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.

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A062821 Number of divisors of totient of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 4, 3, 6, 4, 4, 4, 5, 4, 6, 4, 6, 4, 4, 4, 6, 6, 6, 6, 6, 4, 8, 5, 6, 5, 8, 6, 9, 6, 8, 5, 8, 6, 8, 6, 8, 4, 4, 5, 8, 6, 6, 8, 6, 6, 8, 8, 9, 6, 4, 5, 12, 8, 9, 6, 10, 6, 8, 6, 6, 8, 8, 8, 12, 9, 8, 9, 12, 8, 8, 6, 8, 8, 4, 8, 7, 8, 8, 8, 8, 8, 12, 6, 12, 4, 12, 6, 12, 8, 12, 8
Offset: 1

Views

Author

Labos Elemer, Jul 20 2001

Keywords

Examples

			The number of divisors of phi(n) can be greater than, less than, or equal to the number of divisors of n:
.
  n   phi(n)  d(phi(n))   d(n)
  ==  ======  =========   ====
  10     4        3     <   4
  11    10        4     >   2
  28    12        6     =   6

Links

Crossrefs

Cf. A000005, A000010, A001620, A276044.

Programs

  • Mathematica
    Array[DivisorSigma[0,EulerPhi[#]]&,110] (* Harvey P. Dale, Jul 13 2012 *)
  • PARI
    a(n) = numdiv(eulerphi(n)); \\ Harry J. Smith, Aug 11 2009

Formula

a(n) = A000005(A000010(n)).
Sum_{k=1..n} a(k) ~ n * exp(c(n) * (log(n)/log(log(n)))^(1/2) * (1 + O(log(log(log(n)))/log(log(n))))), where c(n) is a number in the interval (1/7, 2*sqrt(2))*exp(-gamma/2) and gamma is A001620 (Luca and Pomerance, 2007). - Amiram Eldar, Oct 29 2022

Extensions

Offset corrected by Jaroslav Krizek, Jul 24 2009
Edited by Jon E. Schoenfield, Nov 13 2016

A281627 a(n) is the smallest number k such that sigma(phi(k)) = A062402(k) is the n-th Mersenne prime (A000668(n)), or 0 if no such k exists.

Original entry on oeis.org

3, 5, 17, 85, 4369, 65537, 327685, 1431655765, 2305843009213693952, 618970019642690137449562112, 162259276829213363391578010288128, 170141183460469231731687303715884105728
Offset: 1

Views

Author

Jaroslav Krizek, Feb 11 2017

Keywords

Comments

Conjecture 1: If A062402(n) = A000203(A000010(n)) = sigma(phi(n)) is a prime p for some n, then p is Mersenne prime (A000668); a(n) > 0 for all n.
Conjecture 2: a(n) = the smallest number k such that phi(k) has exactly A000043(n)-1 divisors; see A276044.
Conjecture 3: a(n) = the smallest number k such that phi(k) has exactly A000043(n)-1 prime factors (counted with multiplicity); see A275969.
a(n) <= A000668(n) for n = 1-18; conjecture: a(n) <= A000668(n) for all n.
Equals A002181 (index in A002202 of (intersection of A023194 and A002202)). - Michel Marcus, Feb 12 2017

Links

Crossrefs

Cf. A000010, A000043, A000203, A000668, A062402, A062514, A275969, A276044.
Cf. A002181, A002202, A023194.
Cf. A053576 (includes the first 13 known terms of this sequence).

Programs

  • Magma
    A281627:=func; Set(Sort([A281627(n): n in [SumOfDivisors(EulerPhi(n)): n in[1..1000000] | IsPrime(SumOfDivisors(EulerPhi(n)))]]));
  • PARI
    terms() = {v = readvec("b023194.txt"); for(i=1, #v, if (istotient(v[i], &n), print1(n/2, ", ")););} \\ Michel Marcus, Feb 12 2017
  • PARI
    f(p) = {my(s = invsigma(p), t, minv = 0); for(i = 1 ,#s, t = invphi(s[i]); for(j = 1, #t, if(minv == 0, minv = t[j]); if(t[j] < minv, minv = t[j]))); minv;} \\ using Max Alekseyev's invphi.gp
list(pmax) = forprime(p = 1, pmax, if(isprime(2^p-1), print1(f(2^p-1), ", "))); \\ Amiram Eldar, Dec 23 2024

Extensions

a(8) from Michel Marcus, Feb 12 2017
a(9)-a(12) from Amiram Eldar, Dec 23 2024
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