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.

%I A357917 #14 Oct 25 2022 20:04:48
%S A357917 1,2,4,16,25,81,121,256,484,1296,529,1024,1600,2116,2401,7744,11664,
%T A357917 5041,7225,11236,20164,10201,25600,12769,30976,46656,21025,17161,
%U A357917 44944,51076,29929,84100,73984,36481,75076,107584,54289,63001,87025,69169,101761,126025,215296,256036,252004,295936
%N 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.
%C A357917 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.
%C A357917 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.
%C A357917 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
%H A357917 Robert Israel, <a href="/A357917/b357917.txt">Table of n, a(n) for n = 1..3000</a>
%F A357917 A061468(a(n)) = A000010(a(n)) + A000005(a(n)) = A357916(n).
%e A357917 a(4) = 16 because phi(16) + d(16) = 8 + 5 = 13 = A357916(4), and no smaller number than 16 works.
%p A357917 N:= 10^6:
%p A357917 pmax:=  evalf(N/(exp(gamma)*log(log(N))+3/log(log(N))));
%p A357917 V:= 'V': P:= {3}: V[3]:= 2:
%p A357917 for k from 1 to sqrt(N) do
%p A357917   n:= k^2;
%p A357917   v:= numtheory:-phi(n)+numtheory:-tau(n);
%p A357917   if v <= pmax and isprime(v) and not member(v,P) then
%p A357917     P:= P union {v}; V[v]:= n;
%p A357917   fi
%p A357917 od:
%p A357917 P:= sort(convert(P,list)):
%p A357917 seq(V[p], p=P);
%Y A357917 Cf. A000005, A000010, A061468, A225983, A357916.
%K A357917 nonn
%O A357917 1,2
%A A357917 _J. M. Bergot_ and _Robert Israel_, Oct 19 2022