A085986 -id:A085986 - cp's OEIS frontend

A177492 Products of squares of 2 or more distinct primes.

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

A179694 Numbers of the form p^6*q^3 where p and q are distinct primes.

Original entry on oeis.org

1728, 5832, 8000, 21952, 85184, 91125, 125000, 140608, 250047, 314432, 421875, 438976, 778688, 941192, 970299, 1560896, 1601613, 1906624, 3176523, 3241792, 3581577, 4410944, 5000211, 5088448, 5359375, 6644672

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, Prime Signatures

Cf. A006881, A007304, A065036, A085986, A085987, A092759, A178739, A179642, A179643, A179644, A179645, A179646, A179664, A179665, A179666, A179667, A179668, A179669, A179670, A179671, A179672, A179688, A179689, A179690, A179691, A179692, A179693.

Cf. A085541, A085966, A085969.

f[n_]:=Sort[Last/@FactorInteger[n]]=={3,6}; Select[Range[10^6], f]

list(lim)=my(v=List(),t);forprime(p=2, (lim\8)^(1/6), t=p^6;forprime(q=2, (lim\t)^(1/3), if(p==q, next);listput(v,t*q^3))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011

from sympy import primepi, integer_nthroot, primerange
def A179694(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 f(x): return n+x-sum(primepi(integer_nthroot(x//p**6,3)[0]) for p in primerange(integer_nthroot(x,6)[0]+1))+primepi(integer_nthroot(x,9)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Sum_{n>=1} 1/a(n) = P(3)*P(6) - P(9) = A085541 * A085966 - A085969 = 0.000978..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

a(n) = A054753(n)^3. - R. J. Mathar, May 05 2023

A179702 Numbers of the form p^4*q^5 where p and q are two distinct primes.

Original entry on oeis.org

2592, 3888, 20000, 50000, 76832, 151875, 253125, 268912, 468512, 583443, 913952, 1361367, 2576816, 2672672, 3557763, 4170272, 5940688, 6940323, 7503125, 8954912, 10504375, 13045131, 20295603, 22632992, 22717712, 29552672, 30074733

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

nonn

Subsequence of A046312 and of A137493. - R. J. Mathar, Jul 27 2010

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, List of Prime Signatures

Cf. A006881, A007304, A065036, A085986, A085987, A092759, A178739, A179642, A179643, A179644, A179645, A179646, A179664, A179665, A179666, A179667, A179668, A179669, A179670, A179671, A179672, A179688, A179689, A179690, A179691, A179692, A179693, A179694, A179695, A179696, A179698, A179699, A179700.

Cf. A085964, A085965, A085969.

fQ[n_] := Sort[Last /@ FactorInteger @n] == {4, 5}; Select[ Range@ 31668000, fQ] (* fixed by Robert G. Wilson v, Aug 26 2010 *)
lst = {}; Do[ If[p != q, AppendTo[lst, Prime@p^4*Prime@q^5]], {p, 12}, {q, 10}]; Take[ Sort@ Flatten@ lst, 27] (* Robert G. Wilson v, Aug 26 2010 *)
Take[Union[First[#]^4 Last[#]^5&/@Flatten[Permutations/@Subsets[ Prime[ Range[30]],{2}],1]],30] (* Harvey P. Dale, Jan 01 2012 *)

list(lim)=my(v=List(),t);forprime(p=2, (lim\16)^(1/5), t=p^5;forprime(q=2, (lim\t)^(1/4), if(p==q, next);listput(v,t*q^4))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

from sympy import primepi, integer_nthroot, primerange
def A179702(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 f(x): return n+x-sum(primepi(integer_nthroot(x//p**5,4)[0]) for p in primerange(integer_nthroot(x,5)[0]+1))+primepi(integer_nthroot(x,9)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

Sum_{n>=1} 1/a(n) = P(4)*P(5) - P(9) = A085964 * A085965 - A085969 = 0.000748..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

Edited and extended by Ray Chandler and R. J. Mathar, Jul 26 2010

A189991 Numbers with prime factorization p^4*q^4.

