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.

A324708 Lesser of tri-unitary amicable numbers pair: numbers (m, n) such that tsigma(m) = tsigma(n) = m + n, where tsigma(n) is the sum of the tri-unitary divisors of n (A324706).

Original entry on oeis.org

114, 594, 1140, 5940, 8640, 10744, 12285, 13500, 44772, 60858, 62100, 67095, 67158, 79296, 79650, 79750, 118500, 142310, 143808, 177750, 185368, 298188, 308220, 356408, 377784, 462330, 545238, 600392, 608580, 609928, 624184, 635624, 643336, 643776, 669900
Offset: 1

Views

Author

Amiram Eldar, Mar 11 2019

Keywords

Comments

The larger counterparts are in A324709.

Examples

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

Links

Crossrefs

Cf. A002025, A002952, A126169, A292980, A322541, A324706, A324707, 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]; s[n_] := tsigma[n] - n; seq={}; Do[m=s[n]; If[m>n && s[m]==n, AppendTo[seq, n]] ,{n,1,700000}]; seq

A324709 Larger of tri-unitary amicable numbers pair: numbers (m, n) such that tsigma(m) = tsigma(n) = m + n, where tsigma(n) is the sum of the tri-unitary divisors of n (A324706).

Original entry on oeis.org

126, 846, 1260, 8460, 11760, 10856, 14595, 17700, 49308, 83142, 62700, 71145, 73962, 83904, 107550, 88730, 131100, 168730, 149952, 196650, 203432, 306612, 365700, 399592, 419256, 548550, 721962, 669688, 831420, 686072, 691256, 712216, 652664, 661824, 827700
Offset: 1

Views

Author

Amiram Eldar, Mar 11 2019

Keywords

Comments

The terms are ordered according to their lesser counterparts (A324708).

Examples

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

Links

Crossrefs

Cf. A002046, A002953, A126170, A292981, A322542, A324706, A324707, A324708.

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]; s[n_] := tsigma[n] - n; seq={}; Do[m=s[n]; If[m>n && s[m]==n, AppendTo[seq, m]] ,{n,1,700000}]; seq

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

A335385 The number of tri-unitary divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 8, 2, 2, 4, 4, 4, 4, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 8, 4, 8, 4, 4, 2, 8, 2, 4, 4, 4, 4, 8, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4
Offset: 1

Views

Author

Amiram Eldar, Jun 04 2020

Keywords

Comments

A divisor d of k is tri-unitary if the greatest common bi-unitary divisor of d and k/d is 1.
Differs from A037445 at n = 32, 96, 128, 160, 224, ...

Examples

			a(4) = 2 since 4 has 2 tri-unitary divisors, 1 and 4. 2 is not a tri-unitary divisor of 4 since the greatest common bi-unitary divisor of 2 and 4/2 = 2 is 2 and not 1.

Links

Crossrefs

Cf. A222266, A324706, A324707, A324708, A324709.
Cf. A000005, A034444, A037445, A048105, A049419, A286324, A322483.

Programs

  • Mathematica
    f[p_, e_] := If[e == 3 || e == 6, 4, 2]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100]
  • PARI
    a(n) = vecprod(apply(x -> if(x == 3 || x == 6, 4, 2), factor(n)[, 2])); \\ Amiram Eldar, Dec 18 2023

Formula

Multiplicative with a(p^e) = 4 if e = 3 or 6, and a(p^e) = 2 otherwise.

A335387 Tri-unitary harmonic numbers: numbers k such that the harmonic mean of the tri-unitary divisors of k is an integer.

Original entry on oeis.org

1, 6, 45, 60, 90, 270, 420, 630, 2970, 5460, 8190, 9100, 15925, 27300, 36720, 40950, 46494, 47520, 54600, 81900, 95550, 136500, 163800, 172900, 204750, 232470, 245700, 257040, 332640, 409500, 464940, 491400, 646425, 716625, 790398, 791700, 819000, 900900, 929880
Offset: 1

Views

Author

Amiram Eldar, Jun 04 2020

Keywords

Comments

Equivalently, numbers k such that A324706(k) | (k * A335385(k)).
Differs from A063947 from n >= 18.

Examples

			45 is a term since its tri-unitary divisors are {1, 5, 9, 45} and their harmonic mean, 3, in an integer.

Links

Crossrefs

A324707 is a subsequence.
Analogous sequences: A001599 (harmonic numbers), A006