A078779 -id:A078779 - cp's OEIS frontend

A046790 Positive numbers divisible by 8 or by the square of an odd prime.

8, 9, 16, 18, 24, 25, 27, 32, 36, 40, 45, 48, 49, 50, 54, 56, 63, 64, 72, 75, 80, 81, 88, 90, 96, 98, 99, 100, 104, 108, 112, 117, 120, 121, 125, 126, 128, 135, 136, 144, 147, 150, 152, 153, 160, 162, 168, 169, 171, 175, 176, 180, 184, 189, 192, 196, 198, 200, 207, 208

Offset: 1

David W. Wilson, Dec 11 1999

This sequence has many equivalent definitions:
(D1) Positive numbers divisible by 8 or by the square of an odd prime. (We take this as the main definition, since it is the simplest.)
(D2) Moduli m for which there exist affine maps f:x->a*x + b modulo m, with a > 1, such that f has order m in the affine group. (For example, 8 is a term because f:x->(5x+1) mod 8 is a map with order 8 in the group of affine maps mod 8: the smallest power of f equal to identity is f^8. The maps x->x+1 always have this property, so are excluded from consideration.) - Emmanuel Amiot, Jul 28 2007
(D3) Numbers k such that A005361(k) < A003557(k). - Anthony Browne, Jun 03 2016
(D4) Numbers i such that there is a smaller positive number j such that (i+j)/2 and sqrt(i*j) are integers. (See A046791 for the smallest choice for j.) - David W. Wilson, Dec 11 1999
(D5) Numbers k such that A008475(k) is different from A001414(k). - Benoit Cloitre, Jan 11 2003
For a proof of the equivalence of definitions (D1)-(D5) see the Don Reble link.
(D6) Numbers m >= 8 having a divisor k^2 >= 4 such that m and m/k^2 are of the same parity. (See A046791 for the largest such k.) - Vladimir Shevelev, Jun 06 2006
(D7) Numbers that can be the semiperimeter of a isosceles triangle with integer sides and area. - Peter Kagey, May 17 2019
Closed under multiplication, which may be used to construct the sequence. - David A. Corneth, Jun 07 2016
Complement of A078779. - Omar E. Pol, Jun 11 2016
m is in this sequence if and only if m does not divide 2*radical(m). - Peter Luschny, Mar 05 2019
Verified up to a(290) = 1000, {a(n)} is identical to the sequence of group orders for which there exists at least one group G such that |Char(G)| is a nontrivial divisor of |Normal(G)|, where |Char(G)| is the number of characteristic subgroups of G and |Normal(G)| the number of normal subgroups of G. - Miles Englezou, Jul 20 2024

M. F. Hasler, Table of n, a(n) for n = 1..10000 (first 290 terms from N. J. A. Sloane).
Emmanuel Amiot, Autosimilar Melodies, J. Math. Music 2 (2008), no. 3, 157-180. DOI: 10.1080/17459730802598146.
Emmanuel Amiot, Mélodies autosimilaires (Self-Replicating Melodies) (in French).
Don Reble, Proof of equivalence of definitions (D1)-(D5), Jun 06 2016
Mathematics Stack Exchange user "George", Semiperimeter of isosceles Heronian triangles.

Cf. A001414, A003557, A005361, A008475, A046791.

ordreMax[a_, n_]:= Module[{mo, r, s, s0, gcd}, mo=MultiplicativeOrder[a,n]; s= s0=Mod[Sum[a^k,{k,0,mo-1}], n]; Max[Table[gcd=GCD[a-1,b];r=1; While[Mod[s *gcd, n]!=0, s=Mod[s0+a^mos,n];r++ ]; r*mo, {b,0,n-1} ]] ] ordreMax[n_] := Module[{candidats, m,t}, candidats = Select[Range[2,n-1], (GCD[n,# ]==1 && GCD[n, #-1]>1)&]; m=Max[t=Table[ordreMax[a,n], {a, candidats}] ]; {m,Part[candidats,Flatten@Position[t, m] ]}] Module[{resu}, Do[resu=ordreMax[n]; If[First[resu] >=n, Print[n ]], {n,4,200}]] (* This is for definition (D2). Emmanuel Amiot, Jul 28 2007 *)
Select[Range[210], Mod[#, 8] == 0 || AnyTrue[ Divisors[#], DivisorSigma[0, #] == 3 && Mod[#, 4] != 0 &] &] (* Carlos Eduardo Olivieri, Jun 07 2016 *)
Module[{upto=250,prs},prs=Prime[Range[2,PrimePi[Sqrt[upto]]]]^2;Join[ Range[ 8,upto,8],Select[Range[upto],AnyTrue[#/prs,IntegerQ]&]]] // Union (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 18 2018 *)

is(n)={n%8==0||!issquarefree(n>>!bittest(n,0))} \\ M. F. Hasler, Jun 07 2016

print([n for n in (1..208) if not ZZ(n).divides(2*radical(n))])  # Peter Luschny, Mar 05 2019

