A323055 -id:A323055 - cp's OEIS frontend

A332785 Nonsquarefree numbers that are not squareful.

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 204, 207, 208, 212, 220, 224

Offset: 1

Bernard Schott, Feb 24 2020

Sometimes nonsquarefree numbers are misnamed squareful numbers (see 1st comment of A013929). Indeed, every squareful number > 1 is nonsquarefree, but the converse is false. This sequence = A013929 \ A001694 and consists of these counterexamples.
This sequence is not a duplicate: the first 16 terms (<= 68) are the same first 16 terms of A059404, A323055, A242416 and A303946, then 72 is the 17th term of these 4 sequences. Also, the first 37 terms (<= 140) are the same first 37 terms of A317616 then 144 is the 38th term of this last sequence.
From Amiram Eldar, Sep 17 2023: (Start)
Called "hybrid numbers" by Jakimczuk (2019).
These numbers have a unique representation as a product of two numbers > 1, one is squarefree (A005117) and the other is powerful (A001694).
Equivalently, numbers k such that A055231(k) > 1 and A057521(k) > 1.
Equivalently, numbers that have in their prime factorization at least one exponent that is equal to 1 and at least one exponent that is larger than 1.
The asymptotic density of this sequence is 1 - 1/zeta(2) (A229099). (End)

			18 = 2 * 3^2 is nonsquarefree as it is divisible by the square 3^2, but it is not squareful because 2 divides 18 but 2^2 does not divide 18, hence 18 is a term.
72 = 2^3 * 3^2 is nonsquarefree as it is divisible by the square 3^2, but it is also squareful because primes 2 and 3 divide 72, and 2^2 and 3^2 divide also 72, so 72 is not a term.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Powerful Numbers Multiple of a Set of Primes and Hybrid Numbers, 2019.

Cf. A005117 (squarefree), A013929 (nonsquarefree), A001694 (squareful), A052485 (not squareful).

Cf. A059404, A126706, A229099, A242416, A286708, A303946, A317616, A323055 (first terms are the same).

filter:= proc(n) local F;
 F:= ifactors(n)[2][..,2];
 max(F) > 1 and min(F) = 1
end proc:
select(filter, [$1..1000]); # Robert Israel, Sep 15 2024

Select[Range[225], Max[(e = FactorInteger[#][[;;,2]])] > 1 && Min[e] == 1 &] (* Amiram Eldar, Feb 24 2020 *)

isok(m) = !issquarefree(m) && !ispowerful(m); \\ Michel Marcus, Feb 24 2020

from math import isqrt
from sympy import mobius, integer_nthroot
def A332785(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l, j = n-1+squarefreepi(integer_nthroot(x,3)[0])+squarefreepi(x), 0, isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c += j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c-l
    return bisection(f,n,n) # Chai Wah Wu, Sep 14 2024

This sequence is A126706 \ A286708.

Sum_{n>=1} 1/a(n)^s = 1 + zeta(s) - zeta(s)/zeta(2*s) - zeta(2*s)*zeta(3*s)/zeta(6*s), s > 1. - Amiram Eldar, Sep 17 2023

A117571 Expansion of (1+2x^2)/((1-x)(1-x^3)).

Original entry on oeis.org

1, 1, 3, 4, 4, 6, 7, 7, 9, 10, 10, 12, 13, 13, 15, 16, 16, 18, 19, 19, 21, 22, 22, 24, 25, 25, 27, 28, 28, 30, 31, 31, 33, 34, 34, 36, 37, 37, 39, 40, 40, 42, 43, 43, 45, 46, 46, 48, 49, 49, 51, 52, 52, 54, 55, 55, 57, 58, 58, 60, 61, 61, 63, 64, 64, 66, 67, 67, 69, 70, 70, 72

Offset: 0

Paul Barry, Mar 29 2006

Row sums of A116948.
Place n+2 equally-spaced points around a circle, labeled 0,1,2,...,n+1. For each i = 0..n+1 such that 2i != i mod n+2, draw an (undirected) chord from i to 2i mod n+2. Then a(n) is the number of distinct chords. - Kival Ngaokrajang, May 13 2016 (Edited by N. J. A. Sloane, Jun 23 2016)
From Gus Wiseman, Apr 19 2019: (Start)
Also the number of integer partitions of n + 2 with 1 fewer distinct multiplicities than (not necessarily distinct) parts. These are partitions of the form (x,x), (x,y), (x,x,y), or (x,y,y). For example, the a(0) = 1 through a(8) = 9 partitions are the following. The Heinz numbers of these partitions are given by A325270.
  (11)  (21)  (22)   (32)   (33)   (43)   (44)   (54)   (55)
              (31)   (41)   (42)   (52)   (53)   (63)   (64)
              (211)  (221)  (51)   (61)   (62)   (72)   (73)
                     (311)  (411)  (322)  (71)   (81)   (82)
                                   (331)  (332)  (441)  (91)
                                   (511)  (422)  (522)  (433)
                                          (611)  (711)  (442)
                                                        (622)
                                                        (811)
(End)

