A249739 -id:A249739 - cp's OEIS frontend

A277886 If n is squarefree, a(n) = n, else a(n) = A000040(1+A277885(n)) * (n/(A249739(n)^2)).

1, 2, 3, 3, 5, 6, 7, 6, 5, 10, 11, 9, 13, 14, 15, 12, 17, 10, 19, 15, 21, 22, 23, 18, 7, 26, 15, 21, 29, 30, 31, 24, 33, 34, 35, 27, 37, 38, 39, 30, 41, 42, 43, 33, 25, 46, 47, 36, 11, 14, 51, 39, 53, 30, 55, 42, 57, 58, 59, 45, 61, 62, 35, 48, 65, 66, 67, 51, 69, 70, 71, 54, 73, 74, 21, 57, 77, 78, 79, 60, 45

Offset: 1

Antti Karttunen, Nov 08 2016

nonn

If n has non-unitary prime divisors, then divide it by the square of the smallest of them and multiply by a single instance of the next larger prime.

This differs from related A097246 for the first time at n=16. For both sequences A097248 gives the eventual stable points reached when starting iterating from n.

			For n = 12 = 2*2*3, the smallest non-unitary prime divisor (and in this case the only one) is 2, thus we divide with 2^2 and multiply with the next larger prime 3, to get ((2^2 * 3)/(2^2))*3 = 3*3, thus a(12) = 9.
For n = 16 = 2^4, we divide two instances of 2 out and multiply by a single instance of 3 to get 2*2*3 = 12.

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

Cf. A000040, A048675, A097246, A097248, A249739, A277885, A277887, A277888, A277896.

Table[If[SquareFreeQ@ n, n, Prime[1 + PrimePi@ Min[Select[FactorInteger[n][[All, 1]], ! CoprimeQ[#, n/#] &] /. {} -> 0]] (n/If[SquareFreeQ@ n, 1, p = 2; While[! Divisible[n, p^2], p = NextPrime@ p]; p]^2)], {n, 81}] (* Michael De Vlieger, Nov 15 2016 *)

(define (A277886 n) (if (zero? (A277885 n)) n (* (A000040 (+ 1 (A277885 n))) (/ n (expt (A249739 n) 2)))))

If A277885(n) = 0 [when n is squarefree], then a(n) = n, otherwise a(n) = A000040(1+A277885(n)) * (n/(A249739(n)^2)).
Other identities. For all n >= 1:
A048675(a(n)) = A048675(n).

A046027 Smallest 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, 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

A249717 The smallest 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, 2, 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, 2, 2, 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, 2, 5, 2, 5, 2, 2, 2, 5, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 3, 3, 2

Offset: 0

Antti Karttunen, Nov 04 2014

nonn

All such n, for which a(n) < 3, form a subsequence of A249724.

Cf. A249716, A249739, A249724.

Differs from A249718 for the first time at n=36, where a(36) = 2, while A249718(36) = 3.

(define (A249717 n) (A249739 (A249716 n)))

a(n) = A249739(A249716(n)).

A249740 The largest prime whose square divides n, 1 if n is squarefree.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 04 2014

nonn

A249739 gives the corresponding smallest prime.

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

Cf. A003557, A005117, A013929, A006530, A052126, A070003, A241917, A249718.

Differs from A071773 and A249739 for the first time at n=36, where a(36) = 3, while A249739(36) = 2 and A071773(36) = 6.

a[n_] := If[(f = Select[FactorInteger[n], Last[#] > 1 &]) == {}, 1, f[[-1, 1]]]; Array[a, 100] (* Amiram Eldar, Feb 11 2021 *)
Table[If[SquareFreeQ[n],1,Select[FactorInteger[n],#[[2]]>1&][[-1,1]]],{n,120}] (* Harvey P. Dale, Feb 28 2021 *)

(define (A249740 n) (let loop ((n n) (p (A006530 n))) (cond ((= 1 n) n) ((zero? (modulo n (* p p))) p) (else (loop (/ n p) (A006530 (/ n p)))))))
;; Alternative version which is based on the given recurrence, and utilizes the memoizing definec-macro from Antti Karttunen's IntSeq-library:
(definec (A249740 n) (cond ((= n 1) n) ((zero? (A241917 n)) (A006530 n)) (else (A249740 (A052126 n)))))

a(1) = 1, and for n > 1, if A241917(n) = 0 [i.e., n is a term of A070003], a(n) = A006530(n), otherwise a(n) = a(A052126(n)).

a(n) = A006530(A003557(n)). - Amiram Eldar, Feb 11 2021

A277885 a(n) = index of the least non-unitary prime divisor of n or 0 if no such prime-divisor exists.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 3, 0, 2, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 4, 3, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 3, 1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 4, 2, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1

Offset: 1

Antti Karttunen, Nov 08 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from prime indices.

