A007947 -id:A007947 - cp's OEIS frontend

A076479 a(n) = mu(rad(n)), where mu is the Moebius-function (A008683) and rad is the radical or squarefree kernel (A007947).

1, -1, -1, -1, -1, 1, -1, -1, -1, 1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, 1, -1, 1, 1, 1, -1, 1, 1, 1, 1, 1, -1, -1, -1, 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, 1, 1, -1, -1, 1, -1, 1, -1, -1, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Oct 14 2002

Multiplicative: a(1) = 1, a(n) for n >=2 is sign of parity of number of distinct primes dividing n. a(p) = -1, a(pq) = 1, a(pq...z) = (-1)^k, a(p^k) = -1, where p,q,.. z are distinct primes and k natural numbers. - Jaroslav Krizek, Mar 17 2009

a(n) is the unitary Moebius function, i.e., the inverse of the constant 1 function under the unitary convolution defined by (f X g)(n)= sum of f(d)g(n/d), where the sum is over the unitary divisors d of n (divisors d of n such that gcd(d,n/d)=1). - Laszlo Toth, Oct 08 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Zeitschr., Vol. 74 (1960), pp. 66-80.
Jan van de Lune and Robert E. Dressler, Some theorems concerning the number theoretic function omega(n), Journal für die reine und angewandte Mathematik, Vol. 277 (1975), pp. 117-119; alternative link.

Cf. A000005, A000010, A001221, A007947, A008683, A008836, A030230, A065469, A076480, A180403, A226177.

Cf. A174863 (partial sums).

a076479 = a008683 . a007947  -- Reinhard Zumkeller, Jun 01 2013

[(-1)^(#PrimeDivisors(n)): n in [1..100]]; // Vincenzo Librandi, Dec 31 2018

A076479 := proc(n)
    (-1)^A001221(n) ;
end proc:
seq(A076479(n),n=1..80) ; # R. J. Mathar, Nov 02 2016

Table[(-1)^PrimeNu[n], {n,50}] (* Enrique Pérez Herrero, Jan 17 2013 *)

N=66;
mu=vector(N); mu[1]=1;
{ for (n=2,N,
    s = 0;
    fordiv (n,d,
        if (gcd(d,n/d)!=1, next() ); /* unitary divisors only */
        s += mu[d];
    );
    mu[n] = -s;
); };
mu /* Joerg Arndt, May 13 2011 */
/* omitting the line if ( gcd(...)) gives the usual Moebius function */

a(n)=(-1)^omega(n) \\ Charles R Greathouse IV, Aug 02 2013

from math import prod
from sympy.ntheory import mobius, primefactors
def A076479(n): return mobius(prod(primefactors(n))) # Chai Wah Wu, Sep 24 2021

a(n) = A008683(A007947(n)).
a(n) = (-1)^A001221(n). Multiplicative with a(p^e) = -1. - Vladeta Jovovic, Dec 03 2002
a(n) = sign(A180403(n)). - Mats Granvik, Oct 08 2010
Sum_{n>=1} a(n)*phi(n)/n^3 = A065463 with phi()=A000010() [Cohen, Lemma 3.5]. - R. J. Mathar, Apr 11 2011
Dirichlet convolution of A000012 with A226177 (signed variant of A074823 with one factor mu(n) removed). - R. J. Mathar, Apr 19 2011
Sum_{n>=1} a(n)/n^2 = A065469. - R. J. Mathar, Apr 19 2011
a(n) = Sum_{d|n} mu(d)*tau_2(d) = Sum_{d|n} A008683(d)*A000005(d) . - Enrique Pérez Herrero, Jan 17 2013
a(A030230(n)) = -1; a(A030231(n)) = +1. - Reinhard Zumkeller, Jun 01 2013
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 - 2p^(-s)). - Álvar Ibeas, Dec 30 2018
Sum_{n>=1} a(n)/n = 0 (van de Lune and Dressler, 1975). - Amiram Eldar, Mar 05 2021
From Richard L. Ollerton, May 07 2021: (Start)
For n>1, Sum_{k=1..n} a(gcd(n,k))*phi(gcd(n,k))*rad(gcd(n,k))/gcd(n,k) = 0.
For n>1, Sum_{k=1..n} a(n/gcd(n,k))*phi(gcd(n,k))*rad(n/gcd(n,k))*gcd(n,k) = 0. (End)
a(n) = Sum_{d1|n} Sum_{d2|n} mu(d1*d2). - Ridouane Oudra, May 25 2023