Original entry on oeis.org

1296, 10000, 38416, 50625, 194481, 234256, 456976, 1185921, 1336336, 1500625, 2085136, 2313441, 4477456, 6765201, 9150625, 10556001, 11316496, 14776336, 17850625, 22667121, 29986576, 35153041, 45212176, 52200625, 54700816, 57289761, 68574961, 74805201

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 03 2011

nonn

The primes p and q must be distinct, or else the product has factorization p^8 (or q^8, for that matter).

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, List of Prime Signatures
Index to sequences related to prime signature

Cf. A006881, A137488, A179671, A189990.

Cf. A085964, A085968, A085986.

lst = {}; Do[If[Sort[Last/@FactorInteger[n]] == {4, 4}, Print[n]; AppendTo[lst, n]], {n,55000000}]; lst (* Orlovsky *)
lim = 10^8; pMax = PrimePi[(lim/16)^(1/4)]; Select[Union[Flatten[Table[Prime[i]^4 Prime[j]^4, {i, 2, pMax}, {j, i - 1}]]], # <= lim &] (* Alonso del Arte, May 03 2011 *)
With[{nn=30},Take[Union[Times@@@(Subsets[Prime[Range[nn]],{2}]^4)],nn]] (* Harvey P. Dale, Mar 05 2015 *)

list(lim)=my(v=List(),t);forprime(p=2, lim^(1/8), t=p^4;forprime(q=p+1, (lim\t)^(1/4), listput(v,t*q^4))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A189991(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 f(x): return int(n+x+(t:=primepi(s:=isqrt(y:=integer_nthroot(x,4)[0])))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)))
    return bisection(f,n,n) # Chai Wah Wu, Feb 22 2025

Sum_{n>=1} 1/a(n) = (P(4)^2 - P(8))/2 = (A085964^2 - A085968)/2 = 0.000933..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

a(n) = A006881(n)^4 = A085986(n)^2. - Chai Wah Wu, Mar 27 2025

A275387 Numbers of ordered pairs of divisors d < e of n such that gcd(d, e) > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 3, 1, 2, 0, 8, 0, 2, 2, 6, 0, 8, 0, 8, 2, 2, 0, 18, 1, 2, 3, 8, 0, 15, 0, 10, 2, 2, 2, 24, 0, 2, 2, 18, 0, 15, 0, 8, 8, 2, 0, 32, 1, 8, 2, 8, 0, 18, 2, 18, 2, 2, 0, 44, 0, 2, 8, 15, 2, 15, 0, 8, 2, 15, 0, 49, 0, 2, 8, 8, 2, 15, 0, 32, 6, 2

Offset: 1

Michel Lagneau, Aug 03 2016

nonn

Number of elements in the set {(x, y): x|n, y|n, x < y, gcd(x, y) > 1}.
Every element of the sequence is repeated indefinitely, for instance:
a(n)=0 if n prime;
a(n)=1 if n = p^2 for p prime (A001248);
a(n)=2 if n is a squarefree semiprime (A006881);
a(n)=3 if n = p^3 for p prime (A030078);
a(n)=6 if n = p^4 for p prime (A030514);
a(n)=8 if n is a number which is the product of a prime and the square of a different prime (A054753);
a(n)=10 if n = p^5 for p prime (A050997);
a(n)=15 if n is in the set {A007304} union {64} = {30, 42, 64, 66, 70,...} = {Sphenic numbers} union {64};
a(n)=18 if n is the product of the cube of a prime (A030078) and a different prime (see A065036);
a(n)=21 if n = p^7 for p prime (A092759);
a(n)=24 if n is square of a squarefree semiprime (A085986);
a(n)=32 if n is the product of the 4th power of a prime (A030514) and a different prime (see A178739);
a(n)=36 if n = p^9 for p prime (A179665);
a(n)=44 if n is the product of exactly four primes, three of which are distinct (A085987);
a(n)=45 if n is a number with 11 divisors (A030629);
a(n)=49 if n is of the form p^2*q^3, where p,q are distinct primes (A143610);
a(n)=50 if n is the product of the 5th power of a prime (A050997) and a different prime (see A178740);
a(n)=55 if n if n = p^11 for p prime(A079395);
a(n)=72 if n is a number with 14 divisors (A030632);
a(n)=80 if n is the product of four distinct primes (A046386);
a(n)=83 if n is a number with 15 divisors (A030633);
a(n)=89 if n is a number with prime factorization pqr^3 (A189975);
a(n)=96 if n is a number that are the cube of a product of two distinct primes (A162142);
a(n)=98 if n is the product of the 7th power of a prime and a distinct prime (p^7*q) (A179664);
a(n)=116 if n is the product of exactly 2 distinct squares of primes and a different prime (p^2*q^2*r) (A179643);
a(n)=126 if n is the product of the 5th power of a prime and different distinct prime of the 2nd power (p^5*q^2) (A179646);
a(n)=128 if n is the product of the 8th power of a prime and a distinct prime (p^8*q) (A179668);
a(n)=150 if n is the product of the 4th power of a prime and 2 different distinct primes (p^4*q*r) (A179644);
a(n)=159 if n is the product of the 4th power of a prime and a distinct prime of power 3 (p^4*q^3) (A179666).
It is possible to continue with a(n) = 162, 178, 209, 224, 227, 238, 239, 260, 289, 309, 320, 333,...

