19, 5, 9, 11, 23, 3, 5, 13, 3, 7, 29, 7, 31, 2, 15, 17, 17, 3, 37, 19, 17, 5, 41, 9, 43, 11, 7, 23, 47, 3, 7, 5, 23, 13, 53, 3, 17, 7, 25, 29, 59, 7, 61, 31, 9, 2, 31, 15, 67, 17, 31, 17, 71, 3, 73, 37, 7, 19, 31, 17, 79, 5, 3, 41, 83, 9, 41, 43, 39, 11, 89

Offset: 1

Antti Karttunen, Sep 10 2018

nonn

Differs from A051664 in the positions given by A013590, thus for the first time at n=105, where a(105) = 35, while A051664(105) = 33 as the 105th cyclotomic polynomial is the first one that has a coefficient other than 1, 0, or -1.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
Wikipedia, Cyclotomic polynomial

Cf. A013590, A013595, A051258, A051664, A318886.

Array[Total@ Abs@ CoefficientList[Cyclotomic[#, x], x] &, 89] (* Michael De Vlieger, Sep 10 2018 *)

A318884(n) = vecsum(apply(abs,Vec(polcyclo(n)))); \\ Antti Karttunen, Sep 10 2018

A086779 Numbers k such that k-th cyclotomic polynomial has exactly 7 nonzero terms.

Original entry on oeis.org

7, 14, 15, 28, 30, 45, 49, 56, 60, 75, 90, 98, 112, 120, 135, 150, 180, 196, 224, 225, 240, 270, 300, 343, 360, 375, 392, 405, 448, 450, 480, 540, 600, 675, 686, 720, 750, 784, 810, 896, 900, 960, 1080, 1125, 1200, 1215, 1350, 1372, 1440, 1500, 1568, 1620

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 03 2003

nonn

Cf. A086761.

Select[Range[2000], Count[CoefficientList[Cyclotomic[#, x], x], 0] == EulerPhi[#] - 6 &] (* T. D. Noe, Feb 13 2012 *)

isok(n) = {my(p = polcyclo(n)); #select(x->x, vector(1+poldegree(p), k, polcoeff(p, k-1))) == 7;} \\ Michel Marcus, Oct 26 2017

{n: A051664(n)=7}. - R. J. Mathar, Sep 15 2012

Extended by T. D. Noe, Feb 13 2012

cp's OEIS Frontend

A087073 Mobius transform of A051664, the number of nonzero terms in the n-th cyclotomic polynomial.

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A070776 Numbers k such that number of terms in the k-th cyclotomic polynomial is equal to the largest prime factor of k.

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A086780 Number of negative terms in n-th cyclotomic polynomial.

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A070537 Numbers k such that the k-th cyclotomic polynomial has more terms than the largest prime factor of k.

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A086765 Number of positive coefficients in n-th cyclotomic polynomial.

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A070536 Number of terms in n-th cyclotomic polynomial minus largest prime factor of n; a(1)=1 by convention.

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A086761 Numbers k such that k-th cyclotomic polynomial has exactly 5 nonzero terms.

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