Let A(x) be the number of a(n) <= x. Then A(x) ~ (1 - 7/Pi^2)*x = 0.2907517...*x as x goes to infinity. - Vladimir Shevelev, Jun 07 2016

Entry revised by N. J. A. Sloane, with help from Don Reble and several OEIS editors, Jun 07 2016

A121684 Union of {8, 9, 18}, S, 2S and 4S, where S = odd squarefree numbers (A056911).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97

Offset: 1

N. J. A. Sloane, Sep 13 2006

nonn

Numbers n such that the cyclic group Z_n is a CI-group.

The asymptotic density of this sequence is 7/Pi^2. - Amiram Eldar, May 10 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000
B. Alspach and M. Mishna, Enumeration of Cayley graphs and digraphs, Discr. Math., 256 (2002), 527-539.
M. Mishna, Home Page
M. Muzychuk, On Adam's conjecture for circulant graphs, Discr. Math. 167 (1997), 497-510.

Cf. A056911, A078779, A121176.

With[{upto=100},Select[Sort[Join[{8,9,18},Flatten[{#,2#,4#}&/@ Select[ Range[1,upto,2],SquareFreeQ]]]],#<=upto&]] (* Harvey P. Dale, Sep 03 2015 *)

A332713 a(n) = Sum_{d|n} phi(d/gcd(d, n/d)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 7, 8, 10, 11, 12, 13, 14, 15, 13, 17, 16, 19, 20, 21, 22, 23, 21, 22, 26, 22, 28, 29, 30, 31, 24, 33, 34, 35, 32, 37, 38, 39, 35, 41, 42, 43, 44, 40, 46, 47, 39, 44, 44, 51, 52, 53, 44, 55, 49, 57, 58, 59, 60, 61, 62, 56, 46, 65, 66, 67, 68, 69, 70

Offset: 1

Ilya Gutkovskiy, Feb 20 2020

Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Cf. A000010, A001616, A010052, A046790 (numbers n such that a(n) < n), A055653, A061884, A078779 (fixed points), A332619, A332686, A332712.

Table[Sum[EulerPhi[d/GCD[d, n/d]], {d, Divisors[n]}], {n, 1, 70}]
A055653[n_] := Sum[Boole[GCD[d, n/d] == 1] EulerPhi[d], {d, Divisors[n]}]; a[n_] := Sum[Boole[IntegerQ[(n/d)^(1/2)]] A055653[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]

a(n) = sumdiv(n, d, eulerphi(d/gcd(d, n/d))); \\ Michel Marcus, Feb 20 2020

Dirichlet g.f.: zeta(s) * zeta(2*s) * zeta(s - 1) * Product_{p prime} (1 - p^(-s) + p^(-2*s) - p^(1 - 2*s)).
a(n) = Sum_{d|n} phi(lcm(d, n/d)/d).
a(n) = Sum_{d|n} A010052(n/d) * A055653(d).
Sum_{k=1..n} a(k) ~ c * Pi^6 * n^2 / 1080, where c = A330523 = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.5358961538283379998085... - Vaclav Kotesovec, Feb 22 2020
From Richard L. Ollerton, May 10 2021: (Start)
a(n) = Sum_{k=1..n} phi(gcd(n,k)/gcd(gcd(n,k),n/gcd(n,k)))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} phi(n/gcd(n,k)/gcd(gcd(n,k),n/gcd(n,k)))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} phi(lcm(gcd(n,k),n/gcd(n,k))/gcd(n,k))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} phi(lcm(gcd(n,k),n/gcd(n,k))*gcd(n,k)/n)/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} A010052(gcd(n,k))*A055653(n/gcd(n,k))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} A010052(n/gcd(n,k))*A055653(gcd(n,k))/phi(n/gcd(n,k)). (End)

A046791 A046790 has several definitions, one of which is: "Numbers i such that there is a smaller positive number j such that (i+j)/2 and sqrt(i*j) are integers". The present sequence gives the smallest choice for j.

