A060687 -id:A060687 - cp's OEIS frontend

A195088 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 4.

32, 96, 144, 160, 216, 224, 243, 324, 352, 400, 416, 480, 486, 544, 608, 672, 720, 736, 784, 928, 992, 1000, 1008, 1056, 1080, 1120, 1184, 1200, 1215, 1248, 1312, 1376, 1504, 1512, 1584, 1620, 1632, 1696, 1701, 1760, 1800, 1824, 1872, 1888, 1936, 1952

Offset: 1

Harvey P. Dale, Sep 08 2011

The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0237194..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - Amiram Eldar, Sep 24 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A060687, A195069, A195086, A195087, A195089, A195090, A195091, A195092, A195093.

Cf. A025487, A046660, A059956, A112526, A257851, A261256, A264959.

a195088 n = a195088_list !! (n-1)
a195088_list = filter ((== 4) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

Select[Range[2000],PrimeOmega[#]-PrimeNu[#]==4&]

is(n)=bigomega(n)-omega(n)==4 \\ Charles R Greathouse IV, Sep 14 2015

A046660(a(n)) = 4. - Reinhard Zumkeller, Nov 29 2015

A195090 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 6.

Original entry on oeis.org

128, 384, 576, 640, 864, 896, 1296, 1408, 1600, 1664, 1920, 1944, 2176, 2187, 2432, 2688, 2880, 2916, 2944, 3136, 3712, 3968, 4000, 4032, 4224, 4320, 4374, 4480, 4736, 4800, 4992, 5248, 5504, 6016, 6048, 6336, 6480, 6528, 6784

Offset: 1

Harvey P. Dale, Sep 08 2011

The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0059189..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - Amiram Eldar, Sep 24 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A060687, A195069, A195086, A195087, A195088, A195089, A195091, A195092, A195093.

Cf. A025487, A046660, A059956, A112526, A257851, A261256, A264959.

a195090 n = a195090_list !! (n-1)
a195090_list = filter ((== 6) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

op(select(n->bigomega(n)-nops(factorset(n))=6, [$1..6784])); # Paolo P. Lava, Jul 03 2018

Select[Range[7000],PrimeOmega[#]-PrimeNu[#]==6&]

is(n)=bigomega(n)-omega(n)==6 \\ Charles R Greathouse IV, Sep 14 2015

A046660(a(n)) = 6. - Reinhard Zumkeller, Nov 29 2015

A195092 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 8.

Original entry on oeis.org

512, 1536, 2304, 2560, 3456, 3584, 5184, 5632, 6400, 6656, 7680, 7776, 8704, 9728, 10752, 11520, 11664, 11776, 12544, 14848, 15872, 16000, 16128, 16896, 17280, 17496, 17920, 18944, 19200, 19683, 19968, 20992, 22016, 24064, 24192

Offset: 1

Harvey P. Dale, Sep 08 2011

The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0014793..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - Amiram Eldar, Sep 24 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A060687, A195069, A195086, A195087, A195088, A195089, A195090, A195091, A195093.

Cf. A025487, A046660, A059956, A112526, A257851, A261256, A264959.

a195092 n = a195092_list !! (n-1)
a195092_list = filter ((== 8) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

Select[Range[25000],PrimeOmega[#]-PrimeNu[#]==8&]

is(n)=bigomega(n)-omega(n)==8 \\ Charles R Greathouse IV, Sep 14 2015

A046660(a(n)) = 8. - Reinhard Zumkeller, Nov 29 2015

A195093 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 9.

Original entry on oeis.org

1024, 3072, 4608, 5120, 6912, 7168, 10368, 11264, 12800, 13312, 15360, 15552, 17408, 19456, 21504, 23040, 23328, 23552, 25088, 29696, 31744, 32000, 32256, 33792, 34560, 34992, 35840, 37888, 38400, 39936, 41984, 44032, 48128, 48384

Offset: 1

Harvey P. Dale, Sep 08 2011

The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0007396..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - Amiram Eldar, Sep 24 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A060687, A195069, A195086, A195087, A195088, A195089, A195090, A195091, A195092.

Cf. A025487, A046660, A059956, A112526, A257851, A261256, A264959.

a195093 n = a195093_list !! (n-1)
a195093_list = filter ((== 9) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

Select[Range[50000],PrimeOmega[#]-PrimeNu[#]==9&]

is(n)=bigomega(n)-omega(n)==9 \\ Charles R Greathouse IV, Sep 14 2015

A046660(a(n)) = 9. - Reinhard Zumkeller, Nov 29 2015

A082293 Numbers having exactly one square divisor > 1.

Original entry on oeis.org

4, 8, 9, 12, 18, 20, 24, 25, 27, 28, 40, 44, 45, 49, 50, 52, 54, 56, 60, 63, 68, 75, 76, 84, 88, 90, 92, 98, 99, 104, 116, 117, 120, 121, 124, 125, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 164, 168, 169, 171, 172, 175, 184, 188, 189, 198, 204, 207, 212

Offset: 1

Reinhard Zumkeller, Apr 08 2003

nonn

Numbers of the form m*p^2, p prime and m squarefree (A005117). [Corrected by Peter Munn, Nov 17 2020]

The asymptotic density of this sequence is (6/Pi^2)*Sum_{n>=1} 1/prime(n)^2 = 0.274933... (A222056). - Amiram Eldar, Jul 07 2020

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

Cf. A005117, A046951, A222056.
Complement of A048111 within A013929.
Subsequence of A252849.
Disjoint union of A048109 and A060687.
A285508 is a subsequence.

Select[Range[2, 200], MemberQ[{2, 3}, (e = Sort[FactorInteger[#][[;; , 2]]])[[-1]]] && (Length[e] == 1 || e[[-2]] == 1) &] (* Amiram Eldar, Jul 07 2020 *)

is(n)=my(f=vecsort(factor(n)[,2],,4)); #f && f[1]>1 && f[1]<4 && (#f==1 || f[2]==1) \\ Charles R Greathouse IV, Oct 16 2015

from math import isqrt
from sympy import mobius, primerange
def A082293(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def f(x): return int(n+x-sum(g(x//p**2) for p in primerange(isqrt(x)+1)))
    return bisection(f,n,n) # Chai Wah Wu, Feb 24 2025

A046951(a(n)) = 2.

A072357 Cubefree nonsquares whose factorization into a product of primes contains exactly one square.

Original entry on oeis.org

12, 18, 20, 28, 44, 45, 50, 52, 60, 63, 68, 75, 76, 84, 90, 92, 98, 99, 116, 117, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 171, 172, 175, 188, 198, 204, 207, 212, 220, 228, 234, 236, 242, 244, 245, 260, 261, 268, 275, 276, 279, 284, 292, 294, 306, 308

Offset: 1

Reinhard Zumkeller, Jul 18 2002

nonn

Numbers n such that A001222(n) - A001221(n) = 1 and A001221(n)>1.
Numbers with one or more 1's, exactly one 2 and no 3's or higher in their prime exponents. - Antti Karttunen, Sep 19 2019
From Salvador Cerdá, Mar 08 2016: (Start)
12!+1 = 13^2 * 2834329 is in this sequence.
23!+1 = 47^2 * 79 * 148139754736864591 is also in this sequence. (End)
The asymptotic density of this sequence is (6/Pi^2) * Sum_{p prime} 1/(p*(p+1)) (A271971). - Amiram Eldar, Nov 09 2020

			a(14) = 84 = 7*3*2^2; the following numbers are not terms: 36=6^2, as it is a square; 54=2*3^3, as it is not cubefree; 42=2*3*7, as there is no squared prime; 72=2*6^2, as 72 has two squared prime divisors: 2^2 and 3^2.

Robert Israel, Table of n, a(n) for n = 1..10000 ( 1..100 from Paolo P. Lava)

Cf. A001221, A001222, A054753 (subsequence), A271971, A325981 (conjectured subsequence).

Subsequence of: A004709, A048107, A060687, A067259.

N:= 1000: # to get all terms <= N
Primes:= select(isprime, [$2..floor(N^(1/2))]):
SF:= select(numtheory:-issqrfree, [$2..N/4]):
S:= {seq(op(map(p -> p^2*t, select(s -> igcd(s,t)=1 and s^2*t <= N, Primes))), t = SF)}:
sort(convert(S,list)); # Robert Israel, Mar 08 2016

Select[Range@ 308, And[PrimeNu@ # > 1, PrimeOmega@ # - PrimeNu@ # == 1] &] (* Michael De Vlieger, Mar 09 2016 *)

isok(n) = (omega(n) > 1) && (bigomega(n) - omega(n) == 1); \\ Michel Marcus, Jul 16 2015

A325196 Heinz numbers of integer partitions such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 1.

