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.

A069822 Numbers k for which there is at least one x < k such that sigma(x) = sigma(k).

Original entry on oeis.org

11, 15, 17, 23, 25, 26, 31, 35, 38, 39, 41, 46, 47, 51, 53, 55, 56, 58, 59, 62, 65, 69, 70, 71, 75, 77, 78, 79, 82, 83, 87, 89, 92, 94, 95, 97, 103, 107, 110, 113, 115, 116, 118, 119, 122, 123, 124, 125, 127, 130, 131, 135, 139, 140, 141, 142, 143, 145, 151, 153
Offset: 1

Views

Author

Benoit Cloitre, Apr 28 2002

Keywords

Comments

Numbers k for which A263025(k) > 1. - Antti Karttunen, Dec 08 2017

Links

Crossrefs

Cf. A263025, A296214.
Cf. A296087 (a subsequence).

Programs

  • PARI
    for(s=1,200,if(prod(i=1,s-1, sigma(i)-sigma(s))==0,print1(s,",")))
  • PARI
    for(n=1,16384,y=0;s=sigma(n);for(k=1,(n-1),if(sigma(k)==s,y=1;break)); if(y,i++;write("b069822.txt", i," ", n))); \\ Antti Karttunen, Dec 08 2017
  • PARI
    is(k) = invsigmaMin(sigma(k)) < k; \\ Amiram Eldar, Dec 20 2024, using Max Alekseyev's invphi.gp

A296087 Numbers n such that there is k < n for which A003557(k) = A003557(n), A048250(k) = A048250(n) and A173557(k) = A173557(n).

Original entry on oeis.org

15265, 27962, 30217, 30530, 45795, 50541, 54379, 54905, 57598, 60434, 61060, 64255, 66526, 72357, 72713, 89585, 90651, 91590, 101082, 101949, 108758, 109810, 120868, 122120, 128510, 136555, 137385, 137883, 138761, 144714, 145426, 149739, 151085, 152633, 161386, 163137, 164715, 166315, 179170, 181302, 181543, 182942
Offset: 1

Views

Author

Antti Karttunen, Dec 08 2017

Keywords

Comments

Because Euler phi(n) = A000010(n) = A003557(n) * A173557(n), Dedekind psi(n) = A001615(n) = A003557(n) * A048250(n), and because also sigma(n) (A000203) can be computed from those three elements (see A291750), these numbers form also a subset of the positions of such duplicated occurrences of values computed for those functions. See for example A069822 and A296214.
a(11) = 61060 is the first term that is not squarefree.

Examples

			15265 is a term because A003557(15265) = 1 = A003557(15169), A048250(15265) = 19008 = A048250(15169), A173557(15265) = 11760 = A173557(15169).
27962 is a term because A003557(27962) = 1 = A003557(26355), A048250(27962) = 48384 = A048250(26355), A173557(27962) = 12000 = A173557(26355).

Links

Crossrefs

Cf. A000010, A001615, A003557, A048250, A173557.
Subsequence of A069822 and of A296214.

Programs

  • PARI
    search_up_to = (2^23);
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ This function from Michel Marcus, Oct 31 2017
Anotsubmitted1(n) = (1/2)*(2 + ((A003557(n)+A173557(n))^2) - A003557(n) - 3*A173557(n));
Akaikki3(n) = (1/2)*(2 + ((A048250(n)+Anotsubmitted1(n))^2) - A048250(n) - 3*Anotsubmitted1(n));
om = Map(); m = 0; i=0; for(n = 1, search_up_to, k = Akaikki3(n); if(!mapisdefined(om,k), mapput(om,k,n), i++; write("b296087.txt", i, " ", n)));

A074987 a(n) is the least m not equal to n such that phi(m) = phi(n).

Original entry on oeis.org

2, 1, 4, 3, 8, 3, 9, 5, 7, 5, 22, 5, 21, 7, 16, 15, 32, 7, 27, 15, 13, 11, 46, 15, 33, 13, 19, 13, 58, 15, 62, 17, 25, 17, 39, 13, 57, 19, 35, 17, 55, 13, 49, 25, 35, 23, 94, 17, 43, 25, 64, 35, 106, 19, 41, 35, 37, 29, 118, 17, 77, 31, 37, 51, 104, 25, 134, 51, 92, 35, 142
Offset: 1

Views

Author

Joseph L. Pe, Oct 02 2002

Keywords

Comments

In 1922, Carmichael asked if for any given natural number n there exists a natural number m different from n such that phi(m) = phi(n). A. Schlafly and S. Wagon showed in 1994 that if there is an n such that phi(m) != phi(n) for all m distinct from n, then n must be greater than 10^(10^7). [Improved to 10^(10^10) by Kevin Ford. - Pontus von Brömssen, May 15 2020]
I conjecture that a(n) <= 2n. I have checked this for all n <= 10^4. (It is not possible to do better than the 2n upper bound since a(11) = 2*11.)
For odd n the conjecture is true because phi(n)=phi(2n). - T. D. Noe, Oct 18 2006
From Robert Israel, Aug 12 2016: (Start)
If a(n) > n then a(a(n)) = n.
If n is in A138537 then a(n) = 2*n. (End)
From David A. Corneth, May 12 2018: (Start)
A210719 has values n such that a(n) > n, so a(A210719(n)) = n.
Its complement, A296214, has values n such that a(n) < n. (End)

Examples

			phi(5) = 4 and 8 is the least natural number k different from 5 such phi(k) = 4. Hence phi(5) = 8.

References

  • J. Tattersall, "Elementary Number Theory in Nine Chapters", Cambridge University Press, 2001, pp. 162-163.

Links

Crossrefs

Cf. A138537, A210719, A296214.

Programs

  • Maple
    N:= 1000: # to get a(n) for n <= N
todo:= N;
for n from 1 while todo > 0 do
  v:= numtheory:-phi(n);
  if assigned(R[v]) then
    if n <= N then
      A[n]:= R[v]; todo:= todo-1;
    fi;
    if R[v] <= N and not assigned(A[R[v]])  then
      A[R[v]]:= n; todo:= todo-1;
    fi;
  else
    R[v]:= n
  fi
od:
seq(A[n],n=1..N); # Robert Israel, Aug 12 2016
  • Mathematica
    l = {}; Do[ e = EulerPhi[n]; i = 1; While[e != EulerPhi[i] || n == i, i++ ]; l = Append[l, i], {n, 1, 100}]; l
(* Second program: *)
Module[{nn=300,lst},lst=Table[{n,EulerPhi[n]},{n,nn}];Take[Table[ SelectFirst[ lst,#[[2]] == lst[[k,2]] && #[[1]]!=lst[[k,1]]&],{k,nn}],100]][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 23 2020 *)
  • PARI
    a(n) = my(t=eulerphi(n), m=1); while ((eulerphi(m) != t) || (m==n), m++); m; \\ Michel Marcus, May 15 2020
  • Python
    from sympy import totient
def A074987(n):
  m=1
  while totient(m)!=totient(n) or m==n:
    m+=1
  return m # Pontus von Brömssen, May 15 2020
Showing 1-3 of 3 results.

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