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A372740
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Coreful untouchable numbers: numbers that are not the sum of aliquot coreful divisors (A336563) of any number.
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3
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1, 4, 8, 9, 16, 20, 25, 27, 28, 32, 40, 44, 45, 48, 49, 50, 52, 54, 63, 64, 68, 72, 75, 76, 81, 88, 92, 99, 100, 104, 108, 116, 117, 121, 124, 125, 128, 136, 144, 147, 148, 152, 153, 160, 162, 164, 169, 171, 172, 175, 176, 184, 188, 189, 192, 196, 200, 207, 208
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OFFSET
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1,2
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COMMENTS
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A coreful divisor d of n is a divisor that is divisible by every prime that divides n (see also A307958).
Numbers k such that A372739(k) = 0.
Numbers that are not in the range of A336563.
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.
Includes all the squares of primes (A001248).
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... .
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LINKS
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MATHEMATICA
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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]
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PROG
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(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; }
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, ", "))); }
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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