A252665 -id:A252665 - cp's OEIS frontend

A291834 First positions of records in A252665.

1, 4, 8, 12, 16, 24, 36, 48, 72, 96, 120, 144, 216, 240, 288, 360, 432, 480, 576, 720, 1080, 1440, 2160, 2520, 2880, 3600, 4320, 5040, 7200, 7560, 8640, 10080, 14400, 15120, 20160, 25200, 30240, 40320, 50400, 60480, 75600, 80640, 90720, 100800, 120960, 151200, 181440

Offset: 1

Michael De Vlieger, Sep 03 2017

nonn

Distinct from A033833; first term not in A033833 is a(24) = 2520. There appear to be increasingly many terms a(n) not in A033833 as n increases.
The terms 2520, 7560, 25200, 221760, 665280, 8648640, ... are not in A033833 but are in A002182. The term 3600 is the smallest that is in neither A033833 nor A002182, but in A007416. The term 831600 is the smallest that is in none of the three aforementioned sequences.
Conjectures based on a(n) < 10^7:
1. Numbers in a(n) are products of the first several consecutive primes p.
2. Outside of a(1), the least prime factor of a(n) has multiplicity > 1. This implies no primes, primorials, or squarefree a(n) for n > 1.
3. The greatest prime factor of a(n) generally has multiplicity 1. Note, however, exceptions in a(n) for n = {1, 2, 3, 5, 7, 9, 12, 13, 15, 17, 19, 26, 29, 33, 73, ...}.
4. The multiplicities of prime factors p of m generally decrease or stay the same as p increases.
See "Records and first positions of records in A252665" for more information. - Michael De Vlieger, Sep 03 2017

Michael De Vlieger, Table of n, a(n) for n = 1..76

Cf. A002182, A007416, A033833, A252665, A291833.

With[{s = Array[f[#, #, 5] &, 10^4]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]]

A291833 Records transform of A252665.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 12, 16, 18, 21, 28, 30, 37, 43, 51, 53, 59, 66, 92, 103, 150, 188, 189, 235, 239, 312, 351, 396, 400, 493, 593, 628, 751, 947, 954, 1283, 1433, 1632, 2063, 2074, 2088, 2446, 2629, 3143, 3582, 3952, 4008, 4121, 4602, 5803, 6046, 6323, 6899

Offset: 1

Michael De Vlieger, Sep 03 2017

nonn

See a-file "Records and first positions of records in A252665" in that sequence for more information. - Michael De Vlieger, Sep 03 2017

			A252665(n) for 1 <= n <= 24 is {1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7}; the records are {1, 2, 3, 4, 5, 7}, thus these are the first 6 terms of this sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..76

Cf. A033833, A252665, A291834.

f[n_, i_, t_] := b[n, i, t] = If[n == 1, 1, If[t == 1, Boole[n <= i],
Sum[f[n/d, d, t - 1], {d, Select[Divisors@ n, # <= i &]}]]]; Union@ FoldList[Max, Array[f[#, #, 5] &, 10^5]] (* Michael De Vlieger, Sep 03 2017, after Alois P. Heinz at A218320 *)

A355030 a(n) is the number of possible values of the number of prime divisors (counted with multiplicity) of numbers with n divisors.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 11, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 11, 2, 5, 1, 4, 2, 5, 1, 14, 1, 2, 4, 4, 2, 5, 1, 11, 5, 2, 1, 11, 2

Offset: 1

Amiram Eldar, Jun 16 2022

nonn

First differs from A305254 at n = 40, from A001055 and A252665 at n = 36, from A218320 at n = 32 and from A317791, A318559 and A326334 at n = 30.

			a(2) = 1 since numbers with 2 divisors are primes, i.e., numbers k with the single value Omega(k) = 1.
a(4) = 2 since numbers with 4 divisors are either of the following 2 forms: p1 * p2 with p1 and p2 being distinct primes, or of the form p^3 with p prime.
a(8) = 3 since numbers with 8 divisors are either of the following 3 forms: p1 * p2 * p3 with p1, p2 and p3 being distinct primes, p1 * p2^3, or p1^7.

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

Row lengths of A355029.
Cf. A000005, A001222, A118914, A162247, A355027.
Cf. A001055, A218320, A305254, A317791, A318559, A326334.

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

a(n) <= A001055(n).
a(p) = 1 for p prime.
a(A355031(n)) = n.

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A291834 First positions of records in A252665.

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A291833 Records transform of A252665.

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A355030 a(n) is the number of possible values of the number of prime divisors (counted with multiplicity) of numbers with n divisors.

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