A078615 -id:A078615 - 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

A372619 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = 1/(phi(k)) * Sum_{j=1..n} phi(k*j).

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 2, 5, 6, 1, 3, 5, 9, 10, 1, 2, 5, 7, 13, 12, 1, 3, 4, 9, 11, 17, 18, 1, 2, 6, 6, 13, 14, 23, 22, 1, 3, 4, 10, 11, 17, 20, 31, 28, 1, 2, 5, 6, 14, 13, 23, 24, 37, 32, 1, 3, 5, 9, 10, 20, 19, 31, 33, 45, 42, 1, 2, 5, 7, 13, 12, 26, 23, 37, 37, 55, 46

Offset: 1

Seiichi Manyama, May 07 2024

			Square array T(n,k) begins:
   1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
   2,  3,  2,  3,  2,  3,  2,  3,  2,  3, ...
   4,  5,  5,  5,  4,  6,  4,  5,  5,  5, ...
   6,  9,  7,  9,  6, 10,  6,  9,  7,  9, ...
  10, 13, 11, 13, 11, 14, 10, 13, 11, 14, ...
  12, 17, 14, 17, 13, 20, 12, 17, 14, 18, ...
  18, 23, 20, 23, 19, 26, 19, 23, 20, 24, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=1..10 give: A002088, A049690, A372621, A049690, A372622, A372637, A372638, A049690, A372621, A372639.
Main diagonal gives A070639.
Cf. A000010, A372606.
Cf. A078615, A322360.

T[n_, k_] := Sum[EulerPhi[k*j], {j, 1, n}] / EulerPhi[k]; Table[T[k, n-k+1], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2024 *)

T(n, k) = sum(j=1, n, eulerphi(k*j))/eulerphi(k);

T(n,k) ~ (3/Pi^2) * c(k) * n^2, where c(k) = A078615(k)/A322360(k) is the multiplicative function defined by c(p^e) = p^2/(p^2-1). - Amiram Eldar, May 09 2024

A173615 Numbers k such that rad(k)^2 divides sigma(k).

Original entry on oeis.org

1, 96, 864, 1080, 1782, 6144, 7128, 7776, 17280, 27000, 28512, 54432, 55296, 69984, 87480, 114048, 215622, 276480, 381024, 393216, 432000, 433026, 456192, 497664, 629856, 675000, 862488, 1382400, 1399680, 1677312, 1732104, 1824768, 2187000, 2195424, 2667168

Offset: 1

Michel Lagneau, Feb 22 2010

nonn

rad(n) is the product of the primes dividing n (A007947) and sigma(n) = sum of divisors of n (A000203). Considering the integers k = (2^a)*(3^b), where a+1 = 6*m and b >= 1, we obtain an infinite number of numbers such that rad(k)^2 divides sigma(k).

De Koninck (2000, 2002) asked whether 1 and 1782 are the only numbers k such that rad(k)^2 = sigma(k). - Amiram Eldar, Jan 29 2025

			rad(96)^2 = 6^2 = 36, sigma(96) = 252 and 36 divides 252.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B11, p. 102.

Amiram Eldar, Table of n, a(n) for n = 1..500 (terms 1..300 from Donovan Johnson)
Kevin A. Broughan, Jean-Marie De Koninck, Imre Kátai, and Florian Luca, On integers for which the sum of divisors is the square of the squarefree core, J. Integer Seq., 15 (2012), Article 12.7.5, pp. 1-12. See Final remarks pp. 10-11.
Kevin Broughan Daniel Delbourgo, and Qizhi Zhou, A conjecture of De Koninck regarding particular square values of the sum of divisors function, Journal of Number Theory, Vol. 137 (2014), pp. 50-66.
Yong-Gao Chen and Xin Tong, On a conjecture of de Koninck, Journal of Number Theory, Vol. 154 (2015), pp. 324-364.
Jean-Marie De Koninck, Probelm 000:08, Western Number Theory Problems, 17 & 20 Dec 2000, edited by Gerry Myerson, p. 5.
Jean-Marie De Koninck, Problem 10966, The American Mathematical Monthly, Vol. 109, No. 8 (2002), p. 759; Editorial comment, ibid., Vol. 111, No. 6 (2004), p. 536.
Min Tang and Zhi-Jun Zhou, On a conjecture of De Koninck, INTEGERS, Vol. 18 (2018), Article #A60.
Wacław Sierpiński, Number Of Divisors And Their Sum, Elementary theory of numbers, Warszawa, 1964.
Tomohiro Yamada, On a problem of De Koninck, Moscow Journal of Combinatorics and Number Theory, Vol. 10, No. 3 (2021), pp. 249-260; Correction, ibid., Vol. 10, No. 4 (2021), p. 339; arXiv preprint, arXiv:1906.10001 [math.NT], 2019-2021.

Cf. A000203, A007947, A078615.