A243822 Number of k < n such that rad(k) | n but k does not divide n, where rad = A007947.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2, 1, 0, 0, 4, 0, 2, 1, 3, 0, 3, 0, 3, 0, 2, 0, 10, 0, 0, 2, 4, 1, 5, 0, 4, 2, 3, 0, 11, 0, 3, 2, 4, 0, 5, 0, 6, 2, 3, 0, 8, 1, 3, 2, 4, 0, 14, 0, 4, 2, 0, 1, 14, 0, 4, 2, 12, 0, 6, 0, 5, 3, 4, 1, 15, 0, 4, 0, 5, 0, 16, 1, 5, 3, 3, 0, 20, 1, 4, 3, 5, 1, 8, 0, 7, 2, 6

Offset: 1

Michael De Vlieger, Jun 11 2014

nonn

Former name: number of "semidivisors" of n, numbers m < n that do not divide n but divide n^e for some integer e > 1. See ACM Inroads paper.

			From _Michael De Vlieger_, Aug 11 2024: (Start)
Let S(n) = row n of A162306 and let D(n) = row n of A027750.a(2) = 0 since S(2) \ D(2) = {1, 2} \ {1, 2} is null.
a(10) = 2 since S(10) \ D(10) = {1, 2, 4, 5, 8, 10} \ {1, 2, 5, 10} = {4, 8}.a(16) = 0 since S(16) \ D(16) = {1, 2, 4, 8, 16} \ {1, 2, 4, 8, 16} is null, etc.Table of a(n) and S(n) \ D(n):
   n  a(n)  row n of A272618.
  ---------------------------
   6    1   {4}
  10    2   {4, 8}
  12    2   {8, 9}
  14    2   {4, 8}
  15    1   {9}
  18    4   {4, 8, 12*, 16}
  20    2   {8, 16}
  21    1   {9}
  22    3   {4, 8, 16}
  24    3   {9, 16, 18*}
  26    3   {4, 8, 16}
  28    2   {8, 16}
  30   10   {4, 8, 9, 12, 16, 18, 20, 24, 25, 27}
Terms in A272618 marked with an asterisk are counted by A355432. All other terms are counted by A361235. (End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Exploring Number Bases as Tools, ACM Inroads, March 2012, Vol. 3, No. 1, pp. 4-12.
Michael De Vlieger, Regular and coregular numbers, ResearchGate, 2024.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20

Cf. A000005, A000961, A024619, A027750, A010846, A045763, A162306, A243823, A272618, A304570, A355432, A361235.

Table[Count[Range[n], ?(And[Divisible[n, Times @@ FactorInteger[#][[All, 1]]], ! Divisible[n, #]] &)], {n, 120}] (* _Michael De Vlieger, Aug 11 2024 *)

a(n) = A010846(n) - A000005(n) = card({row n of A162306} \ {row n of A027750}).
a(n) = A045763(n) - A243823(n).
a(n) = (Sum_{1<=k<=n, gcd(n,k)=1} mu(k)*floor(n/k)) - tau(n). - Michael De Vlieger, May 10 2016, after Benoit Cloitre at A010846.
From Michael De Vlieger, Aug 11 2024" (Start)
a(n) = 0 for n in A000961, a(n) > 0 for n in A024619.
a(n) = A051953(n) - A000005(n) + 1 = n - A000010(n) - A000005(n) - A243823(n) + 1.
a(n) = A355432(n) + A361235(n).
a(n) = A355432(n) for n in A360768.
a(n) = A361235(n) for n not in A360768.
a(n) = number of terms in row n of A272618.
a(n) = sum of row n of A304570. (End)

New name from David James Sycamore, Aug 11 2024

A147298 Minimum of rad(m (n - m) n) for 0 < m < n, gcd(m,n) = 1, where rad(k) = A007947(k) = product of prime factors of k.

