A225983 -id:A225983 - cp's OEIS frontend

A357917 a(n) is the least k such that phi(k) + d(k) = A357916(n), where phi(k) = A000010(k) is Euler's totient function, and d(k) = A000005(k) is the number of divisors of k.

1, 2, 4, 16, 25, 81, 121, 256, 484, 1296, 529, 1024, 1600, 2116, 2401, 7744, 11664, 5041, 7225, 11236, 20164, 10201, 25600, 12769, 30976, 46656, 21025, 17161, 44944, 51076, 29929, 84100, 73984, 36481, 75076, 107584, 54289, 63001, 87025, 69169, 101761, 126025, 215296, 256036, 252004, 295936

Offset: 1

J. M. Bergot and Robert Israel, Oct 19 2022

nonn

Numbers k such that A061468(k) = phi(k) + d(k) is prime, and no smaller number gives the same value of A061468, sorted in order of the prime values.
All terms except 2 are squares, because if k > 2, phi(k) is even, and if d(k) is odd, k must be a square.
All numbers in this sequence are elements of A225983. For an example, this excludes all numbers of the form (6*m)^2 but only if m is not divisible by 6. - Thomas Scheuerle, Oct 20 2022

			a(4) = 16 because phi(16) + d(16) = 8 + 5 = 13 = A357916(4), and no smaller number than 16 works.

Robert Israel, Table of n, a(n) for n = 1..3000

Cf. A000005, A000010, A061468, A225983, A357916.

N:= 10^6:
pmax:=  evalf(N/(exp(gamma)*log(log(N))+3/log(log(N))));
V:= 'V': P:= {3}: V[3]:= 2:
for k from 1 to sqrt(N) do
  n:= k^2;
  v:= numtheory:-phi(n)+numtheory:-tau(n);
  if v <= pmax and isprime(v) and not member(v,P) then
    P:= P union {v}; V[v]:= n;
  fi
od:
P:= sort(convert(P,list)):
seq(V[p], p=P);

A061468(a(n)) = A000010(a(n)) + A000005(a(n)) = A357916(n).

A298080 Integers m such that both phi(m) and sigma(m) are coprime to tau(m).

Original entry on oeis.org

1, 2, 4, 16, 25, 64, 81, 100, 121, 256, 289, 484, 529, 729, 841, 1024, 1156, 1296, 1600, 1681, 2116, 2209, 2401, 2809, 3025, 3364, 3481, 4096, 4624, 5041, 5184, 6400, 6724, 6889, 7225, 7921, 8464, 8836, 10201, 11236, 11449, 11664, 12100, 12769, 13225, 13456, 13924

Offset: 1

Michel Marcus, Jan 12 2018

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and Imre Kátai, On the coprimality of some arithmetic functions, Publications de l'Institut Mathématique, 2010 87(101):121-128.

Intersection of A046678 and A225983.

Select[Range[14000], CoprimeQ[EulerPhi[#], (d = DivisorSigma[0, #])] && CoprimeQ[DivisorSigma[1, #], d] &] (* Amiram Eldar, Aug 08 2020 *)

isok(n) = (gcd(eulerphi(n), numdiv(n))==1) && (gcd(sigma(n), numdiv(n)) == 1);

cp's OEIS Frontend

A357917 a(n) is the least k such that phi(k) + d(k) = A357916(n), where phi(k) = A000010(k) is Euler's totient function, and d(k) = A000005(k) is the number of divisors of k.

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