Subsequence of A175200.

for n from 1 to 2000000 do : t1:= ifactors(n)[2] : t2 :=mul(t1[i][1], i=1..nops(t1)): if irem(sigma(n),t2^2) = 0 then print (n): else fi: od :

f[p_, e_] := (p^(e+1) - 1)/(p^2 * (p-1)); q[k_] := IntegerQ[Times @@ f @@@ FactorInteger[k]]; q[1] = True; Select[Range[3*10^6], q] (* Amiram Eldar, Jan 29 2025 *)

isok(n) = my(f=factor(n)); (sigma(f) % factorback(f[, 1])^2) == 0; \\ Michel Marcus, Nov 09 2020

a(30)-a(35) from Donovan Johnson, Jan 14 2012

A338790 a(n) = rad(n)^2 - sigma(n), where rad(n) is the squarefree kernel of n (A007947) and sigma(n) is the sum of divisors of n (A000203).

Original entry on oeis.org

0, 1, 5, -3, 19, 24, 41, -11, -4, 82, 109, 8, 155, 172, 201, -27, 271, -3, 341, 58, 409, 448, 505, -24, -6, 634, -31, 140, 811, 828, 929, -59, 1041, 1102, 1177, -55, 1331, 1384, 1465, 10, 1639, 1668, 1805, 400, 147, 2044, 2161, -88, -8, 7, 2529, 578, 2755, -84, 2953

Offset: 1

Michel Marcus, Nov 09 2020

sign

It is conjectured that only 1 and 1782 satisfy a(x) = 0.

R. K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, B11.

Michel Marcus, Table of n, a(n) for n = 1..10000
K. Broughan, J.-M. De Koninck, I. Kátai, F. Luca, On integers for which the sum of divisors is the square of the squarefree core, J. Integer Seq., 15 (2012), pp. 1-12.
Yong-Gao Chen, and Xin Tong, On a conjecture of de Koninck, Journal of Number Theory, Volume 154, September 2015, Pages 324-364. Beware of typo 1728.

Cf. A000203, A007947, A078615, A326142.

a:= n-> mul(i[1], i=ifactors(n)[2])^2-numtheory[sigma](n):
seq(a(n), n=1..60);  # Alois P. Heinz, Nov 09 2020

a(n) = my(f=factor(n)); factorback(f[, 1])^2 - sigma(f);

a(n) = A007947(n)^2 - A000203(n).

a(n) = A078615(n) - A000203(n).

A339744 Numbers k such that rad(k)^2 < sigma(k), where rad(k) is the squarefree kernel of k (A007947) and sigma(k) is the sum of divisors of k (A000203).

Original entry on oeis.org

4, 8, 9, 16, 18, 24, 25, 27, 32, 36, 48, 49, 54, 64, 72, 80, 81, 96, 100, 108, 112, 121, 125, 128, 135, 144, 160, 162, 169, 192, 196, 200, 216, 224, 225, 243, 250, 256, 288, 289, 320, 324, 343, 352, 360, 361, 375, 384, 392, 400, 405, 416, 432, 441, 448, 450, 480, 484, 486, 500

Offset: 1

Bernard Schott, Dec 15 2020

nonn

Prime powers p^e where p is a prime and e >= 2 (A246547) form a subsequence.
For numbers whose prime factors set is {p_1, p_2, ..., p_r}, there exists a minimal element u such that k is a term iff k >= u. This smallest element u satisfies p_1*p_2*...*p_r < u <= (p_1*p_2*...*p_r)^2. These minimal elements are in A339794.
Table with percentage of terms <= 10^k for k = 1, 2, ..., 8, 9 (first rows coming from b-file):
     +-------+------------------------+----------------------------+
     |   k   |number of terms <= 10^k |percentage of terms <= 10^k |
     |       |                        |             %              |
     +-------+------------------------+----------------------------+
     |   1   |           3            |            30              |
     |   2   |          19            |            19              |
     |   3   |          95            |             9.5            |
     |   4   |         435            |             4.35           |
     |   5   |        1853            |             1.85           |
     |   6   |        7793            |             0.78           |
     |   7   |       32365            |             0.32           |
     |   8   |      131200            |             0.13           |
     |   9   |      527161            |             0.05           |
     |       |                        |                            |
     +-------+------------------------+----------------------------+
The percentage of terms decreases as 10^k increases, and a plausible conjecture is that the asymptotic density of this sequence is 0.

			rad(18)^2 - sigma(18) = (2*3)^2 - (1+2+3+6+9+18) = 36 - 39 = -3 and 18 is a term.
rad(25)^2 - sigma(25) = 5^2 - (1+5+25) = 25 - 31 = -6 and 25 is a term.
rad(40)^2 - sigma(40) = (2*5)^2 - (1+2+4+5+8+10+20+40) = 100 - 90 = 10 and 40 is not a term.

Richard K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, B11, p. 102.

Marius A. Burtea, Table of n, a(n) for n = 1..10000

Cf. A000203, A007947, A078615, A338790, A339794.

