A178739 -id:A178739 - cp's OEIS frontend

A179642 Product of exactly 5 primes, 3 of which are distinct.

120, 168, 180, 252, 264, 270, 280, 300, 312, 378, 396, 408, 440, 450, 456, 468, 520, 552, 588, 594, 612, 616, 680, 684, 696, 700, 702, 728, 744, 750, 760, 828, 882, 888, 918, 920, 945, 952, 980, 984, 1026, 1032, 1044, 1064, 1100, 1116, 1128, 1144, 1160

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

nonn

			120=2^3*3*5, 168=2^3*3*7, 180=2^2*3^2*5, 252=2^2*3^2*7, 264=2^3*3*11, 270=2*3^3*5

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

Union of A189975 and A179643.

Cf. A006881, A007304, A085986, A085987, A178739.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,3} || Sort[Last/@FactorInteger[n]]=={1,2,2}; Select[Range[2000], f]

list(lim)=my(v=List(),t);forprime(p=2,(lim\6)^(1/3),forprime(q=2,sqrt(lim\p^3),if(p==q,next);t=p^3*q;forprime(r=q+1,lim\t,if(p==r,next);listput(v,t*r))));forprime(p=2,sqrt(lim\12),forprime(q=p+1,sqrt(lim\p^2\2),t=(p*q)^2;forprime(r=2,lim\t,if(p==r||q==r,next);listput(v,t*r))));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 19 2011

A189982 Numbers with prime signature (2,1,1,1), i.e., factorization pqr*s^2 with distinct primes p, q, r, s.

Original entry on oeis.org

420, 630, 660, 780, 924, 990, 1020, 1050, 1092, 1140, 1170, 1380, 1386, 1428, 1470, 1530, 1540, 1596, 1638, 1650, 1710, 1716, 1740, 1820, 1860, 1932, 1950, 2070, 2142, 2220, 2244, 2380, 2394, 2436, 2460, 2508, 2550, 2574, 2580, 2604, 2610, 2652, 2660, 2790

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 03 2011

nonn

Theorem 4 in Goldston-Graham-Pintz-Yildirim proves that a(n+1) = a(n) + 1 for infinitely many n. - Charles R Greathouse IV, Jul 17 2015, corrected by M. F. Hasler, Jul 17 2019

T. D. Noe, Table of n, a(n) for n = 1..1000
D. A. Goldston, S. W. Graham, J. Pintz, and C. Y. Yıldırım, Small gaps between almost primes, the parity problem, and some conjectures of Erdos on consecutive integers (2008)
Will Nicholes, List of Prime Signatures
Index to sequences related to prime signature

Part of the list A178739 .. A179696 and A030514 .. A030629, A189975 .. A189990 etc., cf. A101296.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,1,2}; Select[Range[4000],f]

is(n)=vecsort(factor(n)[,2])==[1, 1, 1, 2]~ \\ Charles R Greathouse IV, Jul 17 2015

Definition reworded by M. F. Hasler, Jul 17 2019

A255231 The number of factorizations n = Product_i b_i^e_i, where all bases b_i are distinct, and all exponents e_i are distinct >=1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 7, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 6, 1, 2, 2, 4, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 1, 5, 1, 1, 1, 2, 2, 1, 1, 3

Offset: 1

Saverio Picozzi, Feb 18 2015

nonn

Not multiplicative: a(48) = a(2^4*3) = 5 <> a(2^4)*a(3) = 4*1 = 4. - R. J. Mathar, Nov 05 2016

			From _R. J. Mathar_, Nov 05 2016: (Start)
a(4)=2: 4^1 = 2^2.
a(8)=2: 8^1 = 2^3.
a(9)=2: 9^1 = 3^2.
a(12)=2: 12^1 = 2^2*3^1.
a(16)=4: 16^1 = 4^2 = 2^2*4^1 = 2^4.
a(18)=2: 18^1 = 2*3^2.
a(20)=2: 20^1 = 2^2*5^1.
a(24)=3: 24^1 = 2^2*6^1 = 2^3*3^1.
a(32)=5: 32^1 = 2^1*4^2 = 2^2*8^1 = 2^3*4^1 = 2^5.
a(36)=4: 36^1 = 6^2 = 3^2*4^1 = 2^2*9^1.
a(48)=5: 48^1 = 3^1*4^2 = 2^2*12^1 = 2^3*6^1 = 2^4*3^1.
a(60)=2 : 60^1 = 2^2*15^1.
a(64)=7: 64^1 = 8^2 = 4^3 = 2^2*16^1 = 2^3*8^1 = 2^4*4^1 = 2^6.
a(72)=6 : 72^1 = 3^2*8^1 = 2^1*6^2 = 2^2*18^1 = 2^3*9^1 = 2^3*3^2.
(End)

R. J. Mathar, Table of n, a(n) for n = 1..419
R. J. Mathar, Factorizations of integers into factors with distinct bases and exponents
Index to sequences related to prime signature

Cf. A000688 (b_i not necessarily distinct).