Original entry on oeis.org

2, 6, 6, 10, 30, 42, 14, 6, 30, 66, 66, 78, 182, 210, 30, 34, 102, 114, 190, 210, 462, 322, 138, 30, 130, 30, 42, 174, 870, 186, 30, 66, 510, 210, 210, 222, 1254, 546, 390, 246, 1722, 258, 946, 330, 690, 1410, 282, 42, 70, 510, 390, 742, 210, 330, 770, 570, 1218

Offset: 2

Artur Jasinski, Nov 05 2008

nonn

Function rad(k) is used in ABC conjecture applications.
For biggest values of function rad(m n (n - m)) see A147299.
For numbers m for which rad(m n (n - m)) reached minimal value see A147300.
For numbers m for which rad(m n (n - m)) reached maximal value see A147301.
Sequence in each value Log[n]/Log[A147298(n)] reached records see A147302.

Ivan Neretin, Table of n, a(n) for n = 2..1000

Cf. A007947, A085152, A085153, A147298-A147307.

A147298 := proc(n) local rad, g, L;
rad := n -> mul(k, k in numtheory:-factorset(n)):
g := (n, k) -> `if`(igcd(n, k) = 1, 1, infinity):
L := n -> [seq(g(n,k)*rad(n*k*(n-k)), k=1..n/2)]:
min(L(n)) end: seq(A147298(n), n=2..58); # Peter Luschny, Aug 06 2019

logmax = 0; aa = {}; bb = {}; cc = {}; dd = {}; ee = {}; ff = {}; gg \ = {}; Do[min = 10^100; max = 0; ile = 0; Do[If[GCD[m, n, n - m] == 1, ile = ile + 1; s = m n (n - m); k = FactorInteger[s]; g = 1; Do[g = g k[[p]][[1]], {p, 1, Length[k]}]; If[g > max, max = g; mmax = m]; If[g < min, min = g; mmin = m]], {m, 1, n - 1}]; AppendTo[aa, min]; AppendTo[bb, max]; AppendTo[cc, mmax]; AppendTo[dd, mmin]; AppendTo[gg, ile]; If[(Log[n]/Log[min]) > logmax, logmax = (Log[n]/Log[min]); AppendTo[ee, {N[logmax], n, mmin, min, mmax, max}]; Print[{N[logmax], n, mmin, min, mmax, max}]; AppendTo[ff, n]], {n, 2, 129}]; aa (*Artur Jasinski*)
Table[Min[Times @@ FactorInteger[#][[All, 1]] & /@ ((m = Select[Range[1, n - 1], GCD[n, #] == 1 &])*(n - m)*n)], {n, 2, 58}] (* Ivan Neretin, May 21 2015 *)

A147298(n)= local(m=n^2); for( a=1,n\2, gcd(a,n)>1 && next; A007947(n-a)*A007947(a)A007947(n-a)*A007947(a)); m*A007947(n)

A066503 a(n) = n - squarefree kernel of n, A007947.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 6, 6, 0, 0, 6, 0, 0, 0, 14, 0, 12, 0, 10, 0, 0, 0, 18, 20, 0, 24, 14, 0, 0, 0, 30, 0, 0, 0, 30, 0, 0, 0, 30, 0, 0, 0, 22, 30, 0, 0, 42, 42, 40, 0, 26, 0, 48, 0, 42, 0, 0, 0, 30, 0, 0, 42, 62, 0, 0, 0, 34, 0, 0, 0, 66, 0, 0, 60, 38, 0, 0

