A000469 -id:A000469 - cp's OEIS frontend

A361102 1 together with numbers having at least two distinct prime factors.

1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 02 2023

nonn

This is the union of 1 and A024619. It is the sequence C used in the definition of A360519. Since C is central to the analysis of A360519 it deserves its own entry.

This has the same relationship to A024619 as A000469 does to A120944 for squarefree numbers.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000469, A024619, A120944, A126706, A246655, A360519.

isa := n -> is(irem(ilcm(seq(1..n-1)), n) = 0):
aList := upto -> select(isa, [seq(1..upto)]):
aList(112); # Peter Luschny, May 17 2023

Select[Range[120], Not@*PrimePowerQ] (* Michael De Vlieger, May 17 2023 *)

from sympy import primepi, integer_nthroot
def A361102(n):
    def f(x): return int(n+sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    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
    return bisection(f) # Chai Wah Wu, Aug 31 2024

def A361102List(upto: int) -> list[int]:
    return sorted(Set(1..upto).difference(prime_powers(upto)))
print(A361102List(112))  # Peter Luschny, May 17 2023

From Peter Luschny and Michael De Vlieger, May 17 2023: (Start)
The sequence is the complement of the prime powers in the positive integers, a = A000027 \ A246655.
k is in this sequence <=> k divides lcm(1, 2, ..., k-1). (End)
This sequence is {1} U { A120944 U A126706 } = {1} U A024619. - Michael De Vlieger, May 17 2023

Offset set to 1 by Peter Luschny, May 17 2023

A119313 Numbers with a prime as third-smallest divisor.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 21, 22, 24, 26, 30, 33, 34, 35, 36, 38, 39, 42, 45, 46, 48, 50, 51, 54, 55, 57, 58, 60, 62, 63, 65, 66, 69, 70, 72, 74, 75, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 96, 98, 102, 105, 106, 108, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126

Offset: 1

Reinhard Zumkeller, May 15 2006

nonn

m is a term iff A001221(m) > 1 and (A067029(m) = 1 or A119288(m) < A020639(m)^2).

			a(1) = A087134(3) = 6.
From _Gus Wiseman_, Oct 19 2019: (Start)
The sequence of terms together with their divisors begins:
    6: {1,2,3,6}
   10: {1,2,5,10}
   12: {1,2,3,4,6,12}
   14: {1,2,7,14}
   15: {1,3,5,15}
   18: {1,2,3,6,9,18}
   21: {1,3,7,21}
   22: {1,2,11,22}
   24: {1,2,3,4,6,8,12,24}
   26: {1,2,13,26}
   30: {1,2,3,5,6,10,15,30}
   33: {1,3,11,33}
   34: {1,2,17,34}
   35: {1,5,7,35}
   36: {1,2,3,4,6,9,12,18,36}
   38: {1,2,19,38}
   39: {1,3,13,39}
   42: {1,2,3,6,7,14,21,42}
   45: {1,3,5,9,15,45}
   46: {1,2,23,46}
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Complement of A119314.
Subsequences: A006881, A000469, A008588.
A subset of A002808 and A080257.
Numbers whose third-largest divisor is prime are A328338.
Second-smallest divisor is A020639.
Third-smallest divisor is A292269.
Cf. A000005, A000040, A001221, A020639, A027750, A033676, A060775, A067029, A088725, A119288, A328189.

q:= n-> (l-> nops(l)>2 and isprime(l[3]))(
         sort([numtheory[divisors](n)[]])):
select(q, [$1..200])[];  # Alois P. Heinz, Oct 19 2019

Select[Range[100],Length[Divisors[#]]>2&&PrimeQ[Divisors[#][[3]]]&] (* Gus Wiseman, Oct 15 2019 *)
Select[Range[130], Length[f = FactorInteger[#]] > 1 && (f[[1, 2]] == 1 || f[[1, 1]]^2 > f[[2, 1]]) &] (* Amiram Eldar, Jul 02 2022 *)

Name edited by Gus Wiseman, Oct 19 2019

A303554 Union of the prime powers (p^k, p prime, k >= 0) and numbers that are the product of 2 or more distinct primes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110

Offset: 1

Ilya Gutkovskiy, Apr 26 2018

nonn

			42 is in the sequence because 42 = 2*3*7 (3 distinct prime factors).
81 is in the sequence because 81 = 3^4 (4 prime factors, 1 distinct).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Power
Eric Weisstein's World of Mathematics, Squarefree

Complement of A126706.
Union of A005117 and A246547.
Union of A000469 and A246655.
Union of A000961 and A120944.
Cf. A025475.

