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 1-4 of 4 results.

A129087 Odd doubly abundant numbers (A125639).

Original entry on oeis.org

11025, 34155, 38745, 39585, 41895, 75735, 85995, 99225, 118755, 131355, 135135, 193725, 208845, 218025, 237195, 241395, 241605, 245385, 255645, 271215, 272745, 275625, 276885, 279279, 306495, 307125, 323505, 342225, 347985, 364455, 377685
Offset: 1

Views

Author

Ant King, Apr 02 2007

Keywords

Comments

This sequence contains the odd members of A125639, which (empirically) accounts for only about 0.08% of them.

Examples

			The third odd number which is doubly abundant is 38745. Hence a(3)=38745.

Links

Crossrefs

Cf. A125639, A001065, A005101.

Programs

  • Maple
    filter:= proc(n) local s;
   s:= numtheory:-sigma(n)-n;
   s > n and numtheory:-sigma(s)>2*s
end proc:
select(filter, [seq(i,i=3..400000,2)]); # Robert Israel, Jun 04 2018
  • Mathematica
    s[n_]:=DivisorSigma[1,n]-n;DoublyAbundantNumberQ[k_]:=If[s[k]>k && s[s[k]]>s[k],True,False];Select[Range[500000],OddQ[ # ] && DoublyAbundantNumberQ[ # ] & ]
  • PARI
    isok(n) = (n%2) && ((s=sigma(n)-n) > n) && (sigma(s) > 2*s); \\ Michel Marcus, Jun 05 2018

Formula

Odd numbers k, such that both k and s(k) (A001065) are abundant (A005101).

A371920 Abundant numbers whose abundance is also an abundant number.

Original entry on oeis.org

24, 30, 42, 54, 60, 66, 78, 84, 90, 96, 102, 112, 114, 120, 126, 132, 138, 140, 150, 156, 168, 174, 176, 180, 186, 198, 204, 208, 210, 216, 222, 224, 228, 234, 240, 246, 252, 258, 264, 270, 276, 280, 282, 294, 304, 306, 308, 312, 318, 330, 336, 342, 348, 354, 360
Offset: 1

Views

Author

Amiram Eldar, Apr 12 2024

Keywords

Comments

First differs from A125639 at n = 12.
Numbers k such that A033880(k) > 0 and A033880(A033880(k)) > 0.
This sequence is infinite: if m is divisible by 6 and coprime to 5, then 5*m is a term.
All the multiply-perfect numbers (A007691) that are not 1 or perfect (A000396), i.e., the terms of A166069, are terms of this sequence.

Examples

			24 is a term since A033880(24) = 12 > 0 and A033880(12) = 4 > 0.

Links

Crossrefs

Cf. A033880 (abundance), A000396, A007691, A125639.
Subsequence of A005101.
Subsequences: A141545, A166069, A223610, A223611, A223613.

Programs

  • Mathematica
    ab[n_] := DivisorSigma[1, n] - 2*n; q[n_] := Module[{k = ab[n]}, k > 0 && ab[k] > 0]; Select[Range[360], q]
  • PARI
    ab(n) = sigma(n) - 2*n;
is(n) = {my(k = ab(n)); k > 0 && ab(k) > 0;}

A371950 Weird numbers (A006037) whose sum of aliquot divisors is also a weird number.

Original entry on oeis.org

97930, 132230, 146930, 191030, 205730, 215530, 244930, 259630, 279230, 308630, 362530, 411530, 440930, 524230, 529130, 583030, 597730, 602630, 632030, 646730, 705530, 730030, 808430, 891730, 921130, 955430, 970130, 1014230, 1024030, 1028930, 1102430, 1215130, 1435630
Offset: 1

Views

Author

Amiram Eldar, Apr 14 2024

Keywords

Comments

Terms k of A006037 such that A001065(k) is also a term of A006037.

Examples

			97930 is a term because it is a weird number, and A001065(97930) = sigma(97930) - 97930 = 103670 is also a weird number.

Links

Crossrefs

Subsequence of A006037 and A125639.
Cf. A000203 (sigma), A001065, A371951, A371952.

Programs

  • Mathematica
    With[{weirds = Import["https://oeis.org/a006037/b006037.txt", "Table"][[;; , 2]]}, Select[weirds, (s = DivisorSigma[1, #] - #) <= Last[weirds] && MemberQ[weirds, s] &]]

A125640 Primitive doubly abundant numbers - doubly abundant numbers that are not the multiple of another doubly abundant number.

Original entry on oeis.org

24, 30, 42, 54, 66, 78, 102, 114, 138, 140, 174, 176, 186, 222, 224, 246, 258, 282, 308, 318, 340, 354, 364, 366, 380, 402, 426, 438, 440, 474, 476, 498, 520, 532, 534, 580, 582, 606, 618, 642, 644, 654, 678, 762, 786, 812, 822, 834, 868, 894
Offset: 1

Views

Author

Gabriel Cunningham (gabriel.cunningham(AT)gmail.com), Nov 28 2006

Keywords

Comments

Are there infinitely many primitive doubly abundant numbers?

Examples

			42 is a primitive doubly abundant number because it is abundant (s(42) = 54), the sum of its proper divisors is abundant (s(54) = 66) and no divisor of 42 is doubly abundant.

Links

Crossrefs

Cf. A005101, A125639.
Cf. A027751.

Programs

  • Haskell
    import Data.List (intersect)
a125640 n = a125640_list !! (n-1)
a125640_list = f a125639_list [] where
   f (x:xs) ys = if null (a027751_row' x `intersect` ys)
                    then x : f xs (x : ys) else f xs ys
-- Reinhard Zumkeller, Oct 31 2015
  • Mathematica
    s[n_] := DivisorSigma[1, n] - n; q[n_] := Module[{s1 = s[n]}, s1 > n && s[s1] > s1]; primQ[n_] := q[n] && !AnyTrue[Most[Divisors[n]], q]; Select[Range[900], primQ] (* Amiram Eldar, Mar 11 2024 *)

Extensions

Data corrected by Reinhard Zumkeller, Oct 31 2015
Showing 1-4 of 4 results.

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