			a(12) = 8 because the divisors of 12 are {1, 2, 3, 4, 6, 12} and GCD(d_i, d_j)>1 for the 8 following pairs of divisors: (2,4), (2,6), (2,12), (3,6), (3,12), (4,6), (4,12) and (6,12).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001248, A006881, A007304, A030078, A030514, A030632, A046386, A050997, A054753, A063647, A065036, A066446, A079395, A085986, A085987, A092759, A143610, A162142, A178739, A178740, A179644, A179646, A179664, A189975.

Cf. A333976 (same with d <= e).

with(numtheory):nn:=100:
for n from 1 to nn do:
x:=divisors(n):n0:=nops(x):it:=0:
for i from 1 to n0 do:
  for j from i+1 to n0 do:
   if gcd(x[i],x[j])>1
    then
    it:=it+1:
    else
   fi:
  od:
od:
  printf(`%d, `,it):
od:

Table[Sum[Sum[(1 - KroneckerDelta[GCD[i, k], 1]) (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k - 1}], {k, n}], {n, 100}] (* Wesley Ivan Hurt, Jan 01 2021 *)

a(n)=my(d=divisors(n)); sum(i=2,#d, sum(j=1,i-1, gcd(d[i],d[j])>1)) \\ Charles R Greathouse IV, Aug 03 2016

a(n)=my(f=factor(n)[,2],t=prod(i=1,#f,f[i]+1)); t*(t-1)/2 - (prod(i=1,#f,2*f[i]+1)+1)/2 \\ Charles R Greathouse IV, Aug 03 2016

a(n) = A066446(n) - A063647(n).

a(n) = Sum_{d1|n, d2|n, d1Wesley Ivan Hurt, Jan 01 2021

A323090 Number of strict factorizations of n using elements of A007916 (numbers that are not perfect powers).

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 2, 1, 2, 2, 0, 1, 2, 1, 2, 2, 2, 1, 2, 0, 2, 0, 2, 1, 5, 1, 0, 2, 2, 2, 3, 1, 2, 2, 2, 1, 5, 1, 2, 2, 2, 1, 2, 0, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 7, 1, 2, 2, 0, 2, 5, 1, 2, 2, 5, 1, 4, 1, 2, 2, 2, 2, 5, 1, 2, 0, 2, 1, 7, 2, 2, 2

Offset: 1

Gus Wiseman, Jan 04 2019

nonn

			The a(72) = 4 factorizations are (2*3*12), (3*24), (6*12), (72). Missing from this list and not strict are (2*2*2*3*3), (2*2*3*6), (2*6*6), (2*2*18), while missing from the list and using perfect powers are (2*36), (2*4*9), (3*4*6), (4*18), (8*9).

Positions of 0's are A246547.
Positions of 1's are A000040.
Positions of 2's are A084227.
Positions of 3's are A085986.
Positions of 4's are A143610.
Cf. A001055, A001597, A007916, A025147, A045778, A052410, A303707, A323054, A323087, A323088, A323089.

radQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]==1];
facssr[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facssr[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],radQ]}]];
Table[Length[facssr[n]],{n,100}]

A379336 Numbers k such that there exists a divisor pair (d, d/k) such that one neither divides nor is coprime to the other.