Offset: 1

Reinhard Zumkeller, Jan 04 2002

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A005117, A007947, A065463.

a[n_] := n - Times @@ FactorInteger[n][[All, 1]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 01 2021 *)

{ for (n=1, 1000, f=factor(n); k=1; for(i=1, matsize(f)[1], k*=f[i, 1]); write("b066503.txt", n, " ", n - k) ) } \\ Harry J. Smith, Feb 18 2010

a(n) = n - A007947(n).
a(A005117(n)) = 0.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 - A065463 = 0.295557... . - Amiram Eldar, Dec 05 2023

A048803 a(n) = Product_{k=1..n} rad(k), where rad(n) is the product of distinct prime factors of n, cf. A007947.

Original entry on oeis.org

1, 1, 2, 6, 12, 60, 360, 2520, 5040, 15120, 151200, 1663200, 9979200, 129729600, 1816214400, 27243216000, 54486432000, 926269344000, 5557616064000, 105594705216000, 1055947052160000, 22174888095360000, 487847538097920000, 11220493376252160000, 67322960257512960000

Offset: 0

Christian G. Bower, Apr 15 1999

Squarefree factorials: a(1) = 1, a(n+1) = a(n)* largest squarefree divisor of (n+1). - Amarnath Murthy, Nov 28 2004
LCM over all partitions of n of the product of the part sizes in the partition. - Franklin T. Adams-Watters, May 04 2010
a(n) is the product of the lcm of the set of prime factors of k over the range k = 1..n. - Peter Luschny, Jun 10 2011
a(n) is a divisor of n! and n!/a(n) = A085056(n). - Robert FERREOL, Aug 09 2021
In consequence of the definition, pseudo-binomial coefficients a(m+n)/(a(m)*a(n)) are natural numbers for all whole numbers m and n, and this is the minimal increasing sequence (for n >= 1) with that property. In consequence of the comment of Adams-Watters, the corresponding pseudo-multinomial coefficients are natural numbers as well. - Hal M. Switkay, May 26 2024

Paul-Jean Cahen and Jean-Luc Chabert, Integer-valued Polynomials, AMS, Providence, RI, 1997. Math. Rev. 98a:13002. See p. 246.

Reinhard Zumkeller, Table of n, a(n) for n = 0..500
Abdelmalek Bedhouche and Bakir Farhi, On some products taken over the prime numbers, arXiv:2207.07957 [math.NT], 2022. See rho_n p. 3.
Bakir Farhi, On the derivatives of the integer-valued polynomials, arXiv:1810.07560 [math.NT], 2018.
Index entries for sequences related to lcm's

Partial products of A007947.

a048803 n = a048803_list !! n
a048803_list = scanl (*) 1 a007947_list
-- Reinhard Zumkeller, Jul 01 2013

A048803 := proc(n) local i; mul(ilcm(op(numtheory[factorset](i))), i=1..n) end; seq(A048803(i),i=0..22); # Peter Luschny, Jun 10 2011
a := n -> mul(NumberTheory:-Radical(i), i=1..n): # Peter Luschny, Mar 14 2022

a[0] = 1; a[n_] := a[n] = a[n-1] First @ Select[Reverse @ Divisors[n], SquareFreeQ, 1]; Array[a,22,0] (* Jean-François Alcover, May 04 2011 *)
A048803[n_] := Times @@ ResourceFunction["IntegerRadical"][Range[1, n]];
Table[A048803[n], {n, 0, 24}]  (* Peter Luschny, Aug 18 2025 *)

a(n)=local(f); f=n>=0; if(n>1, forprime(p=2,n,f*=p^(n\p))); f

from functools import cache
@cache
def a_rec(n):
    if n == 0: return 1
    return radical(n) * a_rec(n - 1)
print([a_rec(n) for n in range(23)]) # Peter Luschny, Dec 12 2023

a(0) = 1, a(1) = 1; for n > 1, a(n) = lcm( 1, 2, ..., n, a(1)*a(n-1), a(2)*a(n-2), ..., a(n-1)*a(1) ). [Original name.]
a(n) = Product_{p prime} p^floor(n/p). See Farhi link p. 16. - Michel Marcus, Oct 18 2018
For n >=1, a(n) = lcm(1^floor(n/1),2^floor(n/2),...,n^floor(n/n)). - Robert FERREOL, Aug 05 2021
Rephrasing Murthy's comment: a(n) = a(n-1) * A007947(n). - Hal M. Switkay, Dec 31 2024

Entry improved by comments from Michael Somos, Nov 24 2001

New name based on a comment of Amarnath Murthy by Peter Luschny, Aug 18 2025

A078310 a(n) = n*rad(n) + 1, where rad = A007947 (squarefree kernel).

Original entry on oeis.org

2, 5, 10, 9, 26, 37, 50, 17, 28, 101, 122, 73, 170, 197, 226, 33, 290, 109, 362, 201, 442, 485, 530, 145, 126, 677, 82, 393, 842, 901, 962, 65, 1090, 1157, 1226, 217, 1370, 1445, 1522, 401, 1682, 1765, 1850, 969, 676, 2117, 2210, 289, 344, 501, 2602, 1353

Offset: 1

Reinhard Zumkeller, Nov 23 2002

nonn

A112526(a(n) - 1) = 1, see also A224866. - Reinhard Zumkeller, Jul 23 2013

Increase each exponent in the prime factorization by one, then add 1 to the new product. - M. F. Hasler, Jan 22 2017

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

Smallest, greatest factor: A078311, A078312, number of factors: A078313, A078314, min, max exponent: A078315, A078316, number, sum of divisors: A078317, A078318, sum of prime factors: A078319, A078320, Euler's totient: A078321, squarefree kernel: A078322, arithmetic derivative: A078323.

Cf. primes: A078324, squarefree: A078325, squareful: A078326.

a078310 n = n * a007947 n + 1
-- Reinhard Zumkeller, Jul 23 2013, Oct 19 2011

a:= n-> 1+n*mul(i[1], i=ifactors(n)[2]):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 22 2017

A078310[n_] := n*Times @@ FactorInteger[n][[All, 1]] + 1; Array[A078310, 50] (* G. C. Greubel, Apr 25 2017 *)

rad(n)=my(f=factor(n)[,1]);prod(i=1,#f,f[i])
a(n)=n*rad(n)+1 \\ Charles R Greathouse IV, Jul 09 2013

a(n)={n=factor(n);n[,2]+=vectorv(matsize(n)[1],i,1);factorback(n)+1} \\ M. F. Hasler, Jan 22 2017

a(n)=prod(k=1,matsize(n=factor(n))[1],n[k,1]^(n[k,2]+1))+1 \\ M. F. Hasler, Jan 22 2017

a(n) = A064549(n)+1.

A080259 Numbers whose squarefree kernel is not a primorial number, i.e., A007947(a(n)) is not in A002110.

Original entry on oeis.org

3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87

Offset: 1

Labos Elemer, Mar 19 2003

nonn

Complement to A055932.
From Michael De Vlieger, Feb 06 2024: (Start)
Odd prime power p^m, m >= 1 is in the sequence since its squarefree kernel p is odd and not a primorial. Therefore 3^3, 5^2, etc. are in the sequence.
Odd squarefree composite k is in the sequence since its squarefree kernel is odd and thus not a primorial. Therefore 15 and 33 are in the sequence.
Numbers k such that A053669(k) < A006530(k) are in the sequence since the condition A053669(k) < A006530(k) implies the squarefree kernel is not a primorial, etc. (End)

			From _Michael De Vlieger_, Jan 23 2024: (Start)
1 is not in the sequence because its squarefree kernel is 1, the product of the 0 primes that divide 1 (the "empty product") and therefore the same as A002110(0), the 0th primorial.
2 is not in the sequence since its squarefree kernel is 2, the smallest prime, hence the same as A002110(1) = 2.
4 is not in the sequence since its squarefree kernel is 2 = A002110(1).
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A002110, A006530, A007947, A053669, A055932.

