cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A049208 Perfect square associated with A048699.

Original entry on oeis.org

0, 4, 16, 9, 36, 16, 64, 49, 64, 144, 25, 25, 64, 100, 196, 25, 81, 81, 196, 100, 49, 121, 49, 256, 169, 196, 49, 225, 441, 225, 81, 400, 49, 484, 49, 169, 676, 900, 121, 400, 169, 361, 81, 676, 1764, 361, 441, 1024, 289, 441, 81, 1156, 121, 529, 1156, 676, 121, 289
Offset: 1

Views

Author

Enoch Haga

Keywords

Crossrefs

Cf. A048699.

Extensions

Name updated and a(1)=0 inserted by Sean A. Irvine, Jul 21 2021

A048698 Nonprime numbers k such that sum of aliquot divisors of k is a cube.

Original entry on oeis.org

1, 10, 49, 56, 69, 76, 122, 133, 568, 578, 1001, 1018, 1243, 1324, 1431, 1611, 1685, 1819, 1994, 2296, 2323, 3344, 3403, 3627, 3641, 3763, 3981, 4336, 5482, 8186, 9077, 9641, 10113, 10688, 13471, 14188, 14509, 14727, 15940, 16697, 17141, 17619, 19241, 19637
Offset: 1

Views

Author

Enoch Haga

Keywords

Comments

The sum of the aliquot divisors of a prime is exactly 1. - Martin Renner, Oct 13 2011

Examples

			a(4) = 56: the aliquot divisors 1,2,4,7,8,14,28 sum to 64, a cube.

Links

Crossrefs

Cf. A020477, A048699, A194948.

Programs

  • Maple
    a := []; for n from 1 to 1000 do if sigma(n) <> n+1 and type( simplify((sigma(n)-n)^(1/3)), `integer`) then a := [op(a), n]; fi; od: a;
  • Mathematica
    Select[Range[20000], !PrimeQ[#] && IntegerQ @ Surd[DivisorSigma[1, #] - #, 3] &] (* Amiram Eldar, Feb 23 2020 *)
  • PARI
    c=0; for(n=1, 13127239, if(isprime(n)==0, if(ispower(sigma(n)-n, 3), c++; write("b048698.txt", c " " n)))) /* Donovan Johnson, Mar 10 2013 */

A049226 Composite numbers n such that the sum of divisors of n, sigma(n), divided by the number of divisors, d(n) and sigma(n) minus n are both rational squares.

Original entry on oeis.org

119, 527, 1196, 3591, 5831, 6887, 12319, 15407, 18575, 33271, 47959, 51119, 56853, 63119, 65151, 116399, 176911, 328151, 373319, 437999, 438311, 520319, 568519, 724687, 734111, 851927, 957551, 1059191, 1140071, 1437599, 1760831, 1813511, 2320919, 3354479, 3383420
Offset: 1

Views

Author

Enoch Haga

Keywords

Comments

The prime numbers with this property are primes of the form 2*k^2 - 1 (A066436). - Amiram Eldar, Aug 15 2019
The first terms for which the ratio sigma(n)/d(n) is not an integer are 267910912, 1398459816, and 1703794876. - Giovanni Resta, Aug 30 2019

Examples

			a(27) = 957551 is a term since the sum of its 16 divisors is sigma(957551) = 1166400 and both 1166400/16 = 72900 = 270^2 and 1166400 - 957551 = 208849 = 457^2 are perfect squares.

Links

Crossrefs

Cf. A000005, A000203, A049207, A049227, A049228, A048699, A066436.

Programs

  • Magma
    [m:m in [1..3400000]|not IsPrime(m) and IsSquare(SumOfDivisors(m)/#Divisors(m)) and IsSquare(SumOfDivisors(m)-m)]; // Marius A. Burtea, Aug 15 2019
  • Mathematica
    Select[Range[10^5], CompositeQ[#] && And @@ IntegerQ /@ Sqrt[{(s = DivisorSigma[1, #]) * DivisorSigma[0, #], s - #}] &] (* Amiram Eldar, Aug 15 2019 *)
cnQ[n_]:=With[{sg=DivisorSigma[1,n]},CompositeQ[n]&&AllTrue[{Sqrt[sg/DivisorSigma[0,n]],Sqrt[sg-n]},IntegerQ]]; Select[Range[ 339*10^4],cnQ] (* Harvey P. Dale, Mar 31 2025 *)
  • PARI
    is(n) = my(f = factor(n), s = sigma(f), nd = numdiv(f)); issquare(s/nd) && issquare(s - n) && !isprime(n) \\ David A. Corneth, Aug 15 2019

Extensions

Name and offset corrected by Amiram Eldar, Aug 15 2019

A064116 Composite numbers whose sum of aliquot divisors as well as product of aliquot divisors is a perfect square.