Cf. A001248, A005117, A030078, A030514, A054753, A065036, A085986, A085987, A143610, A178739.

# Count solutions for products if n = dvs_i^exps(i) where i=1..pividx are fixed
Apiv := proc(n,dvs,exps,pividx)
    local dvscnt, expscopy,i,a,expsrt,e ;
    dvscnt := nops(dvs) ;
    a := 0 ;
    if pividx > dvscnt then
        # have exhausted the exponent list: leave of the recursion
        # check that dvs_i^exps(i) is a representation
        if n = mul( op(i,dvs)^op(i,exps),i=1..dvscnt) then
            # construct list of non-0 exponents
            expsrt := [];
            for i from 1 to dvscnt do
                if op(i,exps) > 0 then
                    expsrt := [op(expsrt),op(i,exps)] ;
                end if;
            end do;
            # check that list is duplicate-free
            if nops(expsrt) = nops( convert(expsrt,set)) then
                return 1;
            else
                return 0;
            end if;
        else
            return 0 ;
        end if;
    end if;
    # need a local copy of the list to modify it
    expscopy := [] ;
    for i from 1 to nops(exps) do
        expscopy := [op(expscopy),op(i,exps)] ;
    end do:
    # loop over all exponents assigned to the next base in the list.
    for e from 0 do
        candf := op(pividx,dvs)^e ;
        if modp(n,candf) <> 0 then
            break;
        end if;
        # assign e to the local copy of exponents
        expscopy := subsop(pividx=e,expscopy) ;
        a := a+procname(n,dvs,expscopy,pividx+1) ;
    end do:
    return a;
end proc:
A255231 := proc(n)
    local dvs,dvscnt,exps ;
    if n = 1 then
        return 1;
    end if;
    # candidates for the bases are all divisors except 1
    dvs := convert(numtheory[divisors](n) minus {1},list) ;
    dvscnt := nops(dvs) ;
    # list of exponents starts at all-0 and is
    # increased recursively
    exps := [seq(0,e=1..dvscnt)] ;
    # take any subset of dvs for the bases, i.e. exponents 0 upwards
    Apiv(n,dvs,exps,1) ;
end proc:
seq(A255231(n),n=1..120) ; # R. J. Mathar, Nov 05 2016

a(n)=1 for all n in A005117. a(n)=2 for all n in A001248 and for all n in A054753 and for all n in A085987 and for all n in A030078. a(n)=3 for all n in A065036. a(n)=4 for all n in A085986 and for all n in A030514. a(n)=5 for all n in A178739, all n in A179644 and for all n in A050997. a(n)=6 for all n in A143610, all n in A162142 and all n in A178740. a(n)=7 for all n in A030516. a(n)=9 for all n in A189988 and all n in A189987. a(n)=10 for all n in A092759. a(n) = 11 for all n in A179664. a(n)=12 for all n in A179646. - R. J. Mathar, Nov 05 2016, May 20 2017

Values corrected. Incorrect comments removed. - R. J. Mathar, Nov 05 2016

A343443 If n = Product (p_j^k_j) then a(n) = Product (k_j + 2), with a(1) = 1.

Original entry on oeis.org

1, 3, 3, 4, 3, 9, 3, 5, 4, 9, 3, 12, 3, 9, 9, 6, 3, 12, 3, 12, 9, 9, 3, 15, 4, 9, 5, 12, 3, 27, 3, 7, 9, 9, 9, 16, 3, 9, 9, 15, 3, 27, 3, 12, 12, 9, 3, 18, 4, 12, 9, 12, 3, 15, 9, 15, 9, 9, 3, 36, 3, 9, 12, 8, 9, 27, 3, 12, 9, 27, 3, 20, 3, 9, 12, 12, 9, 27, 3, 18

Offset: 1

Ilya Gutkovskiy, Apr 15 2021

Inverse Moebius transform of A056671.

a(n) depends only on the prime signature of n (see formulas). - Bernard Schott, May 03 2021

Cf. A000005, A001221, A005361, A007425, A025847, A034444, A056671, A064549, A074816, A107758, A107759, A348018, A363194.

a[1] = 1; a[n_] := Times @@ ((#[[2]] + 2) & /@ FactorInteger[n]); Table[a[n], {n, 80}]
a[n_] := Sum[If[GCD[d, n/d] == 1, DivisorSigma[0, d], 0], {d, Divisors[n]}]; Table[a[n], {n, 80}]

a(n) = sumdiv(n, d, if(gcd(d, n/d)==1, numdiv(d))) \\ Andrew Howroyd, Apr 15 2021

for(n=1, 100, print1(direuler(p=2, n, (1 + X - X^2)/(1-X)^2)[n], ", ")) \\ Vaclav Kotesovec, Feb 11 2023

from math import prod
from sympy import factorint
def A343443(n): return prod(e+2 for e in factorint(n).values()) # Chai Wah Wu, Feb 21 2025

a(n) = 2^omega(n) * tau_3(n) / tau(n), where omega = A001221, tau = A000005 and tau_3 = A007425.
a(n) = Sum_{d|n, gcd(d, n/d) = 1} tau(d).
From Bernard Schott, May 03 2021: (Start)
a(p^k) = k+2 for p prime, or signature [k].
a(A006881(n)) =  9 for signature [1, 1].
a(A054753(n)) = 12 for signature [2, 1].
a(A065036(n)) = 15 for signature [3, 1].
a(A085986(n)) = 16 for signature [2, 2].
a(A178739(n)) = 18 for signature [4, 1].
a(A143610(n)) = 20 for signature [3, 2].
a(A007304(n)) = 27 for signature [1, 1, 1]. (End)
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 1/p^s - 1/p^(2*s)). - Vaclav Kotesovec, Feb 11 2023
From Amiram Eldar, Sep 01 2023: (Start)
a(n) = A000005(A064549(n)).
a(n) = A363194(A348018(n)). (End)

A179689 Numbers with prime signature {7,2}, i.e., of form p^7*q^2 with p and q distinct primes.

Original entry on oeis.org

1152, 3200, 6272, 8748, 15488, 21632, 36992, 46208, 54675, 67712, 107163, 107648, 123008, 175232, 215168, 236672, 264627, 282752, 312500, 359552, 369603, 445568, 476288, 574592, 632043, 645248, 682112, 703125, 789507, 798848, 881792, 1013888

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, Numbers with same prime signature.
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.

Cf. A085548, A085967, A085969.

a:= proc(n) option remember; local k;
      for k from 1+ `if` (n=1, 1, a(n-1))
        while sort (map (x-> x[2], ifactors(k)[2]), `>`)<>[7, 2]
      do od; k
    end:
seq (a(n), n=1..32);  # Alois P. Heinz, Jan 23 2011

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,7}; Select[Range[10^6], f]

list(lim)=my(v=List(),t);forprime(p=2, (lim\4)^(1/7), t=p^7;forprime(q=2, sqrt(lim\t), if(p==q, next);listput(v,t*q^2))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A179689(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(isqrt(x//p**7)) for p in primerange(integer_nthroot(x,7)[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(2)*P(7) - P(9) = A085548 * A085967 - A085969 = 0.001741..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

Title edited by Daniel Forgues, Jan 22 2011

A179696 Numbers with prime signature {7,1,1}, i.e., of form p^7qr with p, q and r distinct primes.

Original entry on oeis.org

1920, 2688, 4224, 4480, 4992, 6528, 7040, 7296, 8320, 8832, 9856, 10880, 11136, 11648, 11904, 12160, 14208, 14720, 15232, 15744, 16512, 17024, 18048, 18304, 18560, 19840, 20352, 20608, 21870, 22656, 23424, 23680, 23936, 25728, 25984, 26240, 26752, 27264

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes List of Prime Signatures
OEIS Wiki, Numbers with same prime signature.

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.

a:= proc(n) option remember; local k;
      for k from 1+ `if` (n=1, 1, a(n-1))
        while sort (map (x-> x[2], ifactors(k)[2]), `>`)<>[7, 1, 1]
      do od; k
    end:
seq (a(n), n=1..40); # Alois P. Heinz, Jan 23 2011

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,7}; Select[Range[30000], f]

list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\6)^(1/7), t1=p^7;forprime(q=2, lim\t1, if(p==q, next);t2=t1*q;forprime(r=q+1, lim\t2, if(p==r,next);listput(v,t2*r)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

from math import isqrt
from sympy import primerange, primepi, integer_nthroot
def A179696(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((t:=primepi(s:=isqrt(y:=x//r**7)))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(integer_nthroot(x,7)[0]+1))+sum(primepi(x//p**8) for p in primerange(integer_nthroot(x,8)[0]+1))-primepi(integer_nthroot(x,9)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

Title edited by Daniel Forgues, Jan 22 2011

A369937 Numbers whose maximal exponent in their prime factorization is square.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102

Offset: 1

Amiram Eldar, Feb 06 2024

First differs from A366762 at n = 84, and from A197680, A361177 and A369210 at n = 95.
Numbers k such that A051903(k) is square.
The asymptotic density of this sequence is 1/zeta(2) + Sum_{k>=2} (1/zeta(k^2+1) - 1/zeta(k^2)) = 0.64939447949574562687... .

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

Cf. A000290, A013661, A051903.
Subsequences: A002416, A005117, A030514, A030635, A113849, A178739, A179665, A179692, A197680.
Similar sequences: A368714, A369938, A369939.
Cf. A361177, A366762, A369210.

Select[Range[100], IntegerQ@ Sqrt[Max[FactorInteger[#][[;; , 2]]]] &]

lista(kmax) = for(k = 1, kmax, if(k == 1 || issquare(vecmax(factor(k)[, 2])), print1(k, ", ")));

A179690 Numbers of the form p^2q^2r*s where p, q, r, and s are distinct primes.

Original entry on oeis.org

1260, 1980, 2100, 2340, 2772, 2940, 3060, 3150, 3276, 3300, 3420, 3900, 4140, 4284, 4410, 4788, 4950, 5100, 5148, 5220, 5580, 5700, 5796, 5850, 6468, 6660, 6732, 6900, 7260, 7308, 7350, 7380, 7524, 7644, 7650, 7700, 7740, 7812, 7956, 8460, 8550, 8700

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
Index to sequences related to prime signature

Part of the list A178739 .. A179696 and A030514 .. A030629, A189975 .. A189990 etc., cf. A101296. - M. F. Hasler, Jul 17 2019

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,2,2}; Select[Range[10000], f]

list(lim)=my(v=List(),t1,t2,t3); forprime(p=2,sqrtint(lim\60), t1=p^2; forprime(q=2,sqrtint(lim\(6*t1)), if(q==p, next); t2=q^2*t1; forprime(r=2,lim\(2*t2), if(r==p || r==q, next); t3=r*t2; forprime(s=2,lim\t3, if(s==p || s==q || s==r, next); listput(v, t3*s))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

A179691 Numbers p^5q^2r where p, q, r are 3 distinct primes.

Original entry on oeis.org

1440, 2016, 2400, 3168, 3744, 4704, 4860, 4896, 5472, 5600, 6624, 6804, 7840, 8352, 8800, 8928, 10400, 10656, 10692, 11616, 11808, 12150, 12384, 12636, 13536, 13600, 15200, 15264, 16224, 16524, 16992, 17248, 17568, 18400, 18468, 19296, 19360

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

nonn

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

Part of the list A178739 .. A179696 (and A030514 .. A030629, A189982 .. A189990 etc, cf. A101296). - M. F. Hasler, Jul 17 2019

Subsequence of A175746 (numbers with 36 divisors).

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,2,5}; Select[Range[20000], f]

list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\12)^(1/5), t1=p^5;forprime(q=2, sqrt(lim\t1), if(p==q, next);t2=t1*q^2;forprime(r=2, lim\t2, if(p==r||q==r, next);listput(v,t2*r)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A179691(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(x//(p**5*q**2)) for p in primerange(integer_nthroot(x,5)[0]+1) for q in primerange(isqrt(x//p**5)+1))+sum(primepi(integer_nthroot(x//p**5,3)[0]) for p in primerange(integer_nthroot(x,5)[0]+1))+sum(primepi(isqrt(x//p**6)) for p in primerange(integer_nthroot(x,6)[0]+1))+sum(primepi(x//p**7) for p in primerange(integer_nthroot(x,7)[0]+1))-(primepi(integer_nthroot(x,8)[0])<<1)
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

Name improved by M. F. Hasler, Jul 17 2019

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

cp's OEIS Frontend

A179642 Product of exactly 5 primes, 3 of which are distinct.

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A189982 Numbers with prime signature (2,1,1,1), i.e., factorization p*q*r*s^2 with distinct primes p, q, r, s.

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A255231 The number of factorizations n = Product_i b_i^e_i, where all bases b_i are distinct, and all exponents e_i are distinct >=1.

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A343443 If n = Product (p_j^k_j) then a(n) = Product (k_j + 2), with a(1) = 1.

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A179689 Numbers with prime signature {7,2}, i.e., of form p^7*q^2 with p and q distinct primes.

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A179696 Numbers with prime signature {7,1,1}, i.e., of form p^7*q*r with p, q and r distinct primes.

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A369937 Numbers whose maximal exponent in their prime factorization is square.

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A189982 Numbers with prime signature (2,1,1,1), i.e., factorization pqr*s^2 with distinct primes p, q, r, s.

A179696 Numbers with prime signature {7,1,1}, i.e., of form p^7qr with p, q and r distinct primes.

A179690 Numbers of the form p^2q^2r*s where p, q, r, and s are distinct primes.

A179691 Numbers p^5q^2r where p, q, r are 3 distinct primes.