Original entry on oeis.org

3, 4, 9, 10, 12, 15, 18, 20, 42, 45, 50, 60, 63, 70, 75, 84, 90, 100, 105, 126, 140, 150, 294, 315, 330, 350, 420, 441, 462, 490, 495, 525, 550, 588, 630, 660, 693, 700, 735, 770, 825, 882, 924, 980, 990, 1050, 1100, 1155, 1386, 1470, 1540, 1650, 2730, 3234

Offset: 1

Gus Wiseman, Apr 11 2019

nonn

The enumeration of these partitions by sum is given by A325191.

			The sequence of terms together with their prime indices begins:
    3: {2}
    4: {1,1}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   42: {1,2,4}
   45: {2,2,3}
   50: {1,3,3}
   60: {1,1,2,3}
   63: {2,2,4}
   70: {1,3,4}
   75: {2,3,3}
   84: {1,1,2,4}
   90: {1,2,2,3}
  100: {1,1,3,3}
  105: {2,3,4}
  126: {1,2,2,4}

Gus Wiseman, Young diagrams for their first 60 terms.
FindStat, St000783: The maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram
FindStat, St000384: The maximal part of the shifted composition of an integer partition.

Cf. A060687, A065770, A256617, A325166, A325169, A325179, A325181, A325183, A325185, A325188, A325189, A325191, A325195, A325200.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Select[Range[1000],otbmax[primeptn[#]]-otb[primeptn[#]]==1&]

A325191 Number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 1.

Original entry on oeis.org

0, 0, 2, 0, 3, 3, 0, 4, 6, 4, 0, 5, 10, 10, 5, 0, 6, 15, 20, 15, 6, 0, 7, 21, 35, 35, 21, 7, 0, 8, 28, 56, 70, 56, 28, 8, 0, 9, 36, 84, 126, 126, 84, 36, 9, 0, 10, 45, 120, 210, 252, 210, 120, 45, 10, 0, 11, 55, 165, 330, 462

Offset: 0

Gus Wiseman, Apr 11 2019

The Heinz numbers of these partitions are given by A325196.

Under the Bulgarian solitaire step, these partitions form cycles of length >= 2. Length >= 2 means not the length=1 self-loop which occurs from the triangular partition when n is a triangular number. See A074909 for self-loops included. - Kevin Ryde, Sep 27 2019

			The a(2) = 2 through a(12) = 10 partitions (empty columns not shown):
  (2)   (22)   (32)   (322)   (332)   (432)   (4322)   (4332)
  (11)  (31)   (221)  (331)   (422)   (3321)  (4331)   (4422)
        (211)  (311)  (421)   (431)   (4221)  (4421)   (4431)
                      (3211)  (3221)  (4311)  (5321)   (5322)
                              (3311)          (43211)  (5331)
                              (4211)                   (5421)
                                                       (43221)
                                                       (43311)
                                                       (44211)
                                                       (53211)

FindStat, St000380: Half the perimeter of the largest rectangle that fits inside the diagram of an integer partition
FindStat, St000384: The maximal part of the shifted composition of an integer partition
FindStat, St000783: The maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram
Eric Weisstein's World of Mathematics, Graph Distance

Column k=1 of A325200.

Cf. A060687, A065770, A071724, A256617, A325166, A325169, A325178, A325179, A325181, A325187, A325188, A325189, A325195, A325196.

otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[Length[Select[IntegerPartitions[n],otb[#]+1==otbmax[#]&]],{n,0,30}]

a(n) = my(t=ceil(sqrtint(8*n+1)/2), r=n-t*(t-1)/2); if(r==0,0, binomial(t,r)); \\ Kevin Ryde, Sep 27 2019

Positions of zeros are A000217 = n * (n + 1) / 2.

a(n) = A074909(n) - A010054(n). - Kevin Ryde, Sep 27 2019

A350322 Abelian orders m for which there exist exactly 2 groups of order m.