Original entry on oeis.org

2, 1, 4, 2, 6, 1, 3, 2, 4, 10, 5, 12, 1, 2, 6, 14, 7, 4, 2, 3, 20, 1, 22, 10, 6, 2, 11, 4, 26, 12, 28, 13, 30, 1, 5, 14, 2, 15, 34, 4, 3, 6, 38, 17, 10, 2, 42, 1, 19, 7, 44, 20, 46, 21, 12, 4, 22, 2, 23, 52, 6, 14, 1, 58, 26, 60, 2, 3, 5, 62, 10, 28, 4, 29, 66, 30, 68, 11, 31, 70, 2, 1, 6, 74, 33

Offset: 1

David W. Wilson, Dec 11 1999

nonn

Note that A046790 is the complement of A078779. - Omar E. Pol, Jun 11 2016

			From _Vladimir Shevelev_, Jun 07 2016: (Start)
A046790(5)=24 with even squarefree part (6), so a(5) = 6;
A046790(12)=48 with odd squarefree part (3), so a(12) = 3*4=12.
(End)

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A046790.

a(n) = my(n=A046790(n),f=factor(n),p=n%2);f[,2]=f[,2]%2;r=prod(i=1,matsize(f)[1],f[i,1]^f[i,2]);r*=(4^(n%2==0&&r%2==1)) \\ David A. Corneth, Jun 07 2016

Let b(n)=A046790(n). Let k=k(n) be the greatest number whose square divides b(n) and is such that b(n) and b(n)/k^2 are of the same parity. Then a(n) = b(n)/k^2. - Vladimir Shevelev, Jun 07 2016

Or, equivalently, a(n) is the squarefree part s(n) of b(n), if either b(n) is odd or s(n) is even. Otherwise, when b(n) is even, but s(n) is odd, a(n)=4*s(n). - David A. Corneth, Jun 07 2016

Entry revised by N. J. A. Sloane, with help from Don Reble and several OEIS editors. Jun 07 2016

A193551 Smallest number with n as multiplicative projection.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 16, 27, 10, 11, 12, 13, 14, 15, 256, 17, 24, 19, 20, 21, 22, 23, 36, 3125, 26, 19683, 28, 29, 30, 31, 65536, 33, 34, 35, 72, 37, 38, 39, 80, 41, 42, 43, 44, 135, 46, 47, 144, 823543, 160, 51, 52, 53, 216, 55, 112, 57, 58, 59, 60, 61, 62

Offset: 1

Reinhard Zumkeller, Aug 27 2011

nonn

A000026(a(n)) = n and A000026(m) <> n for m < a(n);
a(p^k) = p^(p^(k-1)), p prime, k > 0; the sequence is not multiplicative, but for coprime odd numbers u, v: a(u*v) = a(u) * a(v);
A078779 gives fixed points: a(A078779(n)) = A078779(n).

import Data.List (elemIndex, findIndices)
import Data.Maybe (fromJust)
a193551 n = (fromJust $ elemIndex n a000026_list) + 1

A304410 Numbers k such that k = Product (p_j^e_j) = Product (p_j*(e_j + 1)).

Original entry on oeis.org

1, 8, 9, 72, 13440, 21120, 24960, 29568, 32640, 34944, 36480, 44160, 45696, 49280, 51072, 54912, 55680, 58240, 59520, 61824, 71040, 71808, 76160, 77952, 78720, 80256, 82560, 83328, 84864, 85120, 90240, 91520, 94848, 97152, 99456, 101760, 103040, 110208, 113280, 114816, 115584, 117120, 119680

Offset: 1

Ilya Gutkovskiy, May 12 2018

nonn

Numbers k such that A000005(k)*A007947(k) = k.
Fixed points of A304409.
All terms are refactorable numbers (A033950).

			13440 is a term because 13440 = 2^7*3*5*7 = 2*(7 + 1) * 3*(1 + 1) * 5*(1 + 1) * 7*(1 + 1).

