A284318 -id:A284318 - cp's OEIS frontend

A007947 Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.

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

Offset: 1

R. Muller, Mar 15 1996

Multiplicative with a(p^e) = p.
Product of the distinct prime factors of n.
a(k)=k for k=squarefree numbers A005117. - Lekraj Beedassy, Sep 05 2006
A note on square roots of numbers: we can write sqrt(n) = b*sqrt(c) where c is squarefree. Then b = A000188(n) is the "inner square root" of n, c = A007913(n), b*c = A019554(n) = "outer square root" of n, and a(n) = lcm(a(b),c). Unless n is biquadrateful (A046101), a(n) = lcm(b,c). [Edited by Jeppe Stig Nielsen, Oct 10 2021, and Andrey Zabolotskiy, Feb 12 2025]
a(n) = A128651(A129132(n-1) + 2) for n > 1. - Reinhard Zumkeller, Mar 30 2007
Also the least common multiple of the prime factors of n. - Peter Luschny, Mar 22 2011
The Mobius transform of the sequence generates the sequence of absolute values of A097945. - R. J. Mathar, Apr 04 2011
Appears to be the period length of k^n mod n. For example, n^12 mod 12 has period 6, repeating 1,4,9,4,1,0, so a(12)= 6. - Gary Detlefs, Apr 14 2013
a(n) differs from A014963(n) when n is a term of A024619. - Eric Desbiaux, Mar 24 2014
a(n) is also the smallest base (also termed radix) for which the representation of 1/n is of finite length.  For example a(12) = 6 and 1/12 in base 6 is 0.03, which is of finite length. - Lee A. Newberg, Jul 27 2016
a(n) is also the divisor k of n such that d(k) = 2^omega(n). a(n) is also the smallest divisor u of n such that n divides u^n. - Juri-Stepan Gerasimov, Apr 06 2017

			G.f. = x + 2*x^2 + 3*x^3 + 2*x^4 + 5*x^5 + 6*x^6 + 7*x^7 + 2*x^8 + 3*x^9 + ... - _Michael Somos_, Jul 15 2018

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
Masum Billal, Divisible Sequence and its Characteristic Sequence, arXiv:1501.00609 [math.NT], 2015, theorem 11 page 5.
Henry Bottomley, Some Smarandache-type multiplicative sequences
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Jarosław Grytczuk, Thue type problems for graphs, points and numbers, Discrete Math., 308 (2008), 4419-4429.
Neville Holmes, Integer Sequences [Broken link]
Serge Lang, Old and New Conjectured Diophantine Inequalities, Bull. Amer. Math. Soc., 23 (1990), 37-75. see p. 39.
Wolfdieter Lang, Cantor's List of Real Algebraic Numbers of Heights 1 to 7, arXiv:2307.10645 [math.NT], 2023.
D. H. Lehmer, Euler constants for arithmetical progressions, Collection of articles in memory of Juriĭ Vladimirovič Linnik. Acta Arith. 27 (1975), 125--142. MR0369233 (51 #5468). See N_k on page 131.
Ivar Peterson, The Amazing ABC Conjecture
Paul Tarau, Emulating Primality with Multiset Representations of Natural Numbers, in Theoretical Aspects of Computing, ICTAC 2011, Lecture Notes in Computer Science, 2011, Volume 6916/2011, 218-238
Paul Tarau, Towards a generic view of primality through multiset decompositions of natural numbers, Theoretical Computer Science, Volume 537, 5 June 2014, Pages 105-124.
Wikipedia, Radical of an integer.

See A007913, A062953, A000188, A019554, A003557, A066503, A087207 for other properties related to square and squarefree divisors of n.
More general factorization-related properties, specific to n: A020639, A028234, A020500, A010051, A284318, A000005, A001221, A005361, A034444, A014963, A128651, A267116.
Range of values is A005117.
Cf. A048803, A019565, A024619, A053462, A060735, A073355 (partial sums), A079277, A097945, A129132, A225546.
Bisections: A099984, A099985.
Sequences about numbers that have the same squarefree kernel: A065642, array A284311 (A284457).
A003961, A059896 are used to express relationship between terms of this sequence.
Cf. A001615, A008683, A076479, A078615.

a007947 = product . a027748_row  -- Reinhard Zumkeller, Feb 27 2012

[ &*PrimeDivisors(n): n in [1..100] ]; // Klaus Brockhaus, Dec 04 2008

with(numtheory); A007947 := proc(n) local i,t1,t2; t1 := ifactors(n)[2]; t2 := mul(t1[i][1],i=1..nops(t1)); end;
A007947 := n -> ilcm(op(numtheory[factorset](n))):
seq(A007947(i),i=1..69); # Peter Luschny, Mar 22 2011
A:= n -> convert(numtheory:-factorset(n),`*`):
seq(A(n),n=1..100); # Robert Israel, Aug 10 2014
seq(NumberTheory:-Radical(n), n = 1..78); # Peter Luschny, Jul 20 2021

rad[n_] := Times @@ (First@# & /@ FactorInteger@ n); Array[rad, 78] (* Robert G. Wilson v, Aug 29 2012 *)
Table[Last[Select[Divisors[n],SquareFreeQ]],{n,100}] (* Harvey P. Dale, Jul 14 2014 *)
a[ n_] := If[ n < 1, 0, Sum[ EulerPhi[d] Abs @ MoebiusMu[d], {d, Divisors[ n]}]]; (* Michael Somos, Jul 15 2018 *)
Table[Product[p, {p, Select[Divisors[n], PrimeQ]}], {n, 1, 100}] (* Vaclav Kotesovec, May 20 2020 *)

a(n) = factorback(factorint(n)[,1]); \\ Andrew Lelechenko, May 09 2014

for(n=1, 100, print1(direuler(p=2, n, (1 + p*X - X)/(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

from sympy import primefactors, prod
def a(n): return 1 if n < 2 else prod(primefactors(n))
[a(n) for n in range(1, 51)]  # Indranil Ghosh, Apr 16 2017

def A007947(n): return mul(p for p in prime_divisors(n))
[A007947(n) for n in (1..60)] # Peter Luschny, Mar 07 2017

(define (A007947 n) (if (= 1 n) n (* (A020639 n) (A007947 (A028234 n))))) ;; ;; Needs also code from A020639 and A028234. - Antti Karttunen, Jun 18 2017

If n = Product_j (p_j^k_j) where p_j are distinct primes, then a(n) = Product_j (p_j).
a(n) = Product_{k=1..A001221(n)} A027748(n,k). - Reinhard Zumkeller, Aug 27 2011
Dirichlet g.f.: zeta(s)*Product_{primes p} (1+p^(1-s)-p^(-s)). - R. J. Mathar, Jan 21 2012
a(n) = Sum_{d|n} phi(d) * mu(d)^2 = Sum_{d|n} |A097945(d)|. - Enrique Pérez Herrero, Apr 23 2012
a(n) = Product_{d|n} d^moebius(n/d) (see Billal link). - Michel Marcus, Jan 06 2015
a(n) = n/( Sum_{k=1..n} (floor(k^n/n)-floor((k^n - 1)/n)) ) = e^(Sum_{k=2..n} (floor(n/k) - floor((n-1)/k))*A010051(k)*M(k)) where M(n) is the Mangoldt function. - Anthony Browne, Jun 17 2016
a(n) = n/A003557(n). - Juri-Stepan Gerasimov, Apr 07 2017
G.f.: Sum_{k>=1} phi(k)*mu(k)^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 11 2017
From Antti Karttunen, Jun 18 2017: (Start)
a(1) = 1; for n > 1, a(n) = A020639(n) * a(A028234(n)).
a(n) = A019565(A087207(n)). (End)
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)). - Vaclav Kotesovec, Dec 18 2019
From Peter Munn, Jan 01 2020: (Start)
a(A059896(n,k)) = A059896(a(n), a(k)).
a(A003961(n)) = A003961(a(n)).
a(n^2) = a(n).
a(A225546(n)) = A019565(A267116(n)). (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = A065463/2. - Vaclav Kotesovec, Jun 24 2020
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))^2.
a(n) = Sum_{k=1..n} mu(gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)).
For n>1, Sum_{k=1..n} a(gcd(n,k))*mu(a(gcd(n,k)))*phi(gcd(n,k))/gcd(n,k) = 0.
For n>1, Sum_{k=1..n} a(n/gcd(n,k))*mu(a(n/gcd(n,k)))*phi(gcd(n,k))*gcd(n,k) = 0. (End)
a(n) = (-1)^omega(n) * Sum_{d|n} mu(d)*psi(d), where omega = A001221 and psi = A001615. - Ridouane Oudra, Aug 01 2025

