A355030 -id:A355030 - cp's OEIS frontend

A355031 a(n) is the least number k such that A355030(k) = n, or -1 if no such k exists.

1, 4, 8, 12, 16, 40, 24, 36, 90, 126, 48, 112, 546, 72, 108, 96, 160, 352, 168, 120, 256, 2475, 144, 588, 300, 320, 216, 448, 1216, 240, 810, 420, 288, 1040, 384, 660, 360, 640, 432, 1408, 540, 504, 480, 600, 648, 1176, 792, 672, 1500, 576, 2000, 900, 1824, 1248

Offset: 1

Amiram Eldar, Jun 16 2022

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A001222, A162247, A355029, A355030.

m[n_] := Length[Union[Total[#-1]& /@ f[n]]]; seq[len_, max_] := Module[{s = Table[0, {len}], c = 0, n = 1, k}, While[c < len && n < max, k = m[n]; If[k <= len && s[[k]] == 0, c++; s[[k]] = n]; n++]; s]; seq[60, 10^4] (* using the function f by T. D. Noe at A162247 *)

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 *)

A355032 a(n) is the maximum number of prime signatures of numbers with n divisors that have the same number of prime divisors (counted with multiplicity).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 16 2022

nonn

			a(2) = 1 since the numbers with 2 divisors are all primes and thus have only 1 prime signature.
a(36) = 2 since numbers with 36 divisors have 2 prime signatures, p1^5 * p2^5 and p1 * p2 * p3^8, that correspond to numbers with 10 prime divisors (counted with multiplicity).
a(72) = 3 since numbers with 72 divisors have 3 prime signatures, p1 * p2^5 * p3^5, p1^2 * p2^2 * p3^7 and p1 * p2 * p3 * p4^8, that correspond to numbers with 11 prime divisors (counted with multiplicity).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A001222, A118914, A162247, A355028, A355030, A355031, A355033.

Table[Max[Tally[Total[#-1]& /@ f[n]][[;;,2]]], {n, 1, 100}] (* using the function f by T. D. Noe at A162247 *)

a(A355033(n)) = n.

A355033 a(n) is the least number k such that A355032(k) = n, or -1 if no such k exists.

Original entry on oeis.org

1, 36, 72, 288, 960, 720, 1440, 3456, 2880, 6912, 5760, 10080, 11520, 8640, 24192, 21600, 47520, 17280, 28800, 20160, 62208, 46080, 82944, 34560, 50400, 40320, 57600, 51840, 110880, 126720, 141120, 69120, 60480, 248832, 86400, 80640, 233280, 237600, 103680, 100800

Offset: 1

Amiram Eldar, Jun 16 2022

nonn

Amiram Eldar, Table of n, a(n) for n = 1..250

Cf. A000005, A001222, A162247, A355030, A355031, A355032.

m[n_] := Max[Tally[Total[# - 1] & /@ f[n]][[;; , 2]]]; seq[len_, max_] := Module[{s = Table[0, {len}], c = 0, n = 1, k}, While[c < len && n < max, k = m[n]; If[k <= len && s[[k]] == 0, c++; s[[k]] = n]; n++]; s]; seq[30, 10^6] (* using the function f by T. D. Noe at A162247 *)

cp's OEIS Frontend

A355031 a(n) is the least number k such that A355030(k) = n, or -1 if no such k exists.

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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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A355032 a(n) is the maximum number of prime signatures of numbers with n divisors that have the same number of prime divisors (counted with multiplicity).

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A355033 a(n) is the least number k such that A355032(k) = n, or -1 if no such k exists.

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