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

Author

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

Keywords

• nonn

Comments

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

Examples

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

Links

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

Crossrefs

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

Programs

Maple

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);

Formula

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