A249740 -id:A249740 - cp's OEIS frontend

A249739 The smallest prime whose square divides n, 1 if n is squarefree.

1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 5, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 7, 5, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 5, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 7, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2

Offset: 1

Antti Karttunen, Nov 04 2014

nonn

A249740 gives the corresponding largest prime.

If n belongs to A013929, then a(n)>1. - Robert G. Wilson v, Nov 16 2016

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A003557, A005117, A013929, A020639, A046027, A249717.

Differs from A071773 and A249740 for the first time at n=36, where a(36) = 2, while A249740(36) = 3 and A071773(36) = 6.

Table[If[SquareFreeQ@ n, 1, p = 2; While[! Divisible[n, p^2], p = NextPrime@ p]; p], {n, 120}] (* Michael De Vlieger, Nov 15 2016 *)

a(n) = {f = factor(n/core(n)); vsq = select(x->((x%2) == 0), f[,2], 1); if (#vsq, f[vsq[1], 1], 1);} \\ Michel Marcus, Mar 11 2017

(define (A249739 n) (let loop ((n n) (p (A020639 n))) (cond ((= 1 n) n) ((zero? (modulo n (* p p))) p) (else (loop (/ n p) (A020639 (/ n p)))))))

a(n) = A020639(A003557(n)). - Amiram Eldar, Feb 11 2021

A249718 The largest prime whose square divides the first nonsquarefree number on row n of Pascal's triangle, 1 if all terms on that row are squarefree.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 3, 1, 2, 3, 2, 3, 2, 2, 3, 3, 2, 3, 2, 1, 2, 5, 5, 3, 2, 3, 2, 3, 2, 2, 2, 2, 3, 3, 2, 3, 2, 2, 2, 3, 2, 3, 3, 3, 2, 7, 5, 5, 2, 5, 3, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 5, 2, 5, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 3, 2, 3, 2, 2, 7, 3, 5, 5, 5, 5, 2, 2, 2, 5, 3, 3, 3, 3, 2, 2, 2, 2, 2, 3, 3, 3, 2

Offset: 0

Antti Karttunen, Nov 04 2014

nonn

Cf. A249716, A249740.

Differs from A249717 for the first time at n=36, where a(36) = 3, while A249717(36) = 2.

(define (A249718 n) (A249740 (A249716 n)))

a(n) = A249740(A249716(n)).

A046028 Largest multiple prime factor of the n-th nonsquarefree number (A013929).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Greatest Prime Factor.
Eric Weisstein's World of Mathematics, Squareful.

Cf. A006530, A046027, A013929, A249740.

Cf. A027748, A124010, A212177.

a046028 n = a046028_list !! (n-1)
a046028_list = f 1 where
   f x | null zs   = f (x + 1)
       | otherwise = (fst $ head zs) : f (x + 1)
       where zs = reverse $ filter ((> 1) . snd) $
                  zip (a027748_row x) (a124010_row x)
-- Reinhard Zumkeller, Dec 29 2012

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

from math import isqrt
from sympy import mobius, factorint
def A046028(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,reverse=True) if s[p]>1) # Chai Wah Wu, Jul 22 2024

a(n) = A249740(A013929(n)). - Amiram Eldar, Feb 11 2021

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

A371601 Nonsquarefree numbers whose largest nonunitary prime divisor is smaller than their smallest unitary prime divisor, if it exists.

Original entry on oeis.org

4, 8, 9, 12, 16, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 52, 56, 60, 63, 64, 68, 72, 76, 80, 81, 84, 88, 92, 96, 99, 100, 104, 108, 112, 116, 117, 120, 121, 124, 125, 128, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 164, 168, 169, 171, 172, 175, 176

Offset: 1

Amiram Eldar, Mar 29 2024

Subsequence of A283050 and first differs from it at n = 100: A283050(100) = 300 = 2^2 * 3 * 5^2 is not a term of this sequence.
Powerful numbers and nonpowerful numbers k such that 1 < A249740(k) < A277698(k), or equivalently, 1 < A006530(A057521(k)) < A020639(A055231(k)).
The asymptotic density of this sequence is (6/Pi^2) * Sum_{p prime} f(p)/(p^2-p+1) = 0.32131800923..., where f(p) = Product_{primes q <= p} (q^2-q+1)/(q^2-1).

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

Subsequence of A013929 and A283050.

Cf. A001694, A006530, A020639, A052485, A055231, A057521, A059956, A249740, A277698.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Max[e] > 1 && (Min[e] > 1 || Max[e[[FirstPosition[e, 1][[1]] ;; -1]]] == 1)]; Select[Range[200], q]

is(n) = {my(e = apply(x->if(x > 1, 2, 1), factor(n)[,2])); n > 1 && vecmax(e) > 1 && vecsort(e, , 4) == e;}

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A249739 The smallest prime whose square divides n, 1 if n is squarefree.

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A249718 The largest prime whose square divides the first nonsquarefree number on row n of Pascal's triangle, 1 if all terms on that row are squarefree.

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A046028 Largest multiple prime factor of the n-th nonsquarefree number (A013929).

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

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A371601 Nonsquarefree numbers whose largest nonunitary prime divisor is smaller than their smallest unitary prime divisor, if it exists.

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