A280611 -id:A280611 - cp's OEIS frontend

A280712 Inverse Euler transform of A280611.

2, 1, 0, 1, 0, 4, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 4, 0, 3, 0, 2, 0, 4, 0, 0, 0, 2, 0, 2, 0, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 4, 0, 1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 2, 0, 1, 0, 2, 0, 7, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 9, 0, 0, 0, 0, 0, 2, 0, 1, 0, 2, 0, 2, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 6, 0, 0, 0, 4, 0, 2, 0, 1, 0

Offset: 1

Christopher J. Smyth, Jan 07 2017

a(n) = b(n) for n odd, a(n) = b(n) - b(n/2) for n even >= 2, where b(n) = A014197(n) = the number of m with phi(m) = n.

Note that a(n) = 0 for all odd n > 1, and so a(n) = b(n) for n >= 3, n not a multiple of 4.

			a(4) = #{m:phi(m) = 4} - #{m:phi(m) = 2} = #{5,8,10,12} - #{2,4,6} = 4-3 = 1.

Antti Karttunen, Table of n, a(n) for n = 1..20160
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A014197, A280611.

A014197(n, m=1) = { n==1 && return(1+(m<2)); my(p, q); sumdiv(n, d, if( d>=m && isprime(d+1), sum( i=0, valuation(q=n\d, p=d+1), A014197(q\p^i, p))))}; \\ From A014197
A280712(n) = if(n%2,A014197(n),A014197(n)-A014197(n/2)); \\ Antti Karttunen, Nov 09 2018

Euler transform of sequence = Product_{k>=1} (1-x^k)^(-a(k)) is the g.f. of A280611.

More terms from Antti Karttunen, Nov 09 2018

A014197 Number of numbers m with Euler phi(m) = n.

Original entry on oeis.org

2, 3, 0, 4, 0, 4, 0, 5, 0, 2, 0, 6, 0, 0, 0, 6, 0, 4, 0, 5, 0, 2, 0, 10, 0, 0, 0, 2, 0, 2, 0, 7, 0, 0, 0, 8, 0, 0, 0, 9, 0, 4, 0, 3, 0, 2, 0, 11, 0, 0, 0, 2, 0, 2, 0, 3, 0, 2, 0, 9, 0, 0, 0, 8, 0, 2, 0, 0, 0, 2, 0, 17, 0, 0, 0, 0, 0, 2, 0, 10, 0, 2, 0, 6, 0, 0, 0, 6, 0, 0, 0, 3

Offset: 1

N. J. A. Sloane

Carmichael conjectured that there are no 1's in this sequence. - Jud McCranie, Oct 10 2000
Number of cyclotomic polynomials of degree n. - T. D. Noe, Aug 15 2003
Let v == 0 (mod 24), w = v + 24, and v < k < q < w, where k and q are integer. It seems that, for most values of v, there is no b such that b = a(k) + a(q) and b > a(v) + a(w). The first case where b > a(v) + a(w) occurs at v = 888: b = a(896) + a(900) = 15 + 4, b > a(888) + a(912), or 19 > 8 + 7. The first case where v < n < w and a(n) > a(v) + a(w) occurs at v = 2232: a(2240) > a(2232) + a(2256), or 27 > 7 + 8. - Sergey Pavlov, Feb 05 2017
One elementary result relating to phi(m) is that if m is odd, then phi(m)=phi(2m) because 1 and 2 both have phi value 1 and phi is multiplicative. - Roderick MacPhee, Jun 03 2017

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B39, pp. 144-146.
Joe Roberts, Lure of The Integers, The Mathematical Association of America, 1992, entry 32, page 182.

T. D. Noe, Table of n, a(n) for n = 1..10000
Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions, Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2.
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
R. D. Carmichael, On Euler’s phi-function, Bull. Amer. Math. Soc. 13 (1907), 241-243.
R. D. Carmichael, Note on Euler’s phi-function, Bull. Amer. Math. Soc. 28 (1922), 109-110.
Kevin Ford, The number of solutions of phi(x)=m, arXiv:math/9907204 [math.NT], 1999.
S. Sivasankaranarayana Pillai, On some functions connected with phi(n), Bull. Amer. Math. Soc. 35 (1929), 832-836.
Primefan, Totient Answers For The First 1000 Integers.
Eric Weisstein's World of Mathematics, Totient Function.
Eric Weisstein's World of Mathematics, Totient Valence Function.