Cf. A000005, A000026, A007947, A008478, A033950, A036762, A048768, A078779, A190473, A304409.

a[n_] := Times @@ (#[[1]] (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Select[Range[120000], a[#] == # &]

isok(k) = {my(f = factor(k)); numdiv(f) * vecprod(f[, 1]) == k;} \\ Amiram Eldar, Jan 31 2025

A360523 a(n) = Sum_{d|n} mu(rad(d)) * delta_d(n/d), where rad(n) = A007947(n) and delta_d(n) is the greatest divisor of n that is relatively prime to d.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 5, 7, 4, 10, 4, 12, 6, 8, 12, 16, 7, 18, 8, 12, 10, 22, 10, 23, 12, 24, 12, 28, 8, 30, 27, 20, 16, 24, 14, 36, 18, 24, 20, 40, 12, 42, 20, 28, 22, 46, 24, 47, 23, 32, 24, 52, 24, 40, 30, 36, 28, 58, 16, 60, 30, 42, 58, 48, 20, 66, 32, 44, 24

Offset: 1

Amiram Eldar, Feb 10 2023

Analogous to the Euler totient function (A000010) as A360522 is analogous to A000203.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Mizan R. Khan, A variant of the divisor functions sigma_a(n), JP Journal of Algebra, Number Theory and Applications, Vol. 5, No. 3 (2005), pp. 561-574.

Cf. A000010, A000203, A005117, A007947 (rad), A008683 (mu), A047994, A078779, A360522.

f[p_, e_] := p^e - e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - f[i,2]);}

Multiplicative with a(p^e) = p^e - e.
Dirichlet g.f.: zeta(s-1)*zeta(s)^2 * Product_{p prime} (1 - 3/p^s + 1/p^(2*s-1) + 1/p^(2*s)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - p/((p-1)*(p+1)^2)) = 0.3243742337... .
A000010(n) <= a(n) <= A047994(n) (Khan, 2005).
a(n) = A000010(n) if and only if n is in A078779 (i.e., n is either squarefree or twice a squarefree number).
a(n) = A047994(n) if and only if n is in A005117 (i.e., n is squarefree).

A304194 Numbers k such that k = Product (p_j^e_j) = Product (pi(p_j)*p_j), where pi() = A000720.

Original entry on oeis.org

1, 2, 12, 56, 180, 304, 336, 936, 1696, 1824, 2484, 5040, 5328, 6664, 8384, 8512, 9900, 10176, 13176, 14040, 25632, 26208, 27360, 33372, 33712, 37260, 39808, 39984, 47488, 50304, 51072, 52200, 65232, 69552, 79920, 126900, 128448, 142272, 149184, 152640, 162648, 167776, 184064, 193752, 197640

Offset: 1

Ilya Gutkovskiy, May 07 2018

nonn

Numbers k such that A007947(k)*A156061(k) = k or A156061(k) = A003557(k).

			9900 is a term because 9900 = 2^2 * 3^2 * 5^2 * 11 = prime(1)^2 * prime(2)^2 * prime(3)^2 * prime(5) = 1*prime(1) * 2*prime(2) * 3*prime(3) * 5*prime(5).

Eric Weisstein's World of Mathematics, Distinct Prime Factors
Index entries for sequences computed from indices in prime factorization

Cf. A000720, A003557, A005117, A007947, A008478, A033286, A046022, A048768, A078779, A109298, A156061.

a[n_] := Times @@ (PrimePi[#[[1]]] #[[1]] & /@ FactorInteger[n]); a[1] = 1; Select[Range[200000], a[#] == # &]

isok(n) = {my(f=factor(n)); prod(k=1, #f~, primepi(f[k,1])*f[k,1]) == n;} \\ Michel Marcus, May 08 2018

cp's OEIS Frontend

A046790 Positive numbers divisible by 8 or by the square of an odd prime.

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A121684 Union of {8, 9, 18}, S, 2S and 4S, where S = odd squarefree numbers (A056911).

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A332713 a(n) = Sum_{d|n} phi(d/gcd(d, n/d)).

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A046791 A046790 has several definitions, one of which is: "Numbers i such that there is a smaller positive number j such that (i+j)/2 and sqrt(i*j) are integers". The present sequence gives the smallest choice for j.

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A193551 Smallest number with n as multiplicative projection.

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A304410 Numbers k such that k = Product (p_j^e_j) = Product (p_j*(e_j + 1)).

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A360523 a(n) = Sum_{d|n} mu(rad(d)) * delta_d(n/d), where rad(n) = A007947(n) and delta_d(n) is the greatest divisor of n that is relatively prime to d.

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A304194 Numbers k such that k = Product (p_j^e_j) = Product (pi(p_j)*p_j), where pi() = A000720.

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