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A372740 Coreful untouchable numbers: numbers that are not the sum of aliquot coreful divisors (A336563) of any number.

%I A372740 #7 May 12 2024 11:19:28
%S A372740 1,4,8,9,16,20,25,27,28,32,40,44,45,48,49,50,52,54,63,64,68,72,75,76,
%T A372740 81,88,92,99,100,104,108,116,117,121,124,125,128,136,144,147,148,152,
%U A372740 153,160,162,164,169,171,172,175,176,184,188,189,192,196,200,207,208
%N A372740 Coreful untouchable numbers: numbers that are not the sum of aliquot coreful divisors (A336563) of any number.
%C A372740 A coreful divisor d of n is a divisor that is divisible by every prime that divides n (see also A307958).
%C A372740 Numbers k such that A372739(k) = 0.
%C A372740 Numbers that are not in the range of A336563.
%C A372740 Except for 1, all the terms are not squarefree (A013929), because if k is squarefree (A005117), and there is a prime p such that p|k, then A336563(p*k) = k.
%C A372740 Includes all the squares of primes (A001248).
%C A372740 The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are , 4, 29, 281, 2762, 27690, ... . Apparently, the asymptotic density of this sequence exists and equals 0.27... .
%H A372740 Amiram Eldar, <a href="/A372740/b372740.txt">Table of n, a(n) for n = 1..10000</a>
%t A372740 f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; seq[max_] := Module[{v = Table[0, {max}], i}, Do[i = s[k]; If[0 < i <= max, v[[i]]++], {k, 1, max^2}]; Position[v, _?(# == 0 &)] // Flatten]; seq[200]
%o A372740 (PARI) s(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2] + 1) - 1)/(f[i, 1] - 1) - 1) - n;}
%o A372740 lista(nmax) = {my(v = vector(nmax), i); for(k = 1, nmax^2, i = s(k); if(i > 0 && i <= nmax, v[i]++)); for(k = 1, nmax, if(v[k] == 0, print1(k, ", ")));}
%Y A372740 Cf. A005117, A013929, A307958, A336563, A372739.
%Y A372740 A001248 is a subsequence.
%Y A372740 Similar sequences: A005114, A063948 (unitary), A324276 (bi-unitary), A324277 (infinitary).
%K A372740 nonn
%O A372740 1,2
%A A372740 _Amiram Eldar_, May 12 2024