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.

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A342399 Unitary pseudoperfect numbers k such that no subset of the nontrivial unitary divisors {d|k : 1 < d < k, gcd(d, k/d) = 1} adds up to k.

Original entry on oeis.org

3510, 3770, 5670, 5810, 6790, 7630, 7910, 9590, 9730, 544310, 740870, 2070970, 4017310, 4095190, 5368510, 5569690, 5762330, 5838770, 5855290, 5856130, 5887630, 5902470, 5985770, 6006070, 6039530, 6075370, 6083630, 6181210, 6259610, 6471290, 7038710, 7065730, 7285390
Offset: 1

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Author

Amiram Eldar, Mar 10 2021

Keywords

Comments

Numbers that are the sum of a proper subset of their aliquot unitary divisors but are not the sum of any subset of their nontrivial unitary divisors.
The unitary perfect numbers (A002827) which are a subset of the unitary pseudoperfect numbers (A293188) are excluded from this sequence since otherwise they would all be trivial terms: if k is a unitary perfect number then the sum of the divisors {d|k : 1 < d < k, gcd(d, k/d) = 1} is k-1, so any subset of them has a sum smaller than k.
The unitary pseudoperfect numbers are thus a disjoint union of the unitary perfect numbers, this sequence and A342398.
The unitary abundant numbers (A034683) are a disjoint union of the unitary weird numbers (A064114), this sequence and A342398.

Examples

			3510 is a term since it is a unitary pseudoperfect number, 3510 = 1 + 2 + 5 + 13 + 27 + 54 + 65 + 130 + 135 + 270 + 351 + 702 + 1755, and the set of nontrivial unitary divisors of 3510, {d|3510 : 1 < d < 3510, gcd(d, 3510/d) = 1} = {2, 5, 10, 13, 26, 27, 54, 65, 130, 135, 270, 351, 702, 1755}, has no subset that adds up to 3510.

Links

Crossrefs

The unitary version of A339343.
Subsequence of A034683 and A293188.
Cf. A002827, A064114, A342398.

Programs

  • Mathematica
    q[n_] := Module[{d = Most @ Select[Divisors[n], CoprimeQ[#, n/#] &], x}, Plus @@ d > n && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] > 0 && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, 2, Length[d]}], {x, 0, n}], n] == 0]; Select[Range[10^4], q]

A360524 Numbers k such that A360522(k) = 2*k.

Original entry on oeis.org

6, 12, 198, 240, 264, 270, 396, 540, 6720, 7920, 11880, 13770, 27540, 221760, 337440, 605880, 2500344, 6072570, 11135520, 12145140, 267193080, 441692160, 1112629770, 2225259540, 14575841280, 48955709880
Offset: 1

Views

Author

Amiram Eldar, Feb 10 2023

Keywords

Comments

Analogous to perfect numbers (A000396) with A360522 instead of A000203.
a(27) > 10^11, if it exists.

Examples

			6 is a term since A360522(6) = 12 = 2 * 6.

Crossrefs

Cf. A000203, A360522.
Similar sequences: A000396, A002827, A007357, A054979, A322486, A324707.

Programs

  • Mathematica
    f[p_, e_] := p^e + e; q[n_] := Times @@ f @@@ FactorInteger[n] == 2*n; Select[Range[10^6], q]
  • PARI
    is(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] + f[i,2]) == 2*n;}

A369895 Irregular triangle of iteration steps of A063919 until the end of the terminal cycle is reached, read by rows.

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 7, 1, 8, 1, 9, 1, 10, 8, 1, 11, 1, 12, 8, 1, 13, 1, 14, 10, 8, 1, 15, 9, 1, 16, 1, 17, 1, 18, 12, 8, 1, 19, 1, 20, 10, 8, 1, 21, 11, 1, 22, 14, 10, 8, 1, 23, 1, 24, 12, 8, 1, 25, 1, 26, 16, 1, 27, 1, 28, 12, 8, 1, 29, 1, 30, 42, 54
Offset: 1

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Author

Hartmut F. W. Hoft, Feb 04 2024

Keywords

Examples

			The beginning of the irregular triangle showing 3 terminal cycles ( 1 ), ( 6 ) and ( 30 42 54 ):
  1
  2    1
  3    1
  4    1
  5    1
  6
  7    1
  ...
  14  10   8   1
  ...
  30  42  54
  31  1
  ...
Row 1230 contains a non-monotone iteration that ends in the 5-cycle starting at A097035(3):
1230, 1794, 2238, 2250, 1530, 1710, {1890, 2142, 2178, 1482, 1878 }.