Cf. A000010, A002202, A032446 (bisection), A049283, A051894, A055506, A057635, A057826, A058277 (nonzero terms), A058341, A063439, A066412, A070243 (partial sums), A070633, A071386 (positions of odd terms), A071387, A071388 (positions of primes), A071389 (where prime(n) occurs for the first time), A082695, A097942 (positions of records), A097946, A120963, A134269, A219930, A280611, A280709, A280712, A296655 (positions of positive even terms), A305353, A305656, A319048, A322019.
For records see A131934.
Column 1 of array A320000.

a := function(n)
local S, T, R, max, i, k, r;
S:=[];
for i in DivisorsInt(n)+1 do
    if IsPrime(i)=true then
        S:=Concatenation(S,[i]);
    fi;
od;
T:=[];
for k in [1..Size(S)] do
    T:=Concatenation(T,[S[k]/(S[k]-1)]);
od;
max := n*Product(T);
R:=[];
for r in [1..Int(max)] do
    if Phi(r)=n then
        R:=Concatenation(R,[r]);
    fi;
od;
return Size(R);
end; # Miles Englezou, Oct 22 2024

[#EulerPhiInverse(n): n in [1..100]]; // Marius A. Burtea, Sep 08 2019

with(numtheory): A014197:=n-> nops(invphi(n)): seq(A014197(n), n=1..200);

a[1] = 2; a[m_?OddQ] = 0; a[m_] := Module[{p, nmax, n, k}, p = Select[ Divisors[m]+1, PrimeQ]; nmax = m*Times @@ (p/(p - 1)); n = m; k = 0; While[n <= nmax, If[EulerPhi[n] == m, k++]; n++]; k]; Array[a, 92] (* Jean-François Alcover, Dec 09 2011, updated Apr 25 2016 *)
With[{nn = 116}, Function[s, Function[t, Take[#, nn] &@ ReplacePart[t, Map[# -> Length@ Lookup[s, #] &, Keys@ s]]]@ ConstantArray[0, Max@ Keys@ s]]@ KeySort@ PositionIndex@ Array[EulerPhi, Floor[nn^(3/2)] + 10]] (* Michael De Vlieger, Jul 19 2017 *)

A014197(n,m=1) = { n==1 && return(1+(m<2)); my(p,q); sumdiv(n, d, if( d>=m && isprime(d+1), sum( i=0,valuation(q=n\d,p=d+1), A014197(q\p^i,p))))} \\ M. F. Hasler, Oct 05 2009

a(n) = invphiNum(n); \\ Amiram Eldar, Nov 15 2024 using Max Alekseyev's invphi.gp

from sympy import totient, divisors, isprime, prod
def a(m):
    if m == 1: return 2
    if m % 2: return 0
    X = (x + 1 for x in divisors(m))
    nmax=m*prod(i/(i - 1) for i in X if isprime(i))
    n=m
    k=0
    while n<=nmax:
        if totient(n)==m:k+=1
        n+=1
    return k
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 18 2017, after Mathematica code

Dirichlet g.f.: Sum_{n>=1} a(n)*n^-s = zeta(s)*Product_(1+1/(p-1)^s-1/p^s). - Benoit Cloitre, Apr 12 2003
Limit_{n->infinity} (1/n) * Sum_{k=1..n} a(k) = zeta(2)*zeta(3)/zeta(6) = 1.94359643682075920505707036... (see A082695). - Benoit Cloitre, Apr 12 2003
From Christopher J. Smyth, Jan 08 2017: (Start)
Euler transform = Product_{n>=1} (1-x^n)^(-a(n)) = g.f. of A120963.
Product_{n>=1} (1+x^n)^a(n)
= Product_{n>=1} ((1-x^(2n))/(1-x^n))^a(n)
= Product_{n>=1} (1-x^n)^(-A280712(n))
= Euler transform of A280712 = g.f. of A280611.
(End)
a(A000010(n)) = A066412(n). - Antti Karttunen, Jul 18 2017
From Antti Karttunen, Dec 04 2018: (Start)
a(A000079(n)) = A058321(n).
a(A000142(n)) = A055506(n).
a(A017545(n)) = A063667(n).
a(n) = Sum_{d|n} A008683(n/d)*A070633(d).
a(n) = A056239(A322310(n)).
(End)

A120963 Number of monic polynomials with integer coefficients of degree n with all roots on the unit circle; number of products of cyclotomic polynomials of degree n.

