A331231 -id:A331231 - cp's OEIS frontend

A331201 Numbers k such that the number of factorizations of k into distinct factors > 1 is a prime number.

6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 66, 68, 69, 70, 74, 75, 76, 77, 78, 80, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 98, 99, 100, 102

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

First differs from A080257 in lacking 60.

			Strict factorizations of selected terms:
  (6)    (12)   (24)     (48)     (216)
  (2*3)  (2*6)  (3*8)    (6*8)    (3*72)
         (3*4)  (4*6)    (2*24)   (4*54)
                (2*12)   (3*16)   (6*36)
                (2*3*4)  (4*12)   (8*27)
                         (2*3*8)  (9*24)
                         (2*4*6)  (12*18)
                                  (2*108)
                                  (3*8*9)
                                  (4*6*9)
                                  (2*3*36)
                                  (2*4*27)
                                  (2*6*18)
                                  (2*9*12)
                                  (3*4*18)
                                  (3*6*12)
                                  (2*3*4*9)

The version for strict integer partitions is A035359.
The version for integer partitions is A046063.
The version for set partitions is A051130.
The non-strict version is A330991.
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
Numbers whose number of strict factorizations is odd are A331230.
Numbers whose number of strict factorizations is even are A331231.
The least number with n strict factorizations is A330974(n).
Cf. A001318, A045780, A318286, A328966, A330992, A330993, A330997, A331023/A331024, A331050, A331051, A331200, A331232.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],PrimeQ[Length[strfacs[#]]]&]

A332269 Numbers m with only one divisor d such that sqrt(m) < d < m.

Original entry on oeis.org

6, 8, 10, 14, 15, 16, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 81, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187

Offset: 1

Bernard Schott, May 04 2020

nonn

Equivalently: numbers with only one proper divisor > sqrt(n).
Also: numbers with only one nontrivial divisor d with 1 < d < sqrt(n).
Four subsequences (see examples):
1) Squarefree semiprimes (A006881) p*q with p < q, then this unique divisor is q.
2) Cube of primes p^3 (A030078), then this unique divisor is p^2.
3) Primes^4 (A030514), then this unique divisor is p^3.
4) Numbers with 4 divisors: A030513 = A006881 Union A030078.
For n = 1 to n = 21, we have a(n) = A319238(n) = A331231(n) but a(22) = 65 <> A319238(22) = A331231(22) = 64.
From Marius A. Burtea, May 07 2020: (Start)
The sequence contains terms that are consecutive numbers.
If the numbers 4*k + 1 and 6*k + 1, k >= 1, are prime numbers, then the numbers 12*k + 2 and 12*k + 3 are terms. Examples: (14, 15), (38, 39), (86, 87), (122, 123), (158, 159), (218, 219), (302, 303), ...
If the numbers 6*m + 1, 10*m + 1 and 15*m + 2, m >= 1, are prime numbers, then the numbers 30*m + 3, 30*m + 4 and 30*m + 5 are terms. Examples: (33, 34, 35), (93, 94, 95), (213, 214, 215), (393, 394, 395), (633, 634, 635), ... (End)
There are never more than 3 consecutive terms because one of them would be divisible by 4, and neither 8 nor 16 belong to such a string of 4 consecutive terms.

			The divisors of 15 are {1, 3, 5, 15} and only 5 satisfies sqrt(15) < 5 < 15, hence 15 is a term.
The divisors of 27 are {1, 3, 9, 27} and only 9 satisfies sqrt(27) < 9 < 27, hence 27 is a term.
The divisors of 16 are {1, 2, 4, 8, 16} and only 8 satisfies sqrt(16) < 8 < 16, hence 16 is a term.
The divisors of 28 are {1, 2, 4, 7, 14, 28} but 7 and 14 satisfy sqrt(28) < 7 < 14 < 28, hence 28 is not a term.

Disjoint union of A006881, A030078, and A030514.
Disjoint union of A030513 and A030514.
Cf. A000005, A038548.

[k:k in [1..200]|#[d:d in Divisors(k)|d gt Sqrt(k) and d lt k] eq 1]; // Marius A. Burtea, May 07 2020

Select[Range[200], MemberQ[{4, 5}, DivisorSigma[0, #]] &] (* Amiram Eldar, May 04 2020 *)

isok(m) = #select(x->(x^2 > m), divisors(m)) == 2; \\ Michel Marcus, May 05 2020

m is a term iff tau(m) - A038548(m) = 2 where tau = A000005.

A331230 Numbers k such that the number of factorizations of k into distinct factors > 1 is odd.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 12, 13, 17, 18, 19, 20, 23, 24, 25, 28, 29, 30, 31, 32, 36, 37, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 83, 84, 88, 89, 90, 92, 97, 98, 99, 100, 101, 102

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

First differs from A319237 in lacking 300.

The version for strict integer partitions is A001318.
The version for integer partitions is A052002.
The version for set partitions appears to be A032766.
The non-strict version is A331050.
The version for primes (instead of odds) is A331201.
The even version is A331231.
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
The least number with n strict factorizations is A330974(n).
Cf. A035359, A045780, A318286, A328966, A330991, A330997, A331051, A331200, A331232.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],OddQ[Length[strfacs[#]]]&]

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A331201 Numbers k such that the number of factorizations of k into distinct factors > 1 is a prime number.

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A332269 Numbers m with only one divisor d such that sqrt(m) < d < m.

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A331230 Numbers k such that the number of factorizations of k into distinct factors > 1 is odd.

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