Original entry on oeis.org

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, 5043
Offset: 1

Views

Author

Shyam Sunder Gupta, Sep 09 2001

Keywords

Examples

			75 is a term because the sum of the aliquot divisors of 75 = 1 + 3 + 5 + 15 + 25 = 49 = 7^2 and the product of the aliquot divisors of 75 = 1*3*5*15*25 = 75^2.

Links

Crossrefs

Intersection of A048699 and A064499.
Cf. A001065, A007956.

Programs

  • Mathematica
    Do[d = Delete[ Divisors[n], -1]; If[ !PrimeQ[n] && IntegerQ[ Sqrt[ Apply[ Plus, d]]] && IntegerQ[ Sqrt[ Apply[ Times, d]]], Print[n]], {n, 2, 10^4} ]
spsQ[n_]:=Module[{d=Most[Divisors[n]]},CompositeQ[n]&&AllTrue[{Sqrt[ Total[ d]],Sqrt[Times@@d]},IntegerQ]]; Select[Range[5100],spsQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 14 2018 *)
  • PARI
    isok(k) = { my(s=sigma(k) - k); s>1 && issquare(s) && issquare(vecprod(divisors(k)[1..-2])) } \\ Harry J. Smith, Sep 07 2009

Extensions

More terms from Robert G. Wilson v, Oct 05 2001

A064499 Composite numbers n such that product of aliquot divisors of n is a perfect square.

Original entry on oeis.org

12, 16, 18, 20, 28, 32, 44, 45, 48, 50, 52, 63, 68, 75, 76, 80, 81, 92, 98, 99, 112, 116, 117, 124, 147, 148, 153, 162, 164, 171, 172, 175, 176, 180, 188, 192, 207, 208, 212, 236, 242, 243, 244, 245, 252, 256, 261, 268, 272, 275, 279, 284, 288, 292, 300, 304
Offset: 1

Views

Author

Robert G. Wilson v, Oct 05 2001

Keywords

Links

Crossrefs

Cf. A048699, A007956.

Programs

  • Mathematica
    Do[ If[ !PrimeQ[n] && IntegerQ[ Sqrt[ Apply[ Times, Delete[ Divisors[n], -1]]]], Print[n]], {n, 2, 500} ]
  • PARI
    pad(n)=my(d=divisors(n), p=1); for (i=1, #d-1, p*=d[i]); p
n=0; for (m=2, 10^9, if (!isprime(m) && issquare(pad(m)), write("b064499.txt", n++, " ", m); if (n==1000, break))) \\ Harry J. Smith, Sep 16 2009
  • PARI
    is(n)=!isprime(n) && (ispower(n,4) || numdiv(n)%4==2) && n>1 \\ Charles R Greathouse IV, Oct 17 2015

A383872 Nonprime numbers whose sum of proper divisors is a power of 4.

Original entry on oeis.org

9, 12, 26, 56, 76, 122, 332, 992, 2042, 3344, 4336, 8186, 16256, 32762, 227744, 266176, 269072, 299576, 856544, 2097146, 5385812, 8388602, 16580864, 17895664, 19173944, 33554426, 61008020, 67100672, 201931760, 1074789376, 1108378592, 17179738112, 62472251540, 68700578816
Offset: 1

Views

Author

Hans Ulrich Keller, May 13 2025

Keywords

Comments

Includes 2*p for p in A135535. - Robert Israel, May 13 2025
From David A. Corneth, May 13 2025: (Start)
If a(n) = m*p where p is the largest prime divisor and has multiplicity 1 and s = sigma(m) then p = (4^k - s) / (s - m). Using this, a(44), a(45) and a(46) are at most 4971704751572, 44088037271892 and 44358570286896 respectively.
If a(n) is odd then a(n) is a perfect square. Proof: Suppose a(n) is not a perfect square. Then sigma(a(n)) is even and so sigma(a(n)) - a(n) = 4^k. As sigma(a(n)) - a(n) = 4^k then sigma(a(n)) - a(n) = 1. As a(n) is composite this has no solutions. (End)
For the first 43 terms, a(1) is the only square term and for the other terms, a(n) has a squarefree odd part. However, this is not always true as 44088037271892 (see above) is a term and its odd part is not squarefree. - Chai Wah Wu, May 19 2025

Examples

			12 is not prime; 12 has proper divisors 1, 2, 3, 4, and 6, with a sum of 16. This is a square number as well as a power of 2.

Links

Crossrefs

Intersection of A048699 and A279731.
Cf. A001065, A135535.

