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

A324707 Tri-unitary perfect numbers: numbers k such that tsigma(k) = 2k, where tsigma(k) is the sum of the tri-unitary divisors of k (A324706).

Original entry on oeis.org

6, 60, 90, 36720, 47520, 8173440, 22276800, 126463680, 597542400, 4201148160, 287704872000, 1632485836800
Offset: 1

Views

Author

Amiram Eldar, Mar 11 2019

Keywords

Comments

Also in the sequence is 21623407345626345971712000.
a(13) > 5*10^12. - Giovanni Resta, Mar 14 2019

Examples

			36720 is in the sequence since its sum of tri-unitary divisors is A324706(36720) = 73440 = 2 * 36720.

Crossrefs

Cf. A000396, A002827, A007357, A322486, A324706, A324708, A324709.

Programs

  • Mathematica
    f[p_, e_] := If[e == 3, (p^4-1)/(p-1), If[e==6, (p^8-1)/(p^2-1), p^e+1]]; tsigma[1]=1; tsigma[n_]:= Times @@ f @@@ FactorInteger[n]; Select[Range[50000], tsigma[#]==2# &]

Extensions

a(11)-a(12) from Giovanni Resta, Mar 14 2019

A322541 Lesser of semi-unitary amicable numbers pair: numbers (m, n) such that susigma(m) = susigma(n) = m + n, where susigma(n) is the sum of the semi-unitary divisors of n (A322485).

Original entry on oeis.org

114, 366, 1140, 3660, 3864, 5016, 11040, 15210, 16104, 16536, 18480, 44772, 57960, 67158, 68640, 68880, 142290, 142310, 155760, 196248, 198990, 240312, 248040, 275520, 278160, 308220, 322080, 326424, 339822, 348840, 352632, 366792, 462330, 485760, 607920
Offset: 1

Views

Author

Amiram Eldar, Dec 14 2018

Keywords

Examples

			114 is in the sequence since it is the lesser of the amicable pair (114, 126): susigma(114) = susigma(126) = 114 + 126.

Crossrefs

Cf. A002025, A002952, A322485, A322486, A322542.

Programs

  • Mathematica
    f[p_, e_] := (p^Floor[(e + 1)/2] - 1)/(p - 1) + p^e; s[n_] := If[n == 1, 1, Times @@ (f @@@ FactorInteger[n])] - n; seq = {}; Do[n = s[m]; If[n > m && s[n] == m, AppendTo[seq, m]], {m, 1, 1000000}]; seq
  • PARI
    susigma(n) = {my(f = factor(n)); for (k=1, #f~, my(p=f[k, 1], e=f[k, 2]); f[k, 1] = (p^((e+1)\2) - 1)/(p-1) + p^e; f[k, 2] = 1; ); factorback(f); } \\ A322485
isok(n) = my(m=susigma(n)-n); (m > n) && (susigma(m) == n + m); \\ Michel Marcus, Dec 15 2018

A322542 Larger of semi-unitary amicable numbers pair: numbers (m, n) such that susigma(m) = susigma(n) = m + n, where susigma(n) is the sum of the semi-unitary divisors of n (A322485).

Original entry on oeis.org

126, 378, 1260, 3780, 4584, 5544, 11424, 15390, 16632, 16728, 25296, 49308, 68760, 73962, 88608, 84336, 179118, 168730, 172560, 225096, 256338, 266568, 250920, 297024, 287280, 365700, 374304, 391656, 374418, 387720, 386568, 393528, 548550, 502656, 623280
Offset: 1

Views

Author

Amiram Eldar, Dec 14 2018

Keywords

Comments

The terms are ordered according to the order of their lesser counterparts (A322541).

Examples

			126 is in the sequence since it is the larger of the amicable pair (114, 126): susigma(114) = susigma(126) = 114 + 126.

Crossrefs

Cf. A002025, A002952, A322485, A322486, A322541.

Programs

  • Mathematica
    f[p_, e_] := (p^Floor[(e + 1)/2] - 1)/(p - 1) + p^e; s[n_] := If[n == 1, 1, Times @@ (f @@@ FactorInteger[n])] - n; seq = {}; Do[n = s[m]; If[n > m && s[n] == m, AppendTo[seq, n]], {m, 1, 1000000}]; seq
  • PARI
    susigma(n) = {my(f = factor(n)); for (k=1, #f~, my(p=f[k, 1], e=f[k, 2]); f[k, 1] = (p^((e+1)\2) - 1)/(p-1) + p^e; f[k, 2] = 1; ); factorback(f); } \\ A322485
lista(nn) = {for (n=1, nn, my(m=susigma(n)-n); if ((m > n) && (susigma(m) == n + m), print1(m, ", ")););} \\ Michel Marcus, Dec 15 2018

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;}
Showing 1-5 of 5 results.

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