Original entry on oeis.org

24, 40, 48, 54, 56, 60, 72, 80, 84, 88, 90, 96, 104, 108, 112, 120, 126, 132, 135, 136, 140, 144, 150, 152, 156, 160, 162, 168, 176, 180, 184, 189, 192, 198, 200, 204, 208, 216, 220, 224, 228, 232, 234, 240, 248, 250, 252, 260, 264, 270, 272, 276, 280, 288, 294

Offset: 1

Michael De Vlieger, Dec 24 2024

Both divisors d and d/k are composite, since primes p either divide or are coprime to another number, and all numbers smaller than p are coprime to p.
Proper subset of A126706; contains A378769, which in turn contains A378984.
Consider composite k, m, k != m. Define a "neutral" relation to be such that 1 < gcd(k,m) and not equal to either k or m. Then neither k nor m divides the other, and k and m are not coprime. If k is neutral to m, then m is neutral to k, since order does not matter. Then either the squarefree kernel of one divides the other or it does not. Thus, there are 3 kinds of neutral relation:
Type A: Though gcd(k,m) > 1, k has a factor P that does not divide m, and m has a factor Q that does not divide k.
Type B: rad(k) = rad(m), yet neither k divides m nor m divides k, where rad = A007947 is the squarefree kernel.
Type C: Squarefree kernel of one number divides the other, while the other has a factor that does not divide the former.
A378769, subset of this sequence, contains numbers k that have all 3 types of neutral relation between at least 1 divisor pair (d, k/d) for each.

			a(1) = 24 = 2^3 * 3 = 4*6, both composite; gcd(4,6) = 2, 4 does not divide 6 (type C).
a(2) = 40 = 2^3 * 5 = 4*10, gcd(4,10) = 2 (type C).
a(3) = 48 = 2^4 * 3 = 6*8, gcd(6,8) = 2 (type C).
a(6) = 60 = 2^2 * 3 * 5 = 6*10, gcd(6,10) = 2 (type A).
a(12) = 96 = 2^5 * 3 = 6*16 = 8*12, both type C.
a(38) = 216 = 2^3 * 3^3 = 4*54 (type C) = 9*24 (type C) = 12*18 (type B)
a(1605) = 5400 = 2^3 * 3^3 * 5^2 = 4*1350 (type C) = 24*225 (type A) = 60*90 (type B) = A378769(1).
a(10475) = 32400 = 2^4 * 3^4 * 5^2 = 8*4050 (type C) = 48*675 (type A) = 120*270 (type B) = A378984(1) = A378769(14), etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Chart of (d, a(n)/d) for a(n) = 1..144, showing only the smallest d for each type of neutral relation, where type A is shown in gray, type B in black, and type C in either blue or gold.

Cf. A045763, A085986, A126706, A375055, A376271, A376936, A378767, A378769, A378984.

nn = 300; mm = Floor@ Sqrt[nn]; p = 2; q = 3;
Complement[
  Select[Range[nn], And[#2 > #1 > 1, #2 > 3] & @@ {PrimeNu[#], PrimeOmega[#]} &],
  Union[Reap[
    While[p <= mm, q = NextPrime[p];
      While[p*q <= mm, If[p != q, Sow[p*q]]; q = NextPrime[q]];
        p = NextPrime[p] ] ][[-1, 1]] ]^2 ]

This sequence is A376271 \ A085986 = {k : bigomega(k) > omega(k) > 1, bigomega(k) > 3} \ { k^2 : bigomega(k) = omega(k) = 2 }, where bigomega = A001222 and omega = A001221.

Union of A375055, A376936, and A378767.

A331593 Numbers k that have the same number of distinct prime factors as A225546(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 28, 29, 31, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109, 112, 113, 116, 117, 121, 124, 127, 131, 135, 136, 137, 139, 144, 147, 148, 149

Offset: 1

Antti Karttunen and Peter Munn, Jan 21 2020

nonn

Numbers k for which A001221(k) = A331591(k).
Numbers k that have the same number of terms in their factorization into powers of distinct primes as in their factorization into powers of squarefree numbers with distinct exponents that are powers of 2. See A329332 for a description of the relationship between the two factorizations and A225546.
If k is included, then all such x that A046523(x) = k are also included, i.e., all numbers with the same prime signature as k. Notably, primes (A000040) are included, but squarefree semiprimes (A006881) are not.
k^2 is included if and only if k is included, for example A001248 is included, but A085986 is not.

			There are 2 terms in the factorization of 36 into powers of distinct primes, which is 36 = 2^2 * 3^2 = 4 * 9; but only 1 term in its factorization into powers of squarefree numbers with distinct exponents that are powers of 2, which is 36 = 6^(2^1). So 36 is not included.
There are 2 terms in the factorization of 40 into powers of distinct primes, which is 40 = 2^3 * 5^1 = 8 * 5; and also 2 terms in its factorization into powers of squarefree numbers with distinct exponents that are powers of 2, which is 40 = 10^(2^0) * 2^(2^1) = 10 * 4. So 40 is included.

Cf. A000120, A046523, A225546, A267116, A329332.
Sequences with related definitions: A001221, A331591, A331592.
Subsequences: A000040, A001248, A050376, A054753, A065036, A162142, A225547.
Subsequences of complement: A006881, A056824, A085986, A120944, A177492.

Select[Range@ 150, Equal @@ PrimeNu@ {#, If[# == 1, 1, Apply[Times, Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]]]} &] (* Michael De Vlieger, Jan 26 2020 *)

A331591(n) = if(1==n,0,my(f=factor(n),u=#binary(vecmax(f[, 2])),xs=vector(u),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),xs[i]++)); m<<=1); #select(x -> (x>0),xs));
k=0; n=0; while(k<105, n++; if(omega(n)==A331591(n), k++; print1(n,", ")));

{a(n)} = {k : A001221(k) = A000120(A267116(k))}.

A077448 Numbers k such that Sum_{d|k} mu(d)*mu(k/d)^2 = +1.

Original entry on oeis.org

1, 36, 100, 196, 225, 441, 484, 676, 1089, 1156, 1225, 1444, 1521, 2116, 2601, 3025, 3249, 3364, 3844, 4225, 4761, 5476, 5929, 6724, 7225, 7396, 7569, 8281, 8649, 8836, 9025, 11236, 12321, 13225, 13924, 14161, 14884, 15129, 16641, 17689, 17956

Offset: 1

Benoit Cloitre, Nov 30 2002

nonn

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

Cf. A008683, A030229, A077438, A085986 (subsequence).

Select[Range[135], MoebiusMu[#] == 1 &]^2 (* Amiram Eldar, Jul 16 2020 *)

is(n)=if(!issquare(n,&n),return(0)); my(f=factor(n)[,2]); if(n>1, #f%2==0 && vecmax(f)==1, n==1) \\ Charles R Greathouse IV, Oct 16 2015

a(n) = A030229(n)^2.

Sum_{n>=1} 1/a(n) = 21/(2*Pi^2). - Amiram Eldar, Jul 16 2020

A162144 Products of cubes of 3 distinct primes.

Original entry on oeis.org

27000, 74088, 287496, 343000, 474552, 1061208, 1157625, 1331000, 1481544, 2197000, 2628072, 3652264, 4492125, 4913000, 5268024, 6028568, 6434856, 6859000, 7414875, 10941048, 12167000, 12326391, 13481272, 14886936, 16581375, 17173512, 18821096

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 25 2009

nonn

Numbers of the form p^3*q^3*r^3 where p, q, r are three distinct primes.

The cubic analog of A085986 (squares of 2 distinct primes).

			27000 = 2^3*3^3*5^3. 74088 = 2^3*3^3*7^3. 287496 = 2^3*3^3*11^3.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A006881, A085986, A162142, A162143.

Cf. A085541, A085966, A085969.

fQ[n_]:=Last/@FactorInteger[n]=={1,1,1}; Select[Range[1000], fQ]^3

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A162144(n):
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1),1) for b,m in enumerate(primerange(k+1,isqrt(x//k)+1),a+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
    return bisection(f)**3 # Chai Wah Wu, Aug 30 2024

a(n) = (A007304(n))^3.
A000005(a(n)) = 64.
Sum_{n>=1} 1/a(n) = (P(3)^3 + 2*P(9) - 3*P(3)*P(6))/6 = (A085541^3 + 2*A085969 - 3*A085541*A085966)/6 = 0.0000661486..., where P is the prime zeta function. - Amiram Eldar, Oct 30 2020

Edited by R. J. Mathar, Aug 14 2009

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