A046027 -id:A046027 - 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

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

A283454 The square root of the smallest square referenced in A249025 (Numbers k such that 3^k - 1 is not squarefree).

Original entry on oeis.org

2, 2, 11, 2, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 11, 2, 2, 13, 2, 2, 2, 11, 2, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 11, 2, 13, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 11

Offset: 1

Robert Price, Mar 07 2017

nonn

The terms are the smallest prime whose square divides 3^k-1, when it is not squarefree.

			A249025(3)=5, 3^5-1 = 242 = 2*11*11. 242 is not squarefree the square being 11*11 = 121, the root being 11.

Amiram Eldar, Table of n, a(n) for n = 1..407 (terms 1..121 from Michel Marcus)

Cf. A024023, A046027, A249025, A249739, A283453.

p[n_] := If[(f = Select[FactorInteger[n], Last[#] > 1 &]) == {}, 1, f[[1, 1]]]; p /@ Select[3^Range[100] - 1, !SquareFreeQ[#] &] (* Amiram Eldar, Feb 12 2021 *)

lista(nn) = {for (n=1, nn, if (!issquarefree(k = 3^n-1), f = factor(k/core(k)); vsq = select(x->((x%2) == 0), f[,2], 1); print1(f[vsq[1], 1], ", ");););} \\ Michel Marcus, Mar 11 2017

a(n) = A249739(A024023(A249025(n))). - Amiram Eldar, Feb 12 2021

More terms from Michel Marcus, Mar 11 2017

A283919 The smallest square referenced in A013929 (Numbers that are not squarefree).

Original entry on oeis.org

4, 4, 9, 4, 4, 9, 4, 4, 25, 9, 4, 4, 4, 4, 4, 9, 4, 49, 25, 4, 9, 4, 4, 9, 4, 4, 4, 25, 4, 4, 9, 4, 4, 9, 4, 4, 49, 9, 4, 4, 4, 4, 4, 9, 4, 121, 4, 25, 9, 4, 4, 9, 4, 4, 4, 49, 4, 25, 4, 9, 4, 4, 9, 4, 4, 169, 9, 4, 25, 4, 4, 4, 4, 9, 4, 4, 9, 4, 4, 9, 4, 4

Offset: 1

Robert Price, Mar 17 2017

nonn

			A013929(4) = 12, 12 = 2*2*3, so 12 is not squarefree, the square being 2*2 = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert Price)

Cf. A013929, A046027.

s[n_] := If[(pos = Position[(f = FactorInteger[n])[[;; , 2]], ?(# >= 2 &)]) == {}, 1, f[[pos[[1, 1]], 1]]]; Select[Array[s, 250], # > 1 &]^2 (* _Amiram Eldar, Nov 14 2020 *)

f(n) = fordiv(n, d, if ((d>1) && issquare(d), return (d))); return (1);
apply(f, select(x->!issquarefree(x), [1..200])) \\ Michel Marcus, Nov 14 2020

A384064 a(n) = s(n) divided by the smallest multiple prime factor of s(n), where s = A013929.

Original entry on oeis.org

2, 4, 3, 6, 8, 6, 10, 12, 5, 9, 14, 16, 18, 20, 22, 15, 24, 7, 10, 26, 18, 28, 30, 21, 32, 34, 36, 15, 38, 40, 27, 42, 44, 30, 46, 48, 14, 33, 50, 52, 54, 56, 58, 39, 60, 11, 62, 25, 42, 64, 66, 45, 68, 70, 72, 21, 74, 30, 76, 51, 78, 80, 54, 82, 84, 13, 57, 86

Offset: 1

Michael De Vlieger, Jun 23 2025

a(n) is the largest proper nonunitary divisor of s(n).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where purple additionally signifies powerful numbers that are not prime powers.

Cf. A013929, A046027, A048105.

Map[#/Select[FactorInteger[#], #[[-1]] > 1 &, 1][[1, 1]] &, Select[Range[200], Not @* SquareFreeQ] ] (* or *) Map[Select[Most@ Divisors[#], Not @* CoprimeQ] &, Select[Range[200], Not @* SquareFreeQ] ][[All, -1]]

from math import isqrt
from sympy import mobius, factorint
def A384064(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 m//next(p for p in sorted(s) if s[p]>1) # Chai Wah Wu, Jun 25 2025

a(n) = A013929(n)/A046027(n).

A046026 Smallest prime p dividing n#-1, n#, or n#+1, n squarefree.

Original entry on oeis.org

2, 2, 2, 3, 3, 3, 5, 7, 13, 7, 5, 17, 7, 7, 11, 23, 13, 5, 5, 5, 11, 17, 7, 23, 19, 13, 41, 7, 43, 23, 47, 17, 19, 11, 19, 29, 13, 17, 31, 13, 11, 67, 23, 7, 71, 53, 37, 11, 13, 29, 41, 83, 17, 43, 29, 89, 13, 31, 47, 19, 17, 101, 17, 103, 7, 53, 107, 109, 11, 37, 113, 19

Offset: 1

Eric W. Weisstein

nonn

C. Ashbacher, A Note on the Smarandache Near-To-Primorial Function, Smarandache Notions J. 7 (1996), 46-49.
M. R. Mudge, The Smarandache Near-To-Primorial Function, Abstracts of Papers Presented to the Amer. Math. Soc., 17 (1996), 585

Eric Weisstein's World of Mathematics, Smarandache Near-to-Primorial Function

Cf. A002110 (primorial numbers), A046027, A013929.

Smallest prime p such that n divides one of p#-1, p#, p#+1

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

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

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A283454 The square root of the smallest square referenced in A249025 (Numbers k such that 3^k - 1 is not squarefree).

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A283919 The smallest square referenced in A013929 (Numbers that are not squarefree).

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A384064 a(n) = s(n) divided by the smallest multiple prime factor of s(n), where s = A013929.

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A046026 Smallest prime p dividing n#-1, n#, or n#+1, n squarefree.

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