Select[Range[110], PrimePowerQ[#] || SquareFreeQ[#] &]
Select[Range[110], PrimeNu[#] == 1 || PrimeNu[#] == PrimeOmega[#] &]

from math import isqrt
from sympy import primepi, integer_nthroot, mobius
def A303554(n):
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length()))-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 19 2024

A303606 Powers of composite squarefree numbers that are not squarefree.

Original entry on oeis.org

36, 100, 196, 216, 225, 441, 484, 676, 900, 1000, 1089, 1156, 1225, 1296, 1444, 1521, 1764, 2116, 2601, 2744, 3025, 3249, 3364, 3375, 3844, 4225, 4356, 4761, 4900, 5476, 5929, 6084, 6724, 7225, 7396, 7569, 7776, 8281, 8649, 8836, 9025, 9261, 10000, 10404, 10648, 11025, 11236

Offset: 1

Ilya Gutkovskiy, Apr 26 2018

nonn

			196 is in the sequence because 196 = 2^2*7^2.
4900 is in the sequence because 4900 = 2^2*5^2*7^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Power.
Eric Weisstein's World of Mathematics, Squarefree.

Intersection of A024619 and A072777.
Intersection of A072774 and A126706.
Intersection of A013929 and A182853.
Cf. A000469, A001597, A005117, A120944, A286708.

Select[Range[12000], Length[Union[FactorInteger[#][[All, 2]]]] == 1 && ! SquareFreeQ[#] && ! PrimePowerQ[#] &]
seq[max_] := Module[{sp = Select[Range[Floor@Sqrt[max]], SquareFreeQ[#] && PrimeNu[#] > 1 &], s = {}}, Do[s = Join[s, sp[[k]]^Range[2, Floor@Log[sp[[k]], max]]], {k, 1, Length[sp]}]; Union@s]; seq[10^4] (* Amiram Eldar, Feb 12 2021 *)

from math import isqrt
from sympy import mobius, primepi, integer_nthroot
def A303606(n):
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))-primepi(x))
    def f(x): return n-3+x+(y:=x.bit_length())-sum(g(integer_nthroot(x,k)[0]) for k in range(2,y))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 19 2024

Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((A120944(n)-1)*A120944(n)) = Sum_{k>=2} (zeta(k)/zeta(2*k) - P(k) - 1) = 0.07547719891508850482..., where P(k) is the prime zeta function. - Amiram Eldar, Feb 12 2021

A177492 Products of squares of 2 or more distinct primes.

Original entry on oeis.org

36, 100, 196, 225, 441, 484, 676, 900, 1089, 1156, 1225, 1444, 1521, 1764, 2116, 2601, 3025, 3249, 3364, 3844, 4225, 4356, 4761, 4900, 5476, 5929, 6084, 6724, 7225, 7396, 7569, 8281, 8649, 8836, 9025, 10404, 11025, 11236, 12100, 12321, 12996, 13225, 13924

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 10 2010

nonn

			36=2^2*3^2, 100=2^2*5*2, 196=2^2*7^2,..900=2^2*3^2*5^2,..

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000469, A074985, A085986, A120944, A162142.

q:= n-> not isprime(n) and numtheory[issqrfree](n):
map(x-> x^2, select(q, [$4..120]))[];  # Alois P. Heinz, Aug 02 2024

f1[n_]:=Length[Last/@FactorInteger[n]]; f2[n_]:=Union[Last/@FactorInteger[n]]; lst={};Do[If[f1[n]>1&&f2[n]=={2},AppendTo[lst,n]],{n,0,8!}];lst
Reap[Do[{p, e} = Transpose[FactorInteger[n]]; If[Length[p]>1 && Union[e]=={2}, Sow[n]], {n, 13225}]][[2, 1]]
(* Second program *)
Select[Range[120], And[CompositeQ[#], SquareFreeQ[#]] &]^2 (* Michael De Vlieger, Aug 17 2023 *)

