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 1-3 of 3 results.

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

Views

Author

Labos Elemer, May 07 2002

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

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

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

Views

Author

Labos Elemer, May 03 2002

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

Formula

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

Extensions

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

A318886 The sum of absolute values of the coefficients in the n-th cyclotomic polynomial minus the 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, 28
Offset: 1

Views

Author

Antti Karttunen, Sep 10 2018

Keywords

Comments

See comments in A070536.

Links

Crossrefs

Cf. A006530, A318884.
Differs from A070536 for the first time at n=105, where a(105) = 28, while A070536(105) = 26.

Programs

  • Mathematica
    Array[Total@ Abs@ CoefficientList[Cyclotomic[#, x], x] - FactorInteger[#][[-1, 1]] &, 105] (* Michael De Vlieger, Sep 10 2018 *)
  • PARI
    A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1); \\ From A006530.
A318884(n) = vecsum(apply(abs,Vec(polcyclo(n))));
A318886(n) = (A318884(n) - A006530(n));
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.