Kival Ngaokrajang, Illustration of initial terms
Burkard Polster, Times Tables, Mandelbrot and the Heart of Mathematics, Mathologer video (2015).
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A001045, A116948, A273724.

Cf. A090858, A127002, A323055, A325242, A325243, A325244, A325270.

[1 + Floor(2*n/3) + Floor((n+1)/3) : n in [0..100]]; // Wesley Ivan Hurt, Jul 25 2016

A117571:=n->1 + floor(2*n/3) + floor((n+1)/3): seq(A117571(n), n=0..100); # Wesley Ivan Hurt, Jul 25 2016

CoefficientList[Series[(1 + 2 x^2)/((1 - x) (1 - x^3)), {x, 0, 71}], x] (* Michael De Vlieger, May 13 2016 *)

G.f.: (1+2*x^2)/((1-x)*(1-x^3)).
a(n) = a(n-1) + a(n-3) - a(n-4) for n>3.
a(n) = cos(2*Pi*n/3+Pi/6)/sqrt(3)-sin(2*Pi*n/3+Pi/6)/3+(3n+2)/3.
a(n) = Sum_{k=0..n} 2*A001045(L((n-k+2)/3)) where L(j/p) is the Legendre symbol of j and p.
a(n) = 1 + floor((n+1)/3) + floor(2*n/3). - Wesley Ivan Hurt, Jul 25 2016
a(n) = n+sign((n-1) mod 3). - Wesley Ivan Hurt, Sep 25 2017

A381719 Numbers whose prime indices cannot be partitioned into sets with a common sum.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 184, 188, 189, 192

Offset: 1

Gus Wiseman, Apr 22 2025

nonn

Differs from A059404, A323055, A376250 in lacking 150.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.
Also numbers that cannot be factored into squarefree numbers with a common sum of prime indices (A056239).

			The prime indices of 150 are {1,2,3,3}, and {{3},{3},{1,2}} is a partition into sets with a common sum, so 150 is not in the sequence.

Twice-partitions of this type (sets with a common sum) are counted by A279788.
These multiset partitions (sets with a common sum) are ranked by A326534 /\ A302478.
For distinct block-sums we have A381806, counted by A381990 (complement A381992).
For constant blocks we have A381871 (zeros of A381995), counted by A381993.
Partitions of this type are counted by A381994.
These are the zeros of A382080.
Normal multiset partitions of this type are counted by A382429, see A326518.
The complement counted by A383308.
A000041 counts integer partitions, strict A000009.
A001055 counts factorizations, strict A045778.
A050320 counts factorizations into squarefree numbers, see A381078, A381454.
A050326 counts factorizations into distinct squarefree numbers.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A317141 counts coarsenings of prime indices, refinements A300383.
A381633 counts set systems with distinct sums, see A381634, A293243.
Set multipartitions: A089259, A116540, A270995, A296119, A318360.
Cf. A000720, A000961, A001222, A293511, A321455, A358914, A381635, A381717.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Select[Range[100],Select[mps[prix[#]], SameQ@@Total/@#&&And@@UnsameQ@@@#&]=={}&]

A323056 Numbers with exactly five distinct exponents in their prime factorization, or five distinct parts in their prime signature.

Original entry on oeis.org

174636000, 206388000, 244490400, 261954000, 269892000, 274428000, 288943200, 291060000, 301644000, 309582000, 343980000, 349272000, 365148000, 366735600, 377848800, 383292000, 404838000, 411642000, 412776000, 422301600, 433414800, 449820000, 452466000, 457380000

Offset: 1

Gus Wiseman, Jan 03 2019

nonn

The first term is A006939(5) = 174636000.
Positions of 5's in A071625.
Numbers k such that A001221(A181819(k)) = 5.