from math import isqrt
from sympy import primepi, mobius
def A177492(n):
    def f(x): return n+1+primepi(x)+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n+1, f(n+1)
    while m != k:
        m, k = k, f(k)
    return m**2 # Chai Wah Wu, Aug 02 2024

a(n) = A120944(n)^2. - R. J. Mathar, Dec 06 2010

Definition corrected by R. J. Mathar, Dec 06 2010

A354911 Number of factorizations of n into relatively prime prime-powers.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 1, 0, 2, 2, 1, 0, 5, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 2, 0, 1, 2, 0, 1, 1, 0, 2, 1, 1, 0, 6, 0, 1, 2, 2, 1, 1, 0, 5, 0, 1, 0, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 25 2022

nonn

			The a(n) factorizations for n = 6, 12, 24, 36, 48, 72, 96:
  2*3  3*4    3*8      4*9      3*16       8*9        3*32
       2*2*3  2*3*4    2*2*9    2*3*8      2*4*9      3*4*8
              2*2*2*3  3*3*4    3*4*4      3*3*8      2*3*16
                       2*2*3*3  2*2*3*4    2*2*2*9    2*2*3*8
                                2*2*2*2*3  2*3*3*4    2*3*4*4
                                           2*2*2*3*3  2*2*2*3*4
                                                      2*2*2*2*2*3

Wikipedia, Coprime integers.

This is the relatively prime case of A000688, partitions A023894.
Positions of 0's are A246655 (A000961 includes 1).
For strict instead of relatively prime we have A050361, partitions A054685.
Positions of 1's are A000469 (A120944 excludes 1).
For pairwise coprime instead of relatively prime we have A143731.
The version for partitions instead of factorizations is A356067.
A000005 counts divisors.
A001055 counts factorizations.
A001221 counts distinct prime divisors, with sum A001414.
A001222 counts prime-power divisors.
A289509 lists numbers whose prime indices are relatively prime.
A295935 counts twice-factorizations with constant blocks (type PPR).
A355743 lists numbers with prime-power prime indices, squarefree A356065.
Cf. A000837, A023893, A076610, A085970, A106244, A279784, A318721, A355737, A355742, A356064, A356066.

ufacs[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[ufacs[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Table[Length[Select[ufacs[Select[Divisors[n],PrimePowerQ[#]&],n],GCD@@#<=1&]],{n,100}]

a(n) = A000688(n) if n is nonprime, otherwise a(n) = 0.

A380478 Nonprime numbers k such that A380459(k) has no divisors of form p^p.

Original entry on oeis.org

1, 6, 14, 26, 38, 62, 74, 86, 122, 134, 146, 158, 186, 194, 206, 218, 254, 278, 302, 314, 326, 362, 386, 398, 422, 434, 446, 458, 482, 542, 554, 566, 614, 626, 662, 674, 698, 734, 746, 758, 794, 818, 842, 854, 866, 878, 906, 914, 926, 974, 998, 1046, 1082, 1094, 1142, 1154, 1202, 1214, 1226, 1238, 1262, 1266, 1286

Offset: 1

Antti Karttunen, Feb 03 2025

nonn

See A380468 to find why each term must be squarefree.
After the initial 1 all terms here are even (thus in A039956) because as A276086(k) flips the parity of k, having more than one odd prime factor in k without a single 2 would make A380459(k) a multiple of 4, and thus outside of A048103.
For a similar reason, any term that is not a multiple of 3 may have at most 3 prime factors.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to primorial base.

Cf. A048103, A276086, A380459, A380474 (subsequence), A380477 (characteristic function).
Intersection of A380468 and A018252. (A380468 without primes).
Subsequence of A000469. Subsequence of A039956 (after the initial 1).

PARI
```
is_A380478 = A380477;
```

A073249 Nonprime squarefree numbers n such that both n-1 and n+1 are not squarefree and not prime.