More terms from several people including David W. Wilson

Definition expanded by Jonathan Sondow, Apr 26 2013

A005361 Product of exponents of prime factorization of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1

Offset: 1

Jeffrey Shallit and Olivier Gérard

a(n) depends only on prime signature of n (cf. A025487, A052306). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).
There was a comment here that said "a(n) is the number of nilpotents elements in the ring Z/nZ", but this is false, see A003557.
a(n) is the number of square-full divisors of n. a(n) is also the number of divisors d of n such that d and n have the same prime factors, i.e., A007947(d) = A007947(n). - Laszlo Toth, May 22 2009
Number of divisors u of n such that u|(u^n/n). Row lengths in triangle of A284318. - Juri-Stepan Gerasimov, Apr 05 2017

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
Imanuel Chen and Michael Z. Spivey, Integral Generalized Binomial Coefficients of Multiplicative Functions, Preprint 2015; Summer Research Paper 238, Univ. Puget Sound.
P. Erdős and T. Motzkin, Problem 5735, Amer. Math. Monthly, 78 (1971), 680-681. (Incorrect solution!)
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)
J. Knopfmacher, A prime-divisor function, Proc. Amer. Math. Soc., 40 (1973), 373-377.
MathStackExchange, Dirichlet series for zeta(s)*zeta(2*s)*zeta(3*s)*zeta(6*s)^(-1).
H. N. Shapiro, Problem 5735, Amer. Math. Monthly, 97 (1990), 937.
D. Suryanarayana and R. Sitaramachandra Rao, The number of square-full divisors of an integer, Proc. Amer. Math. Soc., Vol. 34, No. 1 (1972), pp. 79-80.
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A002110, A005117 (indices of ones), A028234, A052306, A067029, A072411, A082695, A284318.
Cf. A360969, A360970, A361132.
Cf. A340065 (Dgf at s=2).

a005361 = product . a124010_row -- Reinhard Zumkeller, Jan 09 2012

A005361 := proc(n)
    local a, p ;
    a := 1 ;
    for p in ifactors(n)[2] do
       a := a*op(2, p) ;
    end do:
    a ;
end proc:
seq(A005361(n),n=1..30) ; # R. J. Mathar, Nov 20 2012
# second Maple program:
a:= n-> mul(i[2], i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 18 2020

Prepend[ Array[ Times @@ Last[ Transpose[ FactorInteger[ # ] ] ]&, 100, 2 ], 1 ]
Array[Times@@Transpose[FactorInteger[#]][[2]]&,80] (* Harvey P. Dale, Aug 15 2012 *)

for(n=1,100, f=factor(n); print1(prod(i=1,omega(f), f[i,2]),",")) \\ edited by M. F. Hasler, Feb 18 2020

a(n)=factorback(factor(n)[,2]) \\ Charles R Greathouse IV, Nov 07 2014

for(n=1, 100, print1(direuler(p=2, n, (1 - X + X^2)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

from math import prod
from sympy import factorint
def a(n): return prod(factorint(n).values())
print([a(n) for n in range(1, 91)]) # Michael S. Branicky, Jul 04 2022

(define (A005361 n) (if (= 1 n) 1 (* (A067029 n) (A005361 (A028234 n))))) ;; Antti Karttunen, Mar 06 2017

n = Product (p_j^k_j) -> a(n) = Product (k_j).
Dirichlet g.f.: zeta(s)*zeta(2s)*zeta(3s)/zeta(6s).
Multiplicative with a(p^e) = e. - David W. Wilson, Aug 01 2001
a(n) = Sum_{d dividing n} floor(rad(d)/rad(n)) where rad(n) is A007947. - Enrique Pérez Herrero, Nov 06 2009
For n > 1: a(n) = Product_{k=1..A001221(n)} A124010(n,k). - Reinhard Zumkeller, Aug 27 2011
a(n) = tau(n/rad(n)), where tau is A000005 and rad is A007947. - Anthony Browne, May 11 2016
a(n) = Sum_{k=1..n}(floor(cos^2(Pi*k^n/n))*floor(cos^2(Pi*n/k))). - Anthony Browne, May 11 2016
From Antti Karttunen, Mar 06 2017: (Start)
For all n >= 1, a(prime^n) = n, a(A002110(n)) = a(A005117(n)) = 1. [From Crossrefs section.]
a(1) = 1; for n > 1, a(n) = A067029(n) * a(A028234(n)).
(End)
Let (b(n)) be multiplicative with b(p^e) = -1 + ( (floor((e-1)/3)+floor(e/3)) mod 4 ) for p prime and e > 0, then b(n) is the Dirichlet inverse of (a(n)). - Werner Schulte, Feb 23 2018
Sum_{i=1..k} a(i) ~ (zeta(2)*zeta(3)/zeta(6)) * k (Suryanarayana and Sitaramachandra Rao, 1972). - Amiram Eldar, Apr 13 2020
More precise asymptotics: Sum_{k=1..n} a(k) ~ 315*zeta(3)*n / (2*Pi^4) + zeta(1/2)*zeta(3/2)*sqrt(n) / zeta(3) + 6*zeta(1/3)*zeta(2/3)*n^(1/3) / Pi^2 [Knopfmacher, 1973]. - Vaclav Kotesovec, Jun 13 2020