Cf. A008578, A028234, A032742, A055396, A067029, A249739.
Cf. A277697.
Cf. A005117 (gives the positions of zeros).

Table[PrimePi@ Min[Select[FactorInteger[n][[All, 1]], ! CoprimeQ[#, n/#] &] /. {} -> 0], {n, 120}] (* Michael De Vlieger, Nov 15 2016 *)

a(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2] > 1, return(primepi(f[i, 1])))); 0;} \\ Amiram Eldar, Jul 28 2024

from sympy import factorint, primepi, isprime, primefactors
def a049084(n): return primepi(n)*(1*isprime(n))
def a055396(n): return 0 if n==1 else a049084(min(primefactors(n)))
def a028234(n):
    f = factorint(n)
    return 1 if n==1 else n/(min(f)**f[min(f)])
def a067029(n):
    f=factorint(n)
    return 0 if n==1 else f[min(f)]
def a(n): return 0 if n==1 else a055396(n) if a067029(n)>1 else a(a028234(n)) # Indranil Ghosh, May 15 2017

(definec (A277885 n) (cond ((= 1 n) 0) ((< 1 (A067029 n)) (A055396 n)) (else (A277885 (A028234 n)))))

a(1) = 0; for n > 1, if A067029(n) > 1, a(n) = A055396(n), otherwise a(n) = a(A028234(n)). [One may use A032742 instead of A028234 for recursing.]
A008578(1+a(n)) = A249739(n).
For n > 1, a(n) + A277697(n) > 0.

A381032 The radix prime of the least significant digit > 1 in the primorial base expansion of n, or 1 if there is no such digit.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 13 2025

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to primorial base.

Cf. A008578, A020639, A249739, A276086, A327860, A328572, A328828, A381033, A381034.
Cf. A088860 (positions of records for values > 1), A276156 (positions of 1's).
Cf. also A053669, A351566.

A381032(n) = { my(p=2); while(n, if((n%p)>1, return(p)); n = n\p; p = nextprime(1+p)); (1); };

a(n) = A008578(1+A328828(n)).
a(n) = A020639(A328572(n)) = A249739(A276086(n)).
For all n, a(n) divides A327860(n).

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

A283670 The single square referenced in A190641.

Original entry on oeis.org

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

Offset: 1

Robert Price, Mar 13 2017

nonn

			A190641(4) = 12, 12 = 2*2*3, so 12 has only one square factor, namely 4.

Robert Price, Table of n, a(n) for n = 1..10000

Cf. A190641, A249739, A283671.

f[n_] := Module[{f = FactorInteger[n], ind}, ind = Position[f[[;;, 2]], ?(# > 1 &)]; If[Length[ind] == 1, f[[ind[[1, 1]],1]]^2, Nothing]]; Array[f, 300] (* _Amiram Eldar, Jul 28 2024 *)

lista(kmax) = for(k = 1, kmax, my(f = factor(k)); if(#select(x -> (x>1), f[,2]) == 1, for(i = 1, #f~, if(f[i,2] > 1, print1(f[i,1]^2, ", "); break)))); \\ Amiram Eldar, Jul 28 2024

a(n) = A283671(n)^2 = A249739(A190641(n))^2. - Amiram Eldar, Jul 28 2024

A283671 Square root of the single square referenced in A190641.

Original entry on oeis.org

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

Offset: 1

Robert Price, Mar 13 2017

nonn

			A190641(4) = 12, 12 = 2*2*3, so 12 has only one square factor, namely 4, and the square root is 2.

Robert Price, Table of n, a(n) for n = 1..10000

Cf. A190641, A249739, A283670.

f[n_] := Module[{f = FactorInteger[n], ind}, ind = Position[f[[;;, 2]], ?(# > 1 &)]; If[Length[ind] == 1, f[[ind[[1, 1]],1]], Nothing]]; Array[f, 300] (* _Amiram Eldar, Jul 28 2024 *)

lista(kmax) = for(k = 1, kmax, my(f = factor(k)); if(#select(x -> (x>1), f[,2]) == 1, for(i = 1, #f~, if(f[i,2] > 1, print1(f[i,1], ", "); break)))); \\ Amiram Eldar, Jul 28 2024

a(n) = sqrt(A283670(n)) = A249739(A190641(n)). - Amiram Eldar, Jul 28 2024

cp's OEIS Frontend

A277886 If n is squarefree, a(n) = n, else a(n) = A000040(1+A277885(n)) * (n/(A249739(n)^2)).

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

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A249717 The smallest 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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A249740 The largest prime whose square divides n, 1 if n is squarefree.

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A277885 a(n) = index of the least non-unitary prime divisor of n or 0 if no such prime-divisor exists.

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A381032 The radix prime of the least significant digit > 1 in the primorial base expansion of n, or 1 if there is no such digit.

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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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A283670 The single square referenced in A190641.

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