Subsequence: A246547.

s:=func; [k:k in [2..500]|s(k)^2 lt DivisorSigma(1,k)]; // Marius A. Burtea, Dec 15 2020

Rad := n -> convert(NumberTheory:-PrimeFactors(n), `*`):
Sigma := n -> NumberTheory:-SumOfDivisors(n):
Is_a := n -> Rad(n)^2 < Sigma(n):
select(Is_a, [`$`(1..500)]); # Peter Luschny, Dec 16 2020

frad2[p_, e_] := p^2; fsig[p_, e_] := (p^(e + 1) - 1)/(p - 1); Select[Range[2, 500], Times @@ frad2 @@@ (f = FactorInteger[#]) < Times @@ fsig @@@ f &] (* Amiram Eldar, Dec 15 2020 *)

isok(k) = factorback(factorint(k)[, 1])^2  < sigma(k); \\ Michel Marcus, Dec 15 2020

More terms from Marius A. Burtea, Dec 15 2020

A374291 Squares of powerful numbers.

Original entry on oeis.org

1, 16, 64, 81, 256, 625, 729, 1024, 1296, 2401, 4096, 5184, 6561, 10000, 11664, 14641, 15625, 16384, 20736, 28561, 38416, 40000, 46656, 50625, 59049, 65536, 82944, 83521, 104976, 117649, 130321, 153664, 160000, 186624, 194481, 234256, 250000, 262144, 279841, 331776

Offset: 1

Amiram Eldar, Jul 02 2024

First differs from A340588 at n = 12.
4-full (or 3-full) squares.
Numbers whose exponents in their prime factorization are all even numbers >= 4.
This sequence is closed under multiplication.
The sequence {A000290(n)*A078615(A000290(n)), n>=1} is a permutation of this sequence, and the sequence {a(n)/A078615(a(n)), n>=1} is a permutation of {A000290(n), n>=1}.
The sequence {A335988(n)*A007947(A335988(n)), n>=1} is a permutation of this sequence, and the sequence {a(n)/A007947(a(n)), n>=1} is a permutation of A335988.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers.

Intersection of A000290 and A036967 (or A036966).
Intersection of A000290 and A337050.
Subsequence of A322449.
Cf. A000290, A001694, A007947, A078615, A335988, A340588.

powQ[n_] := n==1 || AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; Select[Range[600], powQ]^2

is(k) = issquare(k) && ispowerful(sqrtint(k));

from math import isqrt
from sympy import mobius, integer_nthroot
def A374291(n):
    def squarefreepi(n):
        return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2, 3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x, 3)[0])-l
        return c
    return bisection(f,n,n)**2 # Chai Wah Wu, Sep 10 2024

a(n) = A000290(A001694(n)) = A001694(n)^2.
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^2*(p^2-1))) = zeta(4)*zeta(6)/zeta(12) = 15015/(1382*Pi^2) = 1.10082313486953808844... .
Sum_{n>=1} 1/a(n)^s =  Product_{p prime} (1 + 1/(p^(2*s)*(p^(2*s)-1))) = zeta(4*s)*zeta(6*s)/zeta(12*s), for s > 1/4.

A188525 a(n) = rad(rad(n)^2+1), where rad = A007947.

Original entry on oeis.org

2, 5, 10, 5, 26, 37, 10, 5, 10, 101, 122, 37, 170, 197, 226, 5, 290, 37, 362, 101, 442, 485, 530, 37, 26, 677, 10, 197, 842, 901, 962, 5, 1090, 1157, 1226, 37, 1370, 85, 1522, 101, 58, 1765, 370, 485, 226, 2117, 2210, 37, 10, 101, 2602, 677, 2810, 37

Offset: 1

Jonathan Vos Post, Apr 03 2011

			a(7) = rad(rad(7)^2 + 1) = rad(7^2 + 1) = rad(50) = 10.

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

Cf. A007947, A078322.

[ &*PrimeDivisors((&*PrimeDivisors(n))^2+1): n in [1..51] ]; // Bruno Berselli, Apr 04 2011

with(numtheory):
rad:= n-> mul(i, i=factorset(n)):
a:= n-> rad(rad(n)^2+1):
seq(a(n), n=1..70);  # Alois P. Heinz, Apr 03 2011

rad[n_] := Times @@ FactorInteger[n][[All, 1]];
a[n_] := rad[rad[n]^2 + 1];
Array[a, 70] (* Jean-François Alcover, Mar 27 2017 *)

rad(n)=my(f=factor(n)[,1]);prod(i=1,#f,f[i])
a(n)=rad(rad(n)^2+1) \\ Charles R Greathouse IV, Aug 08 2013

a(n) = A007947(A078615(n)+1). - R. J. Mathar, Apr 04 2011

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Python

Sage

Scheme

Formula

Extensions

A372619 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = 1/(phi(k)) * Sum_{j=1..n} phi(k*j).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A173615 Numbers k such that rad(k)^2 divides sigma(k).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A338790 a(n) = rad(n)^2 - sigma(n), where rad(n) is the squarefree kernel of n (A007947) and sigma(n) is the sum of divisors of n (A000203).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

PARI

Formula

A339744 Numbers k such that rad(k)^2 < sigma(k), where rad(k) is the squarefree kernel of k (A007947) and sigma(k) is the sum of divisors of k (A000203).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Extensions

A374291 Squares of powerful numbers.

Views

Author

Keywords

Comments

Links

Crossrefs