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A353124 Numbers k which have a record number of non-divisors < k (i.e., A049820(k)).

%I A353124 #77 May 07 2022 10:13:49
%S A353124 1,3,5,7,9,11,13,17,19,22,23,25,27,29,31,34,35,37,41,43,46,47,49,51,
%T A353124 53,57,58,59,61,65,67,71,73,77,79,82,83,86,87,89,93,94,95,97,101,103,
%U A353124 106,107,109,113,118,119,121,123,125,127,131,134,137,139,142,143
%N A353124 Numbers k which have a record number of non-divisors < k (i.e., A049820(k)).
%C A353124 From _Jon E. Schoenfield_, Apr 30 2022: (Start)
%C A353124 This sequence includes all noncomposite numbers, the squares of all odd primes, and the cube of every odd prime p such that p^3 - 2 is composite.
%C A353124 It also includes every number k of the form p*q, with p and q distinct primes, such that k-2 is composite and k-1 is neither a prime nor the square of a prime.
%C A353124 In general, it includes every number k such that tau(k-j) > tau(k) - j for each j in 1..tau(k)-1.
%C A353124 Terms with larger numbers of divisors occur less frequently. The first terms with 0, 1, 2, 3, and 4 distinct prime factors are 1, 3, 22, 2110, and 17585778, respectively (each of which is squarefree). What is the first term with 5 distinct prime factors?
%C A353124 (End)
%H A353124 Johan Lindgren, <a href="/A353124/b353124.txt">Table of n, a(n) for n = 1..10000</a>
%t A353124 s = {}; fm = -1; Do[f = n - DivisorSigma[0, n]; If[f > fm, fm = f; AppendTo[s, n]], {n, 1, 120}]; s (* _Amiram Eldar_, Apr 25 2022 *)
%o A353124 (C++)
%o A353124 int main() {
%o A353124    int kMax = 500, kRecord = -1;
%o A353124    for ( int k = 1; k < kMax; k++) {
%o A353124       int nonDivisorCount = 0;
%o A353124       for ( int d = 2; d < k; d++ ) { nonDivisorCount += (k % d != 0); }
%o A353124       if ( nonDivisorCount > kRecord ) {
%o A353124          kRecord = nonDivisorCount;
%o A353124          cout << k << "\n";
%o A353124       }
%o A353124    }
%o A353124    return 0;
%o A353124 }
%o A353124 (PARI) f(n) = n - numdiv(n); \\ A049820
%o A353124 lista(nn) = {my(m=-oo, list=List(), fn); for (n=1, nn, if ((fn=f(n)) > m, listput(list, n); m = fn;);); Vec(list);} \\ _Michel Marcus_, Apr 25 2022
%Y A353124 Cf. A049820, A173540.
%K A353124 nonn
%O A353124 1,2
%A A353124 _Johan Lindgren_, Apr 24 2022