A355026 -id:A355026 - cp's OEIS frontend

A355027 a(n) is the number of possible values of numbers of divisors of numbers k with Omega(k) = n.

1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 39, 49, 66, 84, 108, 136, 171, 211, 259, 320, 386, 468, 565, 674, 801, 954, 1117, 1333, 1556, 1831, 2107, 2478, 2838, 3309, 3788, 4396, 4979, 5780, 6511, 7506, 8451, 9696, 10834, 12429, 13846, 15814, 17576, 20030, 22123, 25179

Offset: 0

Amiram Eldar, Jun 16 2022

nonn

			For n = 2, the numbers k with Omega(k) = 2 are either of the form p^2 with p prime, or of the form p1*p2 with p1 and p2 being distinct primes. The corresponding numbers of divisors are 3 and 4, respectively. Since there are 2 possible values, a(2) = 2.
For n = 8, there are 22 prime signatures of numbers k with Omega(k) = 8, corresponding to the number of partitions of 8. However, the prime signatures p1^5 * p2 * p3 * p4 and p1^3 * p2^3 * p3^2 both correspond to the same number of divisors, 48. Therefore, there are only 21 distinct possible values of the number of divisors, and a(8) = 21.

Amiram Eldar, Table of n, a(n) for n = 0..90

Row lengths of A355026.

Cf. A000005, A000041, A001222.

a[n_] := Length @ Union[Times @@ (# + 1) & /@ IntegerPartitions[n]]; Array[a, 50, 0]

a(n) = { my (m=Map()); forpart(p=n, mapput(m,prod(k=1, #p, 1+p[k]),0)); #m } \\ Rémy Sigrist, Jun 17 2022

a(n) <= A000041(n).

A355029 Irregular table read by rows: the n-th row gives the possible values of the number of prime divisors (counted with multiplicity) of numbers with n divisors.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 3, 5, 6, 3, 4, 7, 4, 8, 5, 9, 10, 4, 5, 6, 11, 12, 7, 13, 6, 14, 4, 5, 6, 8, 15, 16, 5, 7, 9, 17, 18, 6, 7, 10, 19, 8, 20, 11, 21, 22, 5, 6, 7, 8, 9, 12, 23, 8, 24, 13, 25, 6, 10, 26, 8, 9, 14, 27, 28, 7, 9, 11, 15, 29, 30, 5, 6, 7, 9, 10, 16, 31

Offset: 1

Amiram Eldar, Jun 16 2022

The n-th row begins with A059975(n) and ends with n-1.

			Table begins:
  0;
  1;
  2;
  2, 3;
  4;
  3, 5;
  6;
  3, 4, 7;
  4, 8;
  5, 9;
  ...
Numbers k with 4 divisors are either of the form p1 * p2 with p1 and p2 being distinct primes, or of the form p^3 with p prime. The corresponding numbers of prime divisors (counted with multiplicity) are 2 and 3, respectively. Therefore, the 4th row is {2, 3}.

Amiram Eldar, Table of n, a(n) for n = 1..6758 (rows 1..1000, flattened)

Cf. A000005, A001222, A059975, A118914, A162247, A355026, A355030 (row lengths).

Table[Union[Total[#-1]& /@ f[n]], {n, 1, 32}] // Flatten (* using the function f by T. D. Noe at A162247 *)

A355028 a(n) is the maximum number of distinct prime signatures of numbers with n prime divisors (counted with multiplicity) that have the same number of divisors.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7, 9, 9, 11, 12, 13, 14, 16, 17, 19, 23, 26, 30, 32, 35, 38, 40, 43, 51, 56, 62, 68, 74, 80, 86, 94, 106, 118, 128, 140, 152, 167, 179, 197, 221, 247, 272, 298, 325, 353, 384, 425, 473, 522, 567, 631

Offset: 0

Amiram Eldar, Jun 16 2022

nonn

			a(8) = 2 since there are 2 prime signatures of numbers k with Omega(k) = 8, p1^5 * p2 * p3 * p4 and p1^3 * p2^3 * p3^2, that correspond to the same number of divisors, 48.

Amiram Eldar, Table of n, a(n) for n = 0..90

Cf. A000005, A001222, A101296, A118914, A355026, A355027.

a[n_] := Max[Tally[Times @@ (# + 1) & /@ IntegerPartitions[n]][[;; , 2]]]; Array[a, 50, 0]

cp's OEIS Frontend

A355027 a(n) is the number of possible values of numbers of divisors of numbers k with Omega(k) = n.

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A355029 Irregular table read by rows: the n-th row gives the possible values of the number of prime divisors (counted with multiplicity) of numbers with n divisors.

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Examples

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Mathematica

A355028 a(n) is the maximum number of distinct prime signatures of numbers with n prime divisors (counted with multiplicity) that have the same number of divisors.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica