A070537 -id:A070537 - cp's OEIS frontend

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

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)));

A324111 Numbers n for which A324108(n) = A324054(n-1) and which are neither prime powers nor of the form 2^i * p^j, where p is an odd prime, with either exponent i or j > 0.

Original entry on oeis.org

1, 87, 174, 348, 696, 1392, 2091, 2784, 4182, 5568, 8364, 11136, 16683, 16728, 22272, 33215, 33366, 33456, 44544, 66430, 66732, 66912, 89088, 132860, 133464, 133824, 178176, 265720, 266928, 267179, 267648, 356352, 531440, 533856, 534358, 535296, 712704, 1062880, 1066877, 1067712, 1068716, 1070592, 1319235, 1425408

Offset: 1

Antti Karttunen, Feb 15 2019

nonn

Setwise difference of A324109 and A070776.
Setwise difference of A070537 and A324110.
If an odd number n > 1 is present, then all 2^k * n are present also. Odd terms > 1 are given in A324112.

			87 is a term, as 87 = 3*29, A324054(3-1) = 4, A324054(29-1) = 156, and A324108(87) = 4*156 = 624 = A324054(87-1).

Antti Karttunen, Table of n, a(n) for n = 1..173

Cf. A070537, A070776, A324109, A324110, A324112.

A000265(n) = (n/2^valuation(n, 2));
A324054(n) = { my(p=2,mp=p*p,m=1); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, if(3==(n%4),mp *= p,m *= (mp-1)/(p-1))); n>>=1); (m); };
A324108(n) = { my(f=factor(n)); prod(i=1, #f~, A324054((f[i,1]^f[i,2])-1)); };
isA324111(n) = ((1!=omega(n))&&(1!=omega(A000265(n)))&&(A324054(n-1)==A324108(n)));
for(n=1,2^20,if(isA324111(n), print1(n,", ")))

A324110 Numbers k such that A324108(k) != A324054(k-1).

Original entry on oeis.org

15, 21, 30, 33, 35, 39, 42, 45, 51, 55, 57, 60, 63, 65, 66, 69, 70, 75, 77, 78, 84, 85, 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, 168, 170, 171, 175, 177, 180, 182, 183, 185, 186, 187, 189, 190, 195, 198, 201

Offset: 1

Antti Karttunen and David A. Corneth, Feb 15 2019

nonn

This is a subsequence of A070537. The missing terms 1, 87, 174, 348, 696, 1392, 2091, ..., are at A324111.

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

Cf. A070537, A324054, A324108, A324109 (complement), A324111.

A324054(n) = { my(p=2,mp=p*p,m=1); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, if(3==(n%4),mp *= p,m *= (mp-1)/(p-1))); n>>=1); (m); };
A324108(n) = { my(f=factor(n)); prod(i=1, #f~, A324054((f[i,1]^f[i,2])-1)); };
isA324110(n) = (A324054(n-1)!=A324108(n));
for(n=1,201,if(isA324110(n), print1(n,", ")))

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

A375745 a(n) is the sum of the vector of the reduced discriminant of the n-th cyclotomic polynomial.

Original entry on oeis.org

1, 1, 4, 4, 16, 4, 36, 16, 36, 16, 100, 16, 144, 36, 72, 64, 256, 36, 324, 64, 156, 100, 484, 64, 400, 144, 324, 144, 784, 72, 900, 256, 420, 256, 648, 144, 1296, 324, 600, 256, 1600, 156, 1764, 400, 648, 484, 2116, 256, 1764, 400, 1056, 576, 2704, 324, 1720, 576

Offset: 1

Darío Clavijo, Aug 26 2024

nonn

Conjecture: a(n) >= phi(n)^2 = A127473(n), and for n>=2 strictly greater iff n is in A070537.

Cf. A127473, A070537, A004124.

a(n) = vecsum(poldiscreduced(polcyclo(n)));

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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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A324111 Numbers n for which A324108(n) = A324054(n-1) and which are neither prime powers nor of the form 2^i * p^j, where p is an odd prime, with either exponent i or j > 0.

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A324110 Numbers k such that A324108(k) != A324054(k-1).

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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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A375745 a(n) is the sum of the vector of the reduced discriminant of the n-th cyclotomic polynomial.

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