Select[Range[120], Nor[IntegerQ@ Log2[#], And[EvenQ[#], Union@ Differences@ PrimePi[FactorInteger[#][[All, 1]]] == {1}]] &] (* Michael De Vlieger, Jan 23 2024 *)

is(n) = {my(f=factor(n)[,1]);n>1&&primepi(f[#f])>#f} \\ David A. Corneth, May 22 2016

from itertools import count, islice
from sympy import primepi, primefactors
def A080259_gen(startvalue=2): # generator of terms >= startvalue
    for k in count(max(startvalue,2)):
        p = list(map(primepi,primefactors(k)))
        if not(min(p)==1 and max(p)==len(p)):
            yield k
A080259_list = list(islice(A080259_gen(),40)) # Chai Wah Wu, Aug 07 2025

{a(n)} = { k : A053669(k) < A006530(k) }. - Michael De Vlieger, Jan 23 2024

Edited by Michael De Vlieger, Jan 23 2024

A355432 a(n) = number of k < n such that rad(k) = rad(n) and k does not divide n, where rad(k) = A007947(k).

Original entry on oeis.org

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

Offset: 1

Michael De Vlieger, Feb 22 2023

nonn

a(n) = 0 for prime powers and squarefree numbers.

			a(1) = 18, since 18/6 >= 3. We note that rad(12) = rad(18) = 6, yet 12 does not divide 18.
a(2) = 24, since 24/6 >= 3. rad(18) = rad(24) = 6 and 24 mod 18 = 6.
a(3) = 36, since 36/6 >= 3. rad(24) = rad(36) = 6 and 36 mod 24 = 12.
a(6) = 54, since 54/6 >= 3. m in {12, 24, 36, 48} are such that rad(m) = rad(54) = 6, but none divides 54, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Plot (k, n) at (x, -y), k = 1..n, n = 1..54, showing k in A126706 in dark blue, n in A360768 in dark red, and for n and nondivisor k such that rad(k) = rad(n), we highlight in large black dots. This sequence counts the number of black dots in row n.
Michael De Vlieger, Condensation of the above plot, showing k = 1..n and only n in A360768 and n <= 54.

Cf. A005361, A007947, A008479, A010846, A013929, A020639, A024619, A027750, A126706, A162306, A243822, A272618, A360589, A360768.

rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]]; Table[Which[PrimePowerQ[n], 0, SquareFreeQ[n], 0, True, r = rad[n]; Count[Select[Range[n], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], _?(And[rad[#] == r, Mod[n, #] != 0] &)]], {n, 120}]

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
a(n) = my(rn=rad(n)); sum(k=1, n-1, if (n % k, rad(k)==rn)); \\ Michel Marcus, Feb 23 2023

a(n) > 0 for n in A360768.
a(n) < A243822(n) < A010846(n).
a(n) = A008479(n) - A005361(n). - Amiram Eldar, Oct 25 2024

A360768 Numbers k that are neither prime powers nor squarefree, such that k/rad(k) >= q, where rad(k) = A007947(k) and prime q = A119288(k).

Original entry on oeis.org

18, 24, 36, 48, 50, 54, 72, 75, 80, 90, 96, 98, 100, 108, 112, 120, 126, 135, 144, 147, 150, 160, 162, 168, 180, 189, 192, 196, 198, 200, 216, 224, 225, 234, 240, 242, 245, 250, 252, 264, 270, 288, 294, 300, 306, 312, 320, 324, 336, 338, 342, 350, 352, 360, 363, 375, 378, 384, 392, 396, 400, 405, 408

Offset: 1

Michael De Vlieger, Feb 22 2023

nonn

Proper subsequence of A126706.