A057723 Sum of positive divisors of n that are divisible by every prime that divides n.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 14, 12, 10, 11, 18, 13, 14, 15, 30, 17, 24, 19, 30, 21, 22, 23, 42, 30, 26, 39, 42, 29, 30, 31, 62, 33, 34, 35, 72, 37, 38, 39, 70, 41, 42, 43, 66, 60, 46, 47, 90, 56, 60, 51, 78, 53, 78, 55, 98, 57, 58, 59, 90, 61, 62, 84, 126, 65, 66, 67, 102, 69, 70

Offset: 1

Leroy Quet, Oct 27 2000

			The divisors of 12 that are divisible by both 2 and 3 are 6 and 12. So a(12) = 6 + 12 = 18.

Ivan Neretin, Table of n, a(n) for n = 1..10000
József Sándor and Krassimir Atanassov, Some new arithmetic functions, Notes on Number Theory and Discrete Mathematics, Volume 30, 2024, Number 4, Pages 851-856. See sigma- function.

Row sums of triangle A284318.
Cf. A000203 (sigma), A007947 (rad), A005361 (number of these divisors).
Cf. A049060 and A060640 (other sigma-like functions).
Cf. A065487, A330596.

[&*PrimeDivisors(n)*SumOfDivisors(n div &*PrimeDivisors(n)): n in [1..70]]; // Vincenzo Librandi, May 14 2015

seq(mul(f[1]*(f[1]^f[2]-1)/(f[1]-1), f = ifactors(n)[2]), n = 1 .. 100); # Robert Israel, May 13 2015

Table[(b = Times @@ FactorInteger[n][[All, 1]])*DivisorSigma[1, n/b], {n, 70}] (* Ivan Neretin, May 13 2015 *)
f[p_, e_] := (p^(e+1)-1)/(p-1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)

a(n) = {my(f = factor(n)); for (i=1, #f~, f[i,2]=1); my(pp = factorback(f)); sumdiv(n, d, if (! (d % pp), d, 0));} \\ Michel Marcus, May 14 2015

If n = Product p_i^e_i then a(n) = Product (p_i + p_i^2 + ... + p_i^e_i).
a(n) = rad(n)*sigma(n/rad(n)) = A007947(n)*A000203(A003557(n)). - Ivan Neretin, May 13 2015
Dirichlet g.f.: zeta(s) * zeta(s-1) * Product(p prime, 1 - p^(-s) + p^(1-2*s)). - Robert Israel, May 13 2015
Sum_{k=1..n} a(k) ~ c * Pi^2 * n^2 / 12, where c = A330596 = Product_{primes p} (1 - 1/p^2 + 1/p^3) = 0.7485352596823635646442150486379106016416403430053244045... - Vaclav Kotesovec, Dec 18 2019
a(n) = Sum_{d|n, rad(d)=rad(n)} d. - R. J. Mathar, Jun 02 2020
Lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k = Product_{p prime}(1 + 1/(p*(p^2-1))) = 1.231291... (A065487). - Amiram Eldar, Jun 10 2020
a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(n/d) * sigma(d). - Ilya Gutkovskiy, Apr 15 2021

A369609 Irregular triangle read by rows where row n lists k <= n such that A007947(k) = A007947(n).

