A068936 -id:A068936 - cp's OEIS frontend

A054411 Numbers k such that Sum_{j} p_j = Sum_{j} e_j where Product_{j} p_j^(e_j) is the prime factorization of k.

1, 4, 27, 48, 72, 108, 162, 320, 800, 1792, 2000, 3125, 3840, 5000, 5760, 6272, 8640, 9600, 10935, 12500, 12960, 14400, 18225, 19440, 21504, 21600, 21952, 24000, 29160, 30375, 31250, 32256, 32400, 36000, 43740, 45056, 48384, 48600, 50625, 54000, 60000, 65610

Offset: 1

Leroy Quet, May 09 2000

nonn

Numbers for which the sum of distinct prime factors equals the sum of exponents in the prime factorization, A008472(n)=A001222(n). - Reinhard Zumkeller, Mar 08 2002

			320 is included because 320 = 2^6 * 5^1 and 2+5 = 6+1.

Giuseppe Coppoletta and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 100 terms from G. Coppoletta)

Cf. A054412, A068935, A068936, A068937, A068938.

f[n_]:=Plus@@First/@FactorInteger[n]==Plus@@Last/@FactorInteger[n]; lst={};Do[If[f[n],AppendTo[lst,n]],{n,0,3*8!}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 19 2010 *)
max = 10^12; Sort@Reap[Sow@1; Do[p = Select[IntegerPartitions[se, All, Prime@ Range@ PrimePi@ se], Sort[#] == Union[#] &]; Do[ np = Length[f]; va = IntegerPartitions[se, {np}, Range[se]]; Do[pe = Permutations[v]; Do[z = Times @@ (f^e); If[z <= max, Sow@z], {e, pe}], {v, va}], {f, p}], {se, 2, Log2[max]}]][[2, 1]] (* Giovanni Resta, May 07 2016 *)

for(n=1,10^6,if(bigomega(n)==sumdiv(n,d,isprime(d)*d),print1(n,",")))

is(n)=my(f=factor(n)); sum(i=1,#f~, f[i,1]-f[i,2])==0 \\ Charles R Greathouse IV, Sep 08 2016

def d(n):
    v=factor(n)[:]; L=len(v); s0=sum(v[j][0] for j in range(L)); s1=sum(v[j][1] for j in range(L))
    return s0-s1
[k for k in (1..100000) if d(k)==0] # Giuseppe Coppoletta, May 07 2016

A068938 Numbers having the sum of distinct prime factors greater than the sum of exponents in prime factorization, A008472(n)>A001222(n).

Original entry on oeis.org

2, 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Reinhard Zumkeller, Mar 08 2002

nonn

			12 is included because 12 = 2^2 * 3^1 and 2+3 > 2+1.

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

Cf. A068935, A068936, A054411, A068937.

sdfQ[n_]:=Module[{fi=Transpose[FactorInteger[n]]},Total[fi[[1]]] > Total[ fi[[2]]]]; Select[Range[80],sdfQ] (* Harvey P. Dale, Jul 23 2013 *)

More terms from David Wasserman, Jun 17 2002

A068935 Numbers having the sum of distinct prime factors less than the sum of exponents in prime factorization, A008472(n) < A001222(n).

Original entry on oeis.org

8, 16, 32, 64, 81, 96, 128, 144, 192, 216, 243, 256, 288, 324, 384, 432, 486, 512, 576, 640, 648, 729, 768, 864, 972, 1024, 1152, 1280, 1296, 1458, 1536, 1600, 1728, 1944, 2048, 2187, 2304, 2560, 2592, 2916, 3072, 3200, 3456, 3584, 3888, 4000, 4096, 4374, 4608

Offset: 1

Reinhard Zumkeller, Mar 08 2002

nonn

			144 is included because 144 = 2^4 * 3^2 and 2+3 < 4+2.

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

Cf. A068936, A054411, A068937, A068938.

okQ[n_]:=Module[{tfi=Transpose[FactorInteger[n]]},Total[First[tfi]]Harvey P. Dale, Jan 17 2011 *)

isok(n) = vecsum(factor(n)[,1]) < bigomega(n); \\ Michel Marcus, Apr 25 2016

More terms from Michel Marcus, Apr 25 2016

A068937 Numbers having the sum of distinct prime factors not less than the sum of exponents in prime factorization, A008472(n)>=A001222(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76

Offset: 1

Reinhard Zumkeller, Mar 08 2002

nonn

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

Cf. A068935, A068936, A054411, A068938.

Select[Range[76], Plus @@ (f = FactorInteger[#])[[;;,1]] >= Plus @@ f[[;;,2]] &] (* Amiram Eldar, Nov 14 2019 *)

More terms from David Wasserman, Jun 17 2002

A090081 Cube root-smooth numbers: numbers k whose largest prime factor does not exceed the cube root of k.

Original entry on oeis.org

1, 8, 16, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 125, 128, 135, 144, 150, 160, 162, 180, 192, 200, 216, 225, 240, 243, 250, 256, 270, 288, 300, 320, 324, 343, 350, 360, 375, 378, 384, 392, 400, 405, 420, 432, 441, 448, 450, 480, 486, 490, 500, 504, 512, 525

Offset: 1

Labos Elemer, Nov 21 2003

What is the asymptotic growth of this sequence?

Answer: a(n) ~ k*n, where k = 1/A175475. That is, about 4.8% of numbers are in this sequence. - Charles R Greathouse IV, Jul 14 2014

			378 = 2 * 3^3 * 7 is a term of the sequence since 7 < 7.23... = 378^(1/3).