Numbers k such that there exists j such that 1 < j < k and rad(j) = rad(k), but j does not divide k.

			a(1) = 18, since 18/6 >= 3. We note that rad(12) = rad(18) = 6, yet 12 does not divide 18.
a(2) = 24, since 24/6 >= 3. Note: rad(18) = rad(24) = 6 and 24 mod 18 = 6.
a(3) = 36, since 36/6 >= 3. Note: rad(24) = rad(36) = 6 and 36 mod 24 = 12.
a(6) = 54, since 54/6 >= 3. Note: m in {12, 24, 36, 48} are such that rad(m) = rad(54) = 6, but none divides 54, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, 1016 pixel square bitmap of indices n = 1..1032256, read left to right, top to bottom, such that A126706(n) in this sequence appears in black and A126706(n) in A360767 in white. Shows a curious "sand ripple" pattern perhaps associated with congruence. (Magnification 3X)
Michael De Vlieger, 1016 pixel square bitmap as described above, at scale 1X.

Cf. A007947, A013929, A024619, A119288, A126706, A355432.

Select[Select[Range[120], Nor[SquareFreeQ[#], PrimePowerQ[#]] &], #1/#2 >= #3 & @@ {#1, Times @@ #2, #2[[2]]} & @@ {#, FactorInteger[#][[All, 1]]} &]

This sequence is { k in A126706 : k/A007947(k) >= A119288(k) }.

A328571 Primorial base expansion of n converted into its prime product form, but with all nonzero digits replaced by 1's: a(n) = A007947(A276086(n)).

Original entry on oeis.org

1, 2, 3, 6, 3, 6, 5, 10, 15, 30, 15, 30, 5, 10, 15, 30, 15, 30, 5, 10, 15, 30, 15, 30, 5, 10, 15, 30, 15, 30, 7, 14, 21, 42, 21, 42, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 7, 14, 21, 42, 21, 42, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 35, 70

Offset: 0

Antti Karttunen, Oct 20 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..32768
Index entries for sequences related to primorial base

Cf. A001221, A007947, A083346, A267263, A276086, A328572.

Cf. A276156 (gives the indices where this coincides with A276086).

rad[n_] := Times @@ FactorInteger[n][[All, 1]];
A276086[n0_] := Module[{m = 1, i = 1, n = n0, p}, While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m];
a[n_] := rad[A276086[n]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Dec 01 2021, after Antti Karttunen in A276086 *)

A328571(n) = { my(m=1, p=2); while(n, m *= (p^!!(n%p)); n = n\p; p = nextprime(1+p)); (m); };

a(n) = A007947(A276086(n)).
a(n) = A276086(n) / A328572(n).
a(A276156(n)) = A276086(A276156(n)). [And at no other points the equality holds]
A001221(a(n)) = A267263(n).
a(n) = A083346(A276086(n)). - Antti Karttunen, Feb 28 2021

cp's OEIS Frontend

A076479 a(n) = mu(rad(n)), where mu is the Moebius-function (A008683) and rad is the radical or squarefree kernel (A007947).

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A243822 Number of k < n such that rad(k) | n but k does not divide n, where rad = A007947.

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A147298 Minimum of rad(m (n - m) n) for 0 < m < n, gcd(m,n) = 1, where rad(k) = A007947(k) = product of prime factors of k.

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A066503 a(n) = n - squarefree kernel of n, A007947.

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A048803 a(n) = Product_{k=1..n} rad(k), where rad(n) is the product of distinct prime factors of n, cf. A007947.

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A078310 a(n) = n*rad(n) + 1, where rad = A007947 (squarefree kernel).

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