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.

A065299 Numbers k such that sigma(k)*phi(k) is squarefree.

Original entry on oeis.org

1, 2, 4, 9, 121, 242, 529, 1058, 2209, 3481, 4418, 5041, 6889, 6962, 10082, 11449, 13778, 17161, 22898, 27889, 32041, 34322, 51529, 55778, 57121, 64082, 96721, 103058, 114242, 120409, 128881, 146689, 175561, 185761, 193442, 196249, 218089
Offset: 1

Views

Author

Labos Elemer, Oct 29 2001

Keywords

Examples

			All solutions are either squares or twice squares. Proper subset of A055008 or A028982. Several squares (of primes) and 2*squares are not here. E.g., 242 is here because phi(242) = 110, sigma(242) = 399, 2*5*11*3*7*19 is squarefree; 18 is not here, since 2*3*3*13 is not squarefree.

Links

Crossrefs

A062354(m) is squarefree.
Cf. A000010, A000203, A008683, A055008, A028982, A049195-A049199, A049149.

Programs

  • Mathematica
    a[x_] := Abs[MoebiusMu[DivisorSigma[1, x]*EulerPhi[x]]] Do[s=as[n]; If[Equal[s, 1], Print[{n, Sqrt[n]}]], {n, 1, 1000000}]
Select[Range[250000],SquareFreeQ[DivisorSigma[1,#]*EulerPhi[#]]&] (* Harvey P. Dale, Jul 15 2015 *)
  • PARI
    n=0; for (m = 1, 10^9, s=abs(moebius(sigma(m)*eulerphi(m))); if (s==1, write("b065299.txt", n++, " ", m); if (n==500, return))) \\ Harry J. Smith, Oct 15 2009
  • PARI
    is(f)=my(n=#f~, v=List()); for(i=1,n, if(f[i,1]>2, listput(v,f[i,1]-1)); if(f[i,2]>2, return(0), f[i,2]>1, listput(v,f[i,1])); listput(v, (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1))); for(i=2,#v, for(j=1,i-1, if(gcd(v[i],v[j])>1, return(0)))); for(i=1,#v, if(!issquarefree(v[i]), return(0))); 1
sq(f)=f[,2]*=2; f
double(f)=if(#f~ && f[1,1]==2, f[1,2]++, f=concat([2,1],f)); f
list(lim)=my(v=List()); forsquarefree(n=1,sqrtint(lim\1), if(is(sq(n[2])), listput(v,n[1]^2))); forsquarefree(n=1,sqrtint(lim\2), if(is(double(sq(n[2]))), listput(v,2*n[1]^2))); Set(v) \\ Charles R Greathouse IV, Feb 05 2018

Formula

Solutions to abs(A008683(A000203(x)*A000010(x))) = 1.

A049197 Squarefree numbers whose Euler totient function is not squarefree.

Original entry on oeis.org

5, 10, 13, 15, 17, 19, 21, 26, 29, 30, 33, 34, 35, 37, 38, 39, 41, 42, 51, 53, 55, 57, 58, 61, 65, 66, 69, 70, 73, 74, 77, 78, 82, 85, 87, 89, 91, 93, 95, 97, 101, 102, 105, 106, 109, 110, 111, 113, 114, 115, 119, 122, 123, 127, 129, 130, 133, 137, 138, 141, 143
Offset: 1

Views

Author

Labos Elemer

Keywords

Comments

Numbers k such that abs(mu(k)) = 1 and mu(phi(k)) = 0.

Examples

			110 = 2*5*11 is squarefree but phi(110) = 40 is divisible by a square, so 110 is in the sequence.

Links

Crossrefs

Cf. A000010, A005117, A013929, A049195, A049196.

Programs

  • Mathematica
    Select[Range[144], SquareFreeQ[#] && !SquareFreeQ[EulerPhi[#]] &] (* Amiram Eldar, Feb 12 2021 *)
  • PARI
    isok(k) = issquarefree(k) && !issquarefree(eulerphi(k)); \\ Michel Marcus, Feb 12 2021

A173617 Numbers k such that phi(tau(k)) = rad(k).

Original entry on oeis.org

1, 4, 8, 32, 36, 192, 288, 768, 972, 1458, 5120, 13122, 326592, 19531250, 22588608, 46137344, 171532242, 110000000000, 132799613957120, 1618481116086272, 506590324238281250, 8358680908399640576, 162805498773679522226642, 198263834416799184651812864, 7852841179377049820122874642432, 4299870835974154129876817272635392
Offset: 1

Views

Author

Michel Lagneau, Feb 22 2010

Keywords

Comments

rad(n) is the product of the primes dividing n (A007947), tau(n) is the number of divisors of n (A000005), and phi(n) is Euler totient function (A000010).
Numbers k such that A163109(k) = A007947(k).
a(18) > 10^10. - Donovan Johnson, Jul 27 2011
From Amiram Eldar, Feb 08 2025: (Start)
1 is the only odd term in this sequence.
The number of terms with any given number of divisors is finite.
There are no terms whose number of divisors d equals 2 or in A049195, or when omega(phi(d)) > bigomega(d), where omega = A001221 and bigomega = A001222.
If p is a Sophie Germain prime (A005384), then 2*p^(2*p) is a term. (End)

Examples

			8 is a term since tau(8) = 4, phi(4) = 2 and rad(8) = 2.
13122 is a term  tau(13122) = 18, phi(18) = 6 and rad(13122) = 6.

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.

Links

Crossrefs

Cf. A000005 (tau), A000010 (phi), A007947 (rad).
Cf. A001221, A001222, A001414, A005384, A049195, A062069, A163109, A173320.

Programs

  • Maple
    with(numtheory):for n from 1 to 1000000 do :t1:= ifactors(n)[2] : t2 :=mul(t1[i][1], i=1..nops(t1)): if phi(tau(n)) = t2 then print (n): else fi : od :
  • Mathematica
    (* First program: see the links section. *)
(* Second program: *)
q[k_] := k == 1 || EvenQ[k] && Module[{f = FactorInteger[k]}, EulerPhi[Times @@ (f[[;;,2]] + 1)] == Times @@ f[[;;, 1]]]; Select[Range[400000], q] (* Amiram Eldar, Feb 08 2025 *)
  • PARI
    isok(k) = if(k == 1, 1, if(k % 2, 0, my(f=factor(k)); eulerphi(numdiv(f)) == vecprod(f[,1]))); \\ Amiram Eldar, Feb 08 2025

Extensions

a(14)-a(17) from Donovan Johnson, Jul 27 2011
a(18)-a(26) from Amiram Eldar, Feb 08 2025
Showing 1-3 of 3 results.

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