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

Cf. A063763, A001248, A048098, A036966, A054744, A059172, A068936, A076405, A076467, A006530, A175475.

filter:= n ->
evalb(max(seq(f[1],f=ifactors(n)[2]))^3 <= n):
select(filter, [$1..1000]); # Robert Israel, Jul 14 2014

ffi[x_] := Flatten[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; lf[x_] := Length[FactorInteger[x]]; ma[x_] := Max[ba[x]]; Do[If[ !Greater[ma[n], gy=n^(1/3)//N]&&!PrimeQ[n], Print[n(*, {gy, ma[n]}*)]], {n, 1, 1000}]
Select[Range[1000], (FactorInteger[#][[-1,1]])^3 <= # &] (* T. D. Noe, Sep 14 2011 *)
Select[Range[1000],FactorInteger[#][[-1,1]]<=CubeRoot[#]&] (* Harvey P. Dale, Jun 30 2025 *)

is(n)=my(f=factor(n)[,1]);f[#f]^3<=n \\ Charles R Greathouse IV, Sep 14 2011

from sympy import primefactors
def ok(n):
    if n==1 or max(primefactors(n))**3<=n: return True
    else: return False
print([n for n in range(1, 1001) if ok(n)]) # Indranil Ghosh, Apr 23 2017

Solutions to A006530(n) <= n^(1/3).

A113645 Numbers k such that sum of exponents in prime factorization of k (i.e., A001222(k)) is >= each prime divisor of k.

Original entry on oeis.org

4, 8, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 80, 81, 96, 108, 120, 128, 144, 160, 162, 180, 192, 200, 216, 240, 243, 256, 270, 288, 300, 320, 324, 360, 384, 400, 405, 432, 448, 450, 480, 486, 500, 512, 540, 576, 600, 640, 648, 672, 675, 720, 729, 750, 768

Offset: 1

Leroy Quet, Jan 15 2006

nonn

			12 = 2^2 *3^1. Since the sum of the prime factorization exponents, 2+1 = 3, is >= the largest prime dividing 12, which is 3, then 12 is included in the sequence.

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

A proper subset of A068936.

Cf. A001222, A006530.

fQ[n_] := Block[{f = FactorInteger@n}, Plus @@ Last /@ f >= Max[First /@ f]]; Select[ Range[2, 800], fQ@ # &] (* Robert G. Wilson v, Jan 16 2006 *)
qu[n_]:=n>1&&Block[{f=Transpose@FactorInteger@n, s}, s=Plus@@f[[2]];s>=Max@f[[1]]]; L ={};Do[If[qu[n], Print[n];AppendTo[L, n]], {n, 1000}];L (* Giovanni Resta, Jan 16 2006 *)

isok(m) = {my(f=factor(m)); #select(x->(x>bigomega(f)), f[,1]~) == 0;} \\ Michel Marcus, Sep 17 2020

A number k is included if A006530(k) <= A001222(k).

More terms from Robert G. Wilson v and Giovanni Resta, Jan 16 2006

A356433 Numbers k such that, in the prime factorization of k, the least common multiple of the exponents equals the least common multiple of the prime factors.

Original entry on oeis.org

1, 4, 27, 72, 108, 192, 576, 800, 1458, 1728, 2916, 3125, 5120, 5832, 6272, 12500, 21600, 25600, 30375, 36000, 46656, 48600, 77760, 84375, 114688, 116640, 121500, 138240, 169344, 225000, 247808, 337500, 384000, 388800, 395136, 583200, 600000, 653184, 691200, 750141, 802816, 823543, 857304, 979776

Offset: 1

Jean-Marc Rebert, Aug 07 2022

nonn

Numbers k such that A072411(k) = A007947(k). - Michel Marcus, Aug 29 2022
Terms p^p, p prime, form the subsequence A051674. - Bernard Schott, Sep 21 2022
Terms p^q * q^p with distinct primes p and q form the subsequence A082949. - Bernard Schott, Feb 01 2023

			576 = 2^6 * 3^2, lcm(2,3) = 6 = lcm(6,2), hence 576 is a term.

Cf. A054411, A054412, A068935, A068936, A068937, A068938.

Cf. A072411, A007947, A051674, A082949.

Select[Range[10^6], Equal @@ LCM @@ FactorInteger[#] &] (* Amiram Eldar, Aug 07 2022 *)

isok(k) = my(f=factor(k)); lcm(f[,1]) == lcm(f[,2]); \\ Michel Marcus, Aug 07 2022

from math import lcm
from sympy import factorint
def ok(n): f = factorint(n); return lcm(*f.keys()) == lcm(*f.values())
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Aug 07 2022

cp's OEIS Frontend

A054411 Numbers k such that Sum_{j} p_j = Sum_{j} e_j where Product_{j} p_j^(e_j) is the prime factorization of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Sage

A068938 Numbers having the sum of distinct prime factors greater than the sum of exponents in prime factorization, A008472(n)>A001222(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A068935 Numbers having the sum of distinct prime factors less than the sum of exponents in prime factorization, A008472(n) < A001222(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A068937 Numbers having the sum of distinct prime factors not less than the sum of exponents in prime factorization, A008472(n)>=A001222(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A090081 Cube root-smooth numbers: numbers k whose largest prime factor does not exceed the cube root of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A113645 Numbers k such that sum of exponents in prime factorization of k (i.e., A001222(k)) is >= each prime divisor of k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A356433 Numbers k such that, in the prime factorization of k, the least common multiple of the exponents equals the least common multiple of the prime factors.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI