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

1, 3, 5, 7, 9, 11, 13, 17, 19, 22, 23, 25, 27, 29, 31, 34, 35, 37, 41, 43, 46, 47, 49, 51, 53, 57, 58, 59, 61, 65, 67, 71, 73, 77, 79, 82, 83, 86, 87, 89, 93, 94, 95, 97, 101, 103, 106, 107, 109, 113, 118, 119, 121, 123, 125, 127, 131, 134, 137, 139, 142, 143

Offset

1,2

Comments

From Jon E. Schoenfield, Apr 30 2022: (Start)

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.

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.

In general, it includes every number k such that tau(k-j) > tau(k) - j for each j in 1..tau(k)-1.

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?

(End)

Links

Johan Lindgren, Table of n, a(n) for n = 1..10000

Mathematica

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 *)

Prog

(C++)
int main() {
   int kMax = 500, kRecord = -1;
   for ( int k = 1; k < kMax; k++) {
      int nonDivisorCount = 0;
      for ( int d = 2; d < k; d++ ) { nonDivisorCount += (k % d != 0); }
      if ( nonDivisorCount > kRecord ) {
         kRecord = nonDivisorCount;
         cout << k << "\n";
      }
   }
   return 0;
}
(PARI) f(n) = n - numdiv(n); \\ A049820
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

Crossrefs

Cf. A049820, A173540.

Sequence in context: A349177 A323550 A165468 * A354149 A371533 A112372

Adjacent sequences: A353121 A353122 A353123 * A353125 A353126 A353127

Keyword

nonn

Author

Johan Lindgren, Apr 24 2022

Status

approved