Crossrefs

Cf. A002827, A063919, A097024, A097030, A097031, A097034, A097035, A318882, A318883.

Programs

  • Mathematica
    a063919[1] = 1; (* function a[] in A063919 by Jean-François Alcover *)
a063919[n_] := Total[Select[Divisors[n], GCD[#, n/#] == 1 &]] - n /; n > 1
iter[k_] := Most[NestWhileList[a063919, k, UnsameQ, All]]
a369895[n_] := Map[iter, Range[n]]
a369895[30] (* irregular triangle *)
Flatten[a369895[30]] (* sequence data *)

A307859 Consider the non-unitary aliquot parts, in ascending order, of a composite number. Take their sum and repeat the process deleting the minimum number and adding the previous sum. The sequence lists the numbers that after some iterations reach a sum equal to themselves.

Original entry on oeis.org

24, 112, 189, 578, 1984, 2125, 3993, 5043, 9583, 19197, 32512, 126445, 149565, 175689, 225578, 236883, 1589949, 1862935, 1928125, 3171174, 5860526, 6149405, 11442047, 16731741, 60634549, 75062535, 134201344, 177816209, 1162143369, 4474779517, 10369035821
Offset: 1

Views

Author

Paolo P. Lava, May 02 2019

Keywords

Examples

			Divisors of 578 are 1, 2, 17, 34, 289, 578. Non-unitary aliquot parts are 17 and 34.
We have:
   17 +  34 =  51;
   34 +  51 =  85;
   51 +  85 = 136;
   85 + 136 = 221;
  136 + 221 = 357;
  221 + 357 = 578.

Links

Crossrefs

Cf. A002827, A034444, A246544, A247012, A247013, A248134, A258142.

Programs

  • Maple
    with(numtheory):P:=proc(q,h) local a,b,c,k,n,t,v; v:=array(1..h);
for n from 1 to q do if not isprime(n) then b:=sort([op(divisors(n))]);
a:=[]; for k from 2 to nops(b)-1 do if gcd(b[k],n/b[k])>1 then
a:=[op(a),b[k]]; fi; od; b:=nops(a); if b>1 then c:=0;
for k from 1 to b do v[k]:=a[k]; c:=c+a[k]: od;
t:=b+1; v[t]:=c; while v[t]
  • Mathematica
    aQ[n_] := CompositeQ[n] && Module[{s = Select[Divisors[n], GCD[#, n/#] != 1 &]}, If[Length[s] < 2, False, While[Total[s] < n, AppendTo[s, Total[s]]; s = Rest[s]]; Total[s] == n]]; Select[Range[10^4], aQ] (* Amiram Eldar, May 07 2019 *)

Extensions

a(20)-a(31) from Amiram Eldar, May 07 2019

A332478 Number that are unitary norm-multiply-perfect numbers in Gaussian integers.

Original entry on oeis.org

1, 10, 12, 20160, 15713280, 137592000, 44289146880
Offset: 1

Views

Author

Amiram Eldar, Feb 13 2020

Keywords

Comments

Numbers k such that their norm of sum of unitary divisors in Gaussian integers, A332474(k), is divisible by their norm, k^2.
The corresponding ratios A332474(a(n))/(a(n)^2) are 1, 4, 1, 5, 5, 2, 5.

Examples

			10 is a term since its sum of unitary divisors in Gaussian integers is -12 + 16*i, whose norm (-12)^2 + 16^2 = 400 is divisible by 10^2 = 100.

Crossrefs

Cf. A002827, A327158, A332318, A332319, A332474.

Programs

  • Mathematica
    f[p_, e_] := If[Abs[p] == 1, 1, (p^e + 1)]; Select[Range[21000], Divisible[Abs[ Times @@ f @@@ FactorInteger[#, GaussianIntegers -> True]]^2, #^2] &]
Previous Showing 41-45 of 45 results.

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