Programs

  • Maple
    filter:= proc(n) local s;
  s:= numtheory:-sigma(n)-n;
  s > 1 and s = 4^padic:-ordp(s,4)
end proc:
select(filter, [$4..10^7]); # Robert Israel, May 13 2025
  • Mathematica
    Zweierpotenzen = {};
Quadratzahlen = {};
Beides = {};
For[k = 1, k <= 50000000, k++,
  SumET = Total[Divisors[k]] - k;
  If[IntegerQ[Log[2, SumET]] && PrimeQ[k] == False,
   AppendTo[Zweierpotenzen, k]];
  If[IntegerQ[Sqrt[SumET]] && PrimeQ[k] == False,
   AppendTo[Quadratzahlen, k ]]];
Beides = Intersection[Zweierpotenzen, Quadratzahlen];
Beides
  • PARI
    isok(k) = if (!ispseudoprime(k), my(s=sigma(k)-k, z); issquare(s) && (ispower(s, , &z) && (z==2))); \\ Michel Marcus, May 13 2025

Extensions

a(27)-a(29) from Michel Marcus, May 13 2025
a(30)-a(34) from Amiram Eldar, May 13 2025

A176996 Numbers n such that sum of divisors, sigma(n), and sum of the proper divisors, sigma(n)-n, are both square.

Original entry on oeis.org

1, 3, 119, 527, 935, 3591, 3692, 6887, 12319, 47959, 65151, 97767, 99116, 202895, 237900, 373319, 438311, 699407, 734111, 851927, 957551, 1032156, 1064124, 1437599, 1443959, 2858687, 3509231, 3699311, 4984199, 7237415
Offset: 1

Views

Author

Claudio Meller, Dec 08 2010

Keywords

Comments

The only prime in this sequence is 3. All prime numbers have the square 1 as the sum of their proper divisors. But since 3 is the only prime of the form n^2 - 1, it is the only prime that satisfies the first condition for inclusion in this sequence.

Examples

			119 has divisors 1, 7, 17, 119; it is in the list because 1+7+17+119 = 144 = 12^2 and 1+7+17 = 25 = 5^2.

Links

Crossrefs

Cf. A006532, which considers all divisors; A048699, which for nonprime numbers considers all divisors except the number itself; A073040, which is the union of A048699 and the prime numbers (A000040).

Programs

  • Mathematica
    Intersection[Select[Range[10^5], IntegerQ[Sqrt[-# + Plus@@Divisors[#]]] &], Select[Range[10^5], IntegerQ[Sqrt[Plus@@Divisors[#]]] &]] (* Alonso del Arte, Dec 08 2010 *)
t = {}; Do[If[And @@ IntegerQ /@ Sqrt[{x = DivisorSigma[1, n], x - n}], AppendTo[t, n]], {n, 10^6}]; t (* Jayanta Basu, Jul 27 2013 *)
sdQ[n_]:=Module[{d=DivisorSigma[1,n]},AllTrue[{Sqrt[d],Sqrt[d-n]}, IntegerQ]]; Select[Range[73*10^5],sdQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 17 2018 *)
  • Sage
    is_A176996 = lambda n: is_square(sigma(n)) and is_square(sigma(n)-n) # D. S. McNeil, Dec 09 2010

Formula

Intersection of A006532 and A073040.

A190665 Numbers k such that sum of aliquot parts of k is a nontrivial power: sigma(k) - k = a^b for integers a>1, b>1.

Original entry on oeis.org

9, 10, 12, 15, 24, 26, 49, 56, 58, 69, 75, 76, 90, 95, 119, 122, 124, 133, 140, 143, 147, 153, 176, 194, 215, 243, 287, 332, 363, 386, 407, 429, 477, 495, 507, 511, 524, 527, 536, 551, 568, 575, 578, 688, 717, 738, 791, 794, 815, 867, 871, 892, 924, 935, 961, 963, 992, 1001, 1018, 1075, 1083, 1159, 1196, 1199, 1243, 1295, 1304, 1324, 1346, 1391, 1415, 1421, 1431, 1532, 1573, 1587
Offset: 1

Views

Author

Antonio Roldán, May 16 2011

Keywords

Comments

Similar to A065496.

Examples

			122: aliquot parts: 1, 2, 61, sum: 1+2+61 = 64 = 8^2
140: sum of aliquot parts: 1+2+4+5+7+10+14+20+28+35+70 = 196 = 14^2.

Crossrefs

Cf. A065496, A001597, A048699.

Programs

  • Maple
    isA001597 := proc(n) for a from 2 do if a^2 > n then return false; end if; for b from 2 do if a^b =n then return true; elif a^b>n then break; end if; end do; end do: end proc:
isA190665 := proc(n) isA001597(numtheory[sigma](n)-n) ; end proc:
for n from 1 to 2000 do if isA190665(n) then printf("%d,",n) ; end if; end do; # R. J. Mathar, May 30 2011
  • Mathematica
    powerQ[n_] := GCD @@ FactorInteger[n][[All, 2]] > 1;
okQ[n_] := powerQ[DivisorSigma[1, n] - n];
Select[Range[2000], okQ] (* Jean-François Alcover, Jul 31 2024 *)
  • PARI
    ypower(n)= { local(f, p=0); f=factor(n); if(gcd(f[, 2])>1,p=1); return(p) }
{  for (n=1, 1000, a=sigma(n)-n; if(ypower(a), print1(n," "))) }
/* Antonio Roldán, Oct 23 2012 */
Showing 1-8 of 8 results.

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