A046028 -id:A046028 - cp's OEIS frontend

A046027 Smallest multiple prime factor of the n-th nonsquarefree number (A013929).

2, 2, 3, 2, 2, 3, 2, 2, 5, 3, 2, 2, 2, 2, 2, 3, 2, 7, 5, 2, 3, 2, 2, 3, 2, 2, 2, 5, 2, 2, 3, 2, 2, 3, 2, 2, 7, 3, 2, 2, 2, 2, 2, 3, 2, 11, 2, 5, 3, 2, 2, 3, 2, 2, 2, 7, 2, 5, 2, 3, 2, 2, 3, 2, 2, 13, 3, 2, 5, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 11, 3, 2, 7, 2, 5, 2, 2, 2, 3

Offset: 1

Eric W. Weisstein

nonn

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

Cf. A013929, A020639, A046028, A249739.

Select[ FactorInteger[#], #[[2]]>1&, 1][[1, 1]]& /@ Select[ Range[300], !SquareFreeQ[#]& ] (* Jean-François Alcover, Nov 06 2012 *)

lista(nn) = apply(x->factor(x)[1,1], apply(x->x/core(x), select(x->!issquarefree(x), [1..nn]))); \\ Michel Marcus, Jun 24 2025

from math import isqrt
from sympy import mobius, factorint
def A046027(n):
    def f(x): return n+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)
    s = factorint(m)
    return next(p for p in sorted(s) if s[p]>1) # Chai Wah Wu, Jul 22 2024

a(n) = A249739(A013929(n)). - Amiram Eldar, Feb 11 2021

A212177 Number of exponents >= 2 in the canonical prime factorization of the n-th nonsquarefree number (A013929(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1

Offset: 1

Matthew Vandermast, Jun 03 2012

Length of second signature of A013929(n) (cf. A212172).

			24 = 2^3*3 has 1 exponent of size 2 or greater in its prime factorization. Since 24 = A013929(8), a(8) = 1.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (1st of 5 pages).

Cf. A013929, A212172, A212174.

Cf. A046028, A056170, A085548, A124010, A229099.

a212177 n = a212177_list !! (n-1)
a212177_list = filter (> 0) a056170_list
-- Reinhard Zumkeller, Dec 29 2012

f[n_] := Module[{c = Count[FactorInteger[n][[;; , 2]], ?(# > 1&)]}, If[n > 1 && c > 0, c, Nothing]]; f[1] = 0; Array[f, 300] (* _Amiram Eldar, Oct 01 2023 *)

from math import isqrt
from sympy import mobius, factorint
def A212177(n):
    def f(x): return n+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 sum(1 for e in factorint(m).values() if e>1) # Chai Wah Wu, Jul 22 2024

a(n) = A056170(A013929(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (Sum_{p prime} 1/p^2)/(1-1/zeta(2)) = A085548 / A229099 = 1.15347789194214704903... . - Amiram Eldar, Oct 01 2023

A083730 Greatest prime^2 factor of n, or a(n)=1 for squarefree n.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 4, 1, 1, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 25, 1, 9, 4, 1, 1, 1, 4, 1, 1, 1, 9, 1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 4, 49, 25, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 1, 9, 4, 1, 1, 1, 4, 1, 1, 1, 9, 1, 1, 25, 4, 1, 1, 1, 4, 9, 1, 1, 4, 1, 1, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 49, 9

Offset: 1

Reinhard Zumkeller, Jun 14 2003

Not multiplicative, for example a(4)*a(9) <> a(36). - R. J. Mathar, Oct 31 2011

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

Cf. A006530, A001248, A005117, A013929, A008833, A046028, A249740.

a[n_] := If[(pos = Position[(f = FactorInteger[n])[[;; , 2]], ?(# >= 2 &)]) == {}, 1, f[[pos[[-1, 1]], 1]]^2];  Array[a, 100] (* _Amiram Eldar, Nov 14 2020 *)

a(n)=my(f=factor(n)); forstep(i=#f~,1,-1, if(f[i,2]>1, return(f[i,1]^2))); 1 \\ Charles R Greathouse IV, Jul 23 2017

a(n) = A249740(n)^2. - Amiram Eldar, Feb 11 2021

cp's OEIS Frontend

A046027 Smallest multiple prime factor of the n-th nonsquarefree number (A013929).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A212177 Number of exponents >= 2 in the canonical prime factorization of the n-th nonsquarefree number (A013929(n)).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

Python

Formula

A083730 Greatest prime^2 factor of n, or a(n)=1 for squarefree n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula