A048741 -id:A048741 - cp's OEIS frontend

A064121 Nonprime numbers n such that the sum of aliquot divisors of n (A001065) and product of aliquot divisors of n (A048741) are both perfect squares.

1, 12, 75, 76, 124, 147, 153, 176, 243, 332, 363, 477, 507, 524, 575, 688, 867, 892, 963, 1075, 1083, 1421, 1532, 1573, 1587, 1611, 1916, 2032, 2075, 2224, 2299, 2401, 2421, 2523, 2572, 2883, 2891, 3100, 3479, 3776, 3888, 4107, 4336, 4527, 4961, 4975

Offset: 1

Robert G. Wilson v, Oct 14 2001

Amiram Eldar, Table of n, a(n) for n = 1..10000

Variant of A064116. - R. J. Mathar, Oct 02 2008

Cf. A064710.

Select[ Range[5000], IntegerQ[ Sqrt[ Apply[ Plus, Delete[ Divisors[ # ], -1]]]] && IntegerQ[ Sqrt[ Apply[ Times, Delete[Divisors[ # ], -1]]]] && ! PrimeQ[ # ] & ]

A066423 Composite numbers n such that the product of proper divisors of the n does not equal n.

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 24, 25, 28, 30, 32, 36, 40, 42, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 121, 124, 126, 128, 130, 132

Offset: 1

Robert G. Wilson v, Dec 26 2001

nonn

A084115(a(n))>1; complement of A084116. - Reinhard Zumkeller, May 12 2003

			The fourth composite number is 9. Its proper or aliquot divisors are 1 and 3. The product of 1 and 3 equals 3 which is not equal to 9. Therefore 9 is in the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A048741, A007422.

Composite[n_] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; Do[m = Composite[n]; If[ Apply[ Times, Drop[ Divisors[m], -1]] != m, Print[m]], {n, 1, 100} ]
Select[Range[150],CompositeQ[#]&&Times@@Most[Divisors[#]]!=#&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 18 2020 *)

is(n)=my(d=numdiv(n)); d>4 || d==3 \\ Charles R Greathouse IV, Oct 15 2015

A048740 Product of divisors of n-th composite number.

Original entry on oeis.org

8, 36, 64, 27, 100, 1728, 196, 225, 1024, 5832, 8000, 441, 484, 331776, 125, 676, 729, 21952, 810000, 32768, 1089, 1156, 1225, 10077696, 1444, 1521, 2560000, 3111696, 85184, 91125, 2116, 254803968, 343, 125000, 2601, 140608, 8503056, 3025, 9834496

Offset: 1

Enoch Haga

			The third composite number is 8. The product of all divisors of 8 is 8*4*2*1 = 64.
Divisors(48) = {1,2,3,4,6,8,12,16,24,48} => product {1,2,3,4,6,8,12,16,24,48} = 254803968.
Divisors(49) = {1,7,49} => product {1,7,49} = 343.
Divisors(50) = {1,2,5,10,25,50} => product {1,2,5,10,25,50} = 125000.

Albert H. Beiler, Recreations in the Theory of Numbers, 2nd ed., pages 10, 23. New York: Dover, 1966. ISBN 0-486-21096-0.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002808, A007955, A048741.

Rest[Times@@Divisors[#]&/@Complement[Range[100], Prime[ Range[ PrimePi[ 100]]]]] (* Harvey P. Dale, Jan 08 2011 *)
pd[n_] := n^(DivisorSigma[0, n]/2); pd /@ Select[Range[100], CompositeQ] (* Amiram Eldar, Sep 07 2019 *)

from math import isqrt
from sympy import divisor_count, composite
def A048740(n): return (lambda m:isqrt(m)**c if (c:=divisor_count(m)) & 1 else m**(c//2))(composite(n)) # Chai Wah Wu, Jun 25 2022

a(n) = A007955(A002808(n)). - Michel Marcus, Sep 07 2019

Corrected by Neven Juric (neven.juric(AT)apis-it.hr), May 25 2006

cp's OEIS Frontend

A064121 Nonprime numbers n such that the sum of aliquot divisors of n (A001065) and product of aliquot divisors of n (A048741) are both perfect squares.

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A066423 Composite numbers n such that the product of proper divisors of the n does not equal n.

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A048740 Product of divisors of n-th composite number.

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