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 7, 2, 4, 8, 3, 9, 10, 11, 6, 12, 13, 14, 15, 2, 4, 8, 16, 17, 6, 12, 18, 19, 10, 20, 21, 22, 23, 6, 12, 18, 24, 5, 25, 26, 3, 9, 27, 14, 28, 29, 30, 31, 2, 4, 8, 16, 32, 33, 34, 35, 6, 12, 18, 24, 36, 37, 38, 39, 10, 20, 40, 41, 42, 43, 22, 44

Offset: 1

Michael De Vlieger, May 09 2024

Differs from A284318 after 27 terms.
Let rad(x) = A007947(x).
Let T(n,k) be the k-th term of row n in this sequence.
Define S(n,k) to be the k-th term in row n of A162306.
T(n,k) = rad(n) * S(n,k), k <= A008479(n).
The number n appears as the last term in row n.

			First rows of the triangle:
  1;
  2;
  3;
  2, 4;
  5;
  6;
  7;
  2, 4, 8;
  3, 9;
  10;
  11;
  6, 12;
  13;
  14;
  15;
  2, 4, 8, 16;
  17;
  6, 12, 18;
  etc.

Michael De Vlieger, Table of n, a(n) for n = 1..12946 (rows n = 1..5000, flattened)
Michael De Vlieger, Fast Mathematica programs that construct A008479, A010846, A162306, and A369609

Cf. A007947, A008479, A162306, A284318.

f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]]; Flatten@ Table[r = f[n]; Select[Range[n], f[#] == r &], {n, 44}]

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
row(n) = my(r=rad(n)); select(x->(rad(x) == r), [1..n]); \\ Michel Marcus, May 11 2024

Row n of this sequence contains row n of A284318.
Length of row n is A008479(n).
For squarefree n, row n = {n}.
For prime power n = p^m, row n = { p^j : j = 1..m }.

A380672 Triangle read by rows where row n lists divisors d | n such that rad(d) != rad(n), where rad = A007947.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 2, 5, 1, 1, 2, 3, 4, 1, 1, 2, 7, 1, 3, 5, 1, 1, 1, 2, 3, 9, 1, 1, 2, 4, 5, 1, 3, 7, 1, 2, 11, 1, 1, 2, 3, 4, 8, 1, 1, 2, 13, 1, 1, 2, 4, 7, 1, 1, 2, 3, 5, 6, 10, 15, 1, 1, 1, 3, 11, 1, 2, 17, 1, 5, 7, 1, 2, 3, 4, 9, 1, 1, 2, 19

Offset: 2

Michael De Vlieger, Feb 13 2025

Row n lists terms in row n of A027750 that do not have the same squarefree kernel as does n.

			D(6) = {1, 2, 3, 6}; of these, {1, 2, 3} are such that rad(d) != rad(6).
D(10) = {1, 2, 5, 10}; of these, {1, 2, 5} are such that rad(d) != rad(10).
D(12) = {1, 2, 3, 4, 6, 12}; of these, {1, 2, 3, 4} are such that rad(d) != rad(12).
D(36) = {1, 2, 3, 4, 6, 9, 12, 18, 36}; of these, {1, 2, 3, 4, 9} are such that rad(d) != rad(36), etc.
Table begins:
   n:  row n
  ---------------
   2:  1;
   3:  1;
   4:  1;
   5:  1;
   6:  1, 2, 3;
   7:  1;
   8:  1;
   9:  1;
  10:  1, 2, 5;
  11:  1;
  12:  1, 2, 3, 4;
  13:  1;
  14:  1, 2, 7;
  15:  1, 3, 5;
  ...

Michael De Vlieger, Table of n, a(n) for n = 2..11751 (rows n = 2..2000, flattened).

Cf. A000005, A005361, A007947, A027750, A183093, A284318, A380819.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; Select[Divisors[n], rad[#] != r &], {n, 2, 40}] // Flatten

Row 1 is empty since rad(1) | rad(1).
The first term of row n is 1 for all n > 1.
n is not in row n since rad(n) = rad(n).
Length of row n = A183093(n) = tau(n) - tau(n/rad(n)).
Let S(n) = row n of A284318 and let D(n) = row n of A027750. Then row n of this sequence is D(n) \ S(n).
For prime p and m > 0, row p^m = {1}, since d | p^m, d > 1, are such that rad(d) = p.
For squarefree composite n, row n = D(n) \ {n} with length 2^(omega(k)-1).

cp's OEIS Frontend

A007947 Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.

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A005361 Product of exponents of prime factorization of n.

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A057723 Sum of positive divisors of n that are divisible by every prime that divides n.

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A369609 Irregular triangle read by rows where row n lists k <= n such that A007947(k) = A007947(n).

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A380672 Triangle read by rows where row n lists divisors d | n such that rad(d) != rad(n), where rad = A007947.

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