Original entry on oeis.org

1, 2, 6, 10, 24, 38, 78, 118, 224, 330, 584, 838, 1420, 2002, 3258, 4514, 7134, 9754, 15010, 20266, 30532, 40798, 60280, 79762, 115966, 152170, 217962, 283754, 401250, 518746, 724866, 930986, 1287306, 1643626, 2250538, 2857450, 3878298, 4899146, 6594822

Offset: 0

Franklin T. Adams-Watters, Jul 19 2006

Also the number of types of crystallographic rotations and reflection-rotations in n-dimensional Euclidean space. - Andrey Zabolotskiy, Jul 08 2017

			The six polynomials of degree 2 consist of 3 irreducible cyclotomic polynomials: x^2+1, x^2+x+1 and x^2-x+1 and 3 products of 2 linear cyclotomic polynomials: x^2+2x+1, x^2-1 and x^2-2x+1.
The six plane crystallographic operations are the identity operation, rotations by 2 Pi/k with k = 2,3,4,6, and a reflection.

Boyd, David W.(3-BC); Montgomery, Hugh L.(1-MI), Cyclotomic partitions. In Number theory (Banff, AB, 1988), 7-25. Walter de Gruyter & Co., Berlin, 1990 ISBN:3-11-011723-1, MR1106647. [Asymptotics]

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
Gaëtan Chenevier, The Characteristic Masses of Niemeier Lattices, arXiv:2002.03707 [math.NT], 2020.
Peter Engel, Louis Michel and Marjorie Senechal, Lattice Geometry, 2004 (see section 1.4.3).
Richard P. Stanley, Some enumerative applications of cyclotomic polynomials, preprint, 2024-2025. See p. 13.
D. Weigel, R. Veysseyre, T. Phan, J. M. Effantin, and Y. Billiet, Crystallography, geometry and physics in higher dimensions. I. Point-symmetry operations, Acta Cryst., A40 (1984), 323-330 (see table 3).

Cf. A014197, A051894, A280611 (variant where repeated roots are not allowed).

A280709 The number of monic integer polynomials of degree n all of whose roots are distinct and of modulus at most 1.

Original entry on oeis.org

1, 3, 6, 10, 16, 24, 38, 58, 86, 122, 172, 236, 328, 448, 606, 802, 1060, 1380, 1806, 2338, 3018, 3846, 4900, 6180, 7816, 9808, 12294, 15274, 18982, 23418, 28938, 35542, 43638, 53226, 64942, 78786, 95686, 115642, 139754, 168022, 202086, 241946

Offset: 0

Christopher J. Smyth, Jan 07 2017

Such polynomials are a product of distinct cyclotomic polynomials, possibly multiplied by z. This follows from a classical result of Kronecker -- see Links.

			a(2)=6 because the six polynomials z^2+z+1, z^2+1, z^2-z+1, z^2-z, z^2+z and z^2-1 are the only ones of the required type.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
L. Kronecker, Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, J. Reine Angew. Math. 53 (1857), 173-175.

Cf. A280611 (variant where all roots must have modulus exactly 1);

Cf. A120963 (variant where multiple roots are allowed).

Table[SeriesCoefficient[(1 + x) Product[(1 + x^EulerPhi@ i), {i, n E^2}], {x, 0, n}], {n, 0, 120}] (* Michael De Vlieger, Jan 10 2017 *)

a(0) = 1 and a(n) = b(n)+b(n-1) for n >= 1, where b(n) = A280611(n).
G.f.: (1+x)*Product_{i>=1} (1+x^phi(i)) = (1+x)*Product_{j>=1} (1+x^j)^A014197(j), where phi(i)=A000010(i) is Euler's totient function.
It is also the Euler transform of A280712 except with its first two terms (2,1) replaced by (3,0).
a(n) ~ exp(sqrt(105*zeta(3)*n/2)/Pi) * (105*zeta(3)/2)^(1/4) / (2*Pi*n^(3/4)). - Vaclav Kotesovec, Sep 02 2021

cp's OEIS Frontend

A280712 Inverse Euler transform of A280611.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A014197 Number of numbers m with Euler phi(m) = n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

Python

Formula

A120963 Number of monic polynomials with integer coefficients of degree n with all roots on the unit circle; number of products of cyclotomic polynomials of degree n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A280709 The number of monic integer polynomials of degree n all of whose roots are distinct and of modulus at most 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A305353 Expansion of Product_{k>=1} (1 - x^phi(k)), where phi is Euler's totient function.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A347524 E.g.f.: exp(Sum_{k>=1} A014197(k)*x^k).

Views

Author

Keywords

Links

Crossrefs

Formula