Original entry on oeis.org

4, 9, 25, 45, 49, 99, 121, 153, 169, 175, 207, 245, 261, 289, 325, 361, 369, 423, 425, 475, 477, 529, 531, 539, 575, 637, 639, 725, 747, 765, 801, 833, 841, 845, 847, 909, 925, 931, 961, 963, 1017, 1035, 1075, 1127, 1175, 1179, 1233, 1305, 1325, 1341, 1369, 1445, 1475

Offset: 1

Jianing Song, Dec 25 2021

nonn

Abelian orders of the form p^2 * q_1 * q_2 * ... * q_s, where p, q_1, q_2, ..., q_s are distinct primes such that p^2 !== 1 (mod q_j), q_i !== 1 (mod p_j), q_i !== 1 (mod q_j) for i != j. In this case there are 2^r groups of order m.
Note that the smallest abelian order with precisely 2^n groups must be the square of a squarefree number.
Except for a(1) = 4, all terms are odd. The terms that are divisible by 3 are of the form 9 * q_1 * q_2 * ... * q_s, where q_i are distinct primes congruent to 5 modulo 6, q_i !== 1 (mod q_j) for i != j.

			For primes p, p^2 is a term since the 2 groups of that order are C_{p^2} and C_p X C_p.
For primes p, q, if p^2 !== 1 (mod q) and q !== 1 (mod p), then p^2*q is a term since the 2 groups of that order are C_{p^2*q} and C_p X C_{p*q}.

Jianing Song, Table of n, a(n) for n = 1..10000

Equals A060687 INTERSECT A051532 = A054395 INTERSECT A051532 = A054395 INTERSECT A060687 = A054395 INTERSECT A013929.
Equals A350152 \ A350323.
Equals A054395 \ A350586.
Subsequence of A350152.
A001248 and A350332 are subsequences.

isA054395(n) = {
  my(p=gcd(n, eulerphi(n)), f);
  if (!isprime(p), return(0));
  if (n%p^2 == 0, return(1 == gcd(p+1, n)));
  f = factor(n); 1 == sum(k=1, matsize(f)[1], f[k, 1]%p==1);
} \\ Gheorghe Coserea's program for A054395
isA350322(n) = isA054395(n) && (bigomega(n)-omega(n)==1)

isA051532(n) = my(f=factor(n), v=vector(#f[, 1])); for(i=1, #v, if(f[i, 2]>2, return(0), v[i]=f[i, 1]^f[i, 2])); for(i=1, #v, for(j=i+1, #v, if(v[i]%f[j, 1]==1 || v[j]%f[i, 1]==1, return(0)))); 1 \\ Charles R Greathouse IV's program for A051532
isA350322(n) = isA051532(n) && (bigomega(n)-omega(n)==1)

def is_ok(m):
    f = factorint(m)
    return (
        sum(f.values()) == len(f) + 1 and
        all((q - 1) % p > 0 for p in f for q in f) and
        (m := next(p for p, e in f.items() if e == 2) ** 2 - 1) and
        all(m % q > 0 for q in f)) # David Radcliffe, Jul 30 2025

A386796 Numbers that have exactly one exponent in their prime factorization that is equal to 2.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 108, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 164, 169, 171, 172, 175, 188, 198, 200, 204, 207, 212, 220, 228, 234, 236, 242, 244, 245, 260, 261, 268

Offset: 1

Amiram Eldar, Aug 02 2025

First differs from its subsequence A060687 at n = 16: a(16) = 72 is not a term of A060687.
Differs from A286228 by having the terms 60, 72, 84, 90, ..., and not having the term 1.
Numbers k such that A369427(k) = 1.
The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^2 + 1/p^3) * Sum_{p prime} (p-1)/(p^3 - p + 1) = 0.22661832022705616779... (the product is A330596) (Elma and Martin, 2024).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ertan Elma and Greg Martin, Distribution of the number of prime factors with a given multiplicity, Canadian Mathematical Bulletin, Vol. 67, No. 4 (2024), pp. 1107-1122; arXiv preprint, arXiv:2406.04574 [math.NT], 2024.
Index entries for sequences computed from indices in prime factorization.

A060687 is a subsequence.
Cf. A286228, A330596, A369427.
Numbers that have exactly one exponent in their prime factorization that is equal to k: A119251 (k=1), this sequence (k=2), A386800 (k=3), A386804 (k=4), A386808 (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 2: A337050 (m=0), this sequence (m=1), A386797 (m=2), A386798 (m=3).

f[p_, e_] := If[e == 2, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[300], s[#] == 1 &]

isok(k) = vecsum(apply(x -> if(x == 2, 1, 0), factor(k)[, 2])) == 1;

cp's OEIS Frontend

A195088 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 4.

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A195090 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 6.

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A195092 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 8.

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A195093 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 9.

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A082293 Numbers having exactly one square divisor > 1.

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A072357 Cubefree nonsquares whose factorization into a product of primes contains exactly one square.

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A325196 Heinz numbers of integer partitions such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 1.

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