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 1-5 of 5 results.

A322609 Numbers k such that s(k) = 2*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

Original entry on oeis.org

24, 54, 112, 150, 294, 726, 1014, 1734, 1984, 2166, 3174, 5046, 5766, 8214, 10086, 11094, 13254, 16854, 19900, 20886, 22326, 26934, 30246, 31974, 32512, 37446, 41334, 47526, 56454, 61206, 63654, 68694, 71286, 76614, 96774, 102966, 112614, 115926, 133206
Offset: 1

Views

Author

Amiram Eldar, Dec 20 2018

Keywords

Comments

This sequence is infinite since 6*p^2 is included for all primes p. Terms that are not of the form 6*p^2: 112, 1984, 19900, 32512, 134201344, ...
Includes 4*k if k is an even perfect number: see A000396. - Robert Israel, Jan 06 2019
From Amiram Eldar, Oct 01 2022: (Start)
24 = 6*prime(1)^2 = 4*A000396(1) is the only term that is common to the two forms that are mentioned above.
19900 is the only term below 10^11 which is not of any of these two forms. Are there any other such terms?
All the known nonunitary perfect numbers (A064591) are also of the form 4*k, where k is an even perfect number.
Equivalently, numbers k such that A325314(k) = -k. (End)

Examples

			24 is a term since A162296(24) = 48 = 2*24.

Links

Crossrefs

Cf. A162296, A325314.
Subsequence of A005101 and A013929.
Numbers k such that A162296(k) = m*k: A005117 (m=0), A001248 (m=1), this sequence (m=2), A357493 (m=3), A357494 (m=4).
Similar sequences: A000396, A002827, A007357, A054979, A064591, A322486, A323757, A324707, A327633.

Programs

  • Maple
    filter:= proc(n) convert(remove(numtheory:-issqrfree,numtheory:-divisors(n)),`+`)=2*n end proc:
select(filter, [$1..200000]); # Robert Israel, Jan 06 2019
  • Mathematica
    s[1]=0; s[n_] := DivisorSigma[1,n] - Times@@(1+FactorInteger[n][[;;,1]]); Select[Range[10000], s[#] == 2# &]
  • PARI
    s(n) = sumdiv(n, d, d*(1-moebius(d)^2)); \\ A162296
isok(n) = s(n) == 2*n; \\ Michel Marcus, Dec 20 2018
  • Python
    from sympy import divisors, factorint
A322609_list = [k for k in range(1,10**3) if sum(d for d in divisors(k,generator=True) if max(factorint(d).values(),default=1) >= 2) == 2*k] # Chai Wah Wu, Sep 19 2021

A357494 Numbers k such that s(k) = 4*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

Original entry on oeis.org

902880, 1534680, 361674720, 767685600, 4530770640, 4941414720, 5405788800, 5517818880, 16993944000, 20429240832, 94820077440
Offset: 1

Views

Author

Amiram Eldar, Oct 01 2022

Keywords

Comments

Analogous to 4-perfect numbers (A027687) with nonsquarefree divisors.
Equivalently, numbers k such that A325314(k) = -3*k.
There are no numbers k below 10^11 such that A162296(k) = m*k for integers m > 4.

Examples

			902880 is a term since A162296(902880) = 3611520 = 4*902880.

Crossrefs

Cf. A162296, A325314.
Subsequence of A013929 and A023198.
Numbers k such that A162296(k) = m*k: A005117 (m=0), A001248 (m=1), A322609 (m=2), A357493 (m=3), this sequence (m=4).
Similar sequence: A027687.

Programs

  • Mathematica
    q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) == 4*n]; Select[Range[2, 2*10^6], q]

A357495 Lesser of a pair of amicable numbers k < m such that s(k) = m and s(m) = k, where s(k) = A162296(k) - k is the sum of aliquot divisors of k that have a square factor.

Original entry on oeis.org

880, 10480, 20080, 24928, 42976, 69184, 110565, 252080, 267712, 489472, 566656, 569240, 603855, 626535, 631708, 687424, 705088, 741472, 786896, 904365, 1100385, 1234480, 1280790, 1425632, 1749824, 1993750, 2012224, 2401568, 2439712, 2496736, 2542496, 2573344, 2671856
Offset: 1

Views

Author

Amiram Eldar, Oct 01 2022

Keywords

Comments

Analogous to amicable numbers (A002025 and A002046) with nonsquarefree divisors.
The larger counterparts are in A357496.
Both members of each pair are necessarily nonsquarefree numbers.

Examples

			880 is a term since s(880) = 1136 and s(1136) = 880.

Links

Crossrefs

Cf. A162296, A325314, A322609, A357493, A357494, A357496.
Subsequence of A013929.
Similar sequences: A002025, A002952, A126165, A126169, A259038, A292980, A322541, A324708, A348343.

Programs

  • Mathematica
    s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) - n]; seq = {}; Do[m = s[n]; If[m > n && s[m] == n, AppendTo[seq, n]], {n, 2, 3*10^6}]; seq

A357496 Greater of a pair of amicable numbers k < m such that s(k) = m and s(m) = k, where s(k) = A162296(k) - k is the sum of aliquot divisors of k that have a square factor.

Original entry on oeis.org

1136, 11696, 22256, 25472, 43424, 73664, 131355, 304336, 267968, 492608, 612704, 674920, 640305, 788697, 691292, 705344, 723392, 813728, 809776, 1117395, 1258335, 1559696, 1518570, 1598368, 1821376, 2218250, 2058944, 2678752, 2744288, 2765024, 2848864, 2610656, 3134224
Offset: 1

Views

Author

Amiram Eldar, Oct 01 2022

Keywords

Comments

Analogous to amicable numbers (A002025 and A002046) with nonsquarefree divisors.
The terms are ordered according to their lesser counterparts (A357495).
Both members of each pair are necessarily nonsquarefree numbers.

Examples

			1136 is a term since s(1136) = 880 and s(880) = 1136.

Links

Crossrefs

Cf. A162296, A325314, A322609, A357493, A357494, A357495.
Subsequence of A013929.
Similar sequences: A002046, A002953, A126166, A126170, A259039, A292981, A322542, A324709, A348344.

Programs

  • Mathematica
    s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) - n]; seq = {}; Do[m = s[n]; If[m > n && s[m] == n, AppendTo[seq, m]], {n, 2, 3*10^6}]; seq

A357497 Nonsquarefree numbers whose harmonic mean of nonsquarefree divisors in an integer.

Original entry on oeis.org

4, 9, 12, 18, 24, 25, 28, 45, 49, 54, 60, 90, 112, 121, 126, 132, 150, 153, 168, 169, 198, 270, 289, 294, 336, 361, 364, 414, 529, 560, 594, 630, 637, 684, 726, 841, 918, 961, 1014, 1140, 1232, 1305, 1350, 1369, 1512, 1521, 1638, 1680, 1681, 1710, 1734, 1849, 1984
Offset: 1

Views

Author

Amiram Eldar, Oct 01 2022

Keywords

Comments

Analogous to harmonic numbers (A001599) with nonsquarefree divisors.
The squares of primes (A001248) are terms since they have a single nonsquarefree divisor.
If p is a prime then 6*p^2 is a term.

Examples

			12 is a term since its nonsquarefree divisors are 4 and 12 and their harmonic mean is 6 which is an integer.

Links

Crossrefs

Cf. A162296, A322609, A357493, A357494, A357495, A357496.
Subsequence of A013929.
Subsequence: A001248.
Similar sequences: A001599 (harmonic numbers), A006086 (unitary), A063947 (infinitary), A286325 (bi-unitary), A319745 (nonunitary), A335387 (tri-unitary).

Programs

  • Mathematica
    q[n_] := Length[d = Select[Divisors[n], ! SquareFreeQ[#] &]] > 0 && IntegerQ[HarmonicMean[d]]; Select[Range[2000], q]
Showing 1-5 of 5 results.

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