			174636000 = 2^5 * 3^4 * 5^3 * 7^2 * 11^1 has five distinct exponents so belongs to the sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000

One distinct exponent: A062770 or A072774.
Two distinct exponents: A323055.
Three distinct exponents: A323024.
Four distinct exponents: A323025.
Five distinct exponents: A323056.
Cf. A001221, A001222, A006939, A051270, A059404, A071625, A118914, A181819, A182855, A323014, A323022, A323024.

Select[Range[300000000],Length[Union[Last/@FactorInteger[#]]]==5&]

is(n) = #Set(factor(n)[, 2]) == 5 \\ David A. Corneth, Jan 12 2019

a(13)-a(24) from Daniel Suteu, Jan 12 2019

A325243 Number of integer partitions of n with exactly two distinct multiplicities.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 9, 12, 19, 26, 39, 47, 70, 89, 115, 148, 189, 235, 294, 362, 450, 558, 669, 817, 980, 1197, 1421, 1709, 2012, 2429, 2836, 3380, 3961, 4699, 5433, 6457, 7433, 8770, 10109, 11818, 13547, 15912, 18109, 21105, 24121, 27959, 31736, 36840, 41670

Offset: 0

Gus Wiseman, Apr 15 2019

nonn

For example, (32211) has two distinct multiplicities (1 and 2) so is counted under a(9).

The Heinz numbers of these partitions are given by A323055.

			The a(4) = 1 through a(9) = 19 partitions:
  (211)  (221)   (411)    (322)     (332)      (441)
         (311)   (3111)   (331)     (422)      (522)
         (2111)  (21111)  (511)     (611)      (711)
                          (2221)    (3221)     (3222)
                          (3211)    (4211)     (3321)
                          (4111)    (5111)     (4221)
                          (22111)   (22211)    (4311)
                          (31111)   (32111)    (5211)
                          (211111)  (41111)    (6111)
                                    (221111)   (22221)
                                    (311111)   (32211)
                                    (2111111)  (33111)
                                               (42111)
                                               (51111)
                                               (321111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)

Column k = 2 of A325242. Dominated by A325267.

Cf. A008284, A062770, A071625, A098859, A116608, A244515, A323055, A325244.

Table[Length[Select[IntegerPartitions[n],Length[Union[Length/@Split[#]]]==2&]],{n,0,30}]

A383080 Numbers k such that sopf(k) does not divide evenly sopfr(k).

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 198

Offset: 1

Torlach Rush, Apr 15 2025

nonn

First differs from A059404 and A323055 at n = 59.
a(n) has a square factor, A008683(a(n)) = 0.
If p and q are distinct primes, p*q^k is in the sequence iff p + q does not divide k - 1. - Robert Israel, Apr 16 2025

			12 is a term because sopf(12)=5 does not evenly divide sopfr(12)=7.
18 is a term because sopf(18)=5 does not evenly divide sopfr(18)=8.
20 is a term because sopf(20)=7 does not evenly divide sopfr(20)=9.

Cf. A001414, A008472, A008683.

Cf. A059404, A323055.

filter:= proc(n) local F,t;
  F:= ifactors(n)[2];
  add(t[1]*t[2],t=F) mod add(t[1],t=F) <> 0
end proc:
select(filter, [$2..300]); # Robert Israel, Apr 16 2025

q[k_] := Module[{f = FactorInteger[k]}, !Divisible[Plus @@ Times @@@ f, Plus @@ f[[;; , 1]]]]; Select[Range[200], q] (* Amiram Eldar, Apr 16 2025 *)

isok(k) = if (k>1, my(f=factor(k)); sum(j=1, #f~, f[j,1]*f[j,2]) % sum(j=1, #f~, f[j,1])); \\ Michel Marcus, Apr 16 2025

def spf(k):
    fl = list(factor(k))
    sr = sum(p * e for p, e in fl)
    sd = sum(p for p, _ in fl)
    return sd, sr
def output(limit=198):
    results = []
    for k in range(2, limit + 1):
        sd, sr = spf(k)
        if 0 < sd and sr % sd != 0:
            results.append(k)
    return results
print(output())

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A332785 Nonsquarefree numbers that are not squareful.

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A117571 Expansion of (1+2*x^2)/((1-x)*(1-x^3)).

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A381719 Numbers whose prime indices cannot be partitioned into sets with a common sum.

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A323056 Numbers with exactly five distinct exponents in their prime factorization, or five distinct parts in their prime signature.

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A325243 Number of integer partitions of n with exactly two distinct multiplicities.

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A383080 Numbers k such that sopf(k) does not divide evenly sopfr(k).

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A117571 Expansion of (1+2x^2)/((1-x)(1-x^3)).