Original entry on oeis.org

26, 51, 55, 91, 161, 170, 235, 249, 295, 305, 339, 341, 362, 377, 413, 415, 451, 485, 489, 530, 551, 559, 579, 595, 629, 638, 649, 651, 665, 667, 685, 687, 703, 721, 723, 737, 749, 849, 851, 874, 917, 926, 949, 951, 955, 962, 989, 1015, 1027, 1057, 1059

Offset: 1

Reinhard Zumkeller, Jul 22 2002

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000469, A005117, A073247, A073250, A073251.

npsQ[n_]:=SquareFreeQ[n]&&NoneTrue[n+{1,0,-1},PrimeQ]&&NoneTrue[n+{1,-1},SquareFreeQ]; Select[Range[2000],npsQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 20 2019 *)
Mean/@SequencePosition[Table[Which[CompositeQ[n]&&SquareFreeQ[n],1,!SquareFreeQ[ n] && CompositeQ[ n],-1,True,0],{n,1100}],{-1,1,-1}] (* Harvey P. Dale, Jun 17 2022 *)

Corrected by Harvey P. Dale, Jan 20 2019

A073250 Nonprime squarefree numbers n such that n+1 is also squarefree and nonprime, but not n-1 and n+2.

Original entry on oeis.org

14, 21, 38, 57, 65, 69, 77, 105, 110, 114, 118, 122, 129, 133, 145, 154, 158, 165, 177, 182, 194, 205, 209, 221, 230, 237, 246, 258, 273, 290, 298, 309, 318, 326, 329, 334, 345, 354, 357, 365, 370, 381, 385, 390, 398, 402, 406, 410, 417, 426, 429, 434, 437

Offset: 1

Reinhard Zumkeller, Jul 22 2002

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000469, A005117, A073248, A073251, A073249.

tQ[n_]:=!PrimeQ[n+1]&&SquareFreeQ[n+1]&&(PrimeQ[n-1]||!SquareFreeQ[n-1])&&(PrimeQ[n+2]||!SquareFreeQ[n+2])
Select[Select[Complement[Range[500],Prime[Range[PrimePi[500]]]],SquareFreeQ],tQ]  (* Harvey P. Dale, Feb 14 2011 *)
SequencePosition[Table[If[SquareFreeQ[n]&&!PrimeQ[n],1,0],{n,500}],{0,1,1,0}][[;;,1]]+1 (* Harvey P. Dale, Feb 27 2023 *)

A073251 Numbers k such that k, k+1 and k+2 are nonprime and squarefree.

Original entry on oeis.org

33, 85, 93, 141, 185, 201, 213, 217, 253, 265, 285, 301, 321, 393, 445, 453, 469, 481, 517, 533, 553, 581, 589, 609, 633, 669, 697, 705, 713, 753, 777, 789, 793, 805, 813, 869, 893, 897, 901, 913, 921, 933, 957, 985, 993, 1001, 1005, 1041, 1045, 1065, 1113

Offset: 1

Reinhard Zumkeller, Jul 22 2002

nonn

k-1 and k+3 are not squarefree. Proof: k is odd, otherwise k or k+2 would be divisible by 4. Thus k+1 is even and not divisible by 4, hence k-1 and k+3 are divisible by 4.

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

Cf. A000469, A005117, A007675, A073249, A073250, A121495.

f[upto_]:=Module[{pp=PrimePi[upto],n},lst=Partition[Complement[Range[upto], Prime[Range[pp]]],3,1];Transpose[Select[lst,And@@SquareFreeQ/@#&]][[1]]]; f[1200] (* Harvey P. Dale, Mar 21 2011 *)

isok1(k) = !isprime(k) && issquarefree(k); \\ A000469
isok(k) = isok1(k) && isok1(k+1) && isok1(k+2); \\ Michel Marcus, Mar 25 2021

Edited by Klaus Brockhaus, Aug 07 2006

cp's OEIS Frontend

A361102 1 together with numbers having at least two distinct prime factors.

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A119313 Numbers with a prime as third-smallest divisor.

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A303554 Union of the prime powers (p^k, p prime, k >= 0) and numbers that are the product of 2 or more distinct primes.

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A303606 Powers of composite squarefree numbers that are not squarefree.

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A177492 Products of squares of 2 or more distinct primes.

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A354911 Number of factorizations of n into relatively prime prime-powers.

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A380478 Nonprime numbers k such that A380459(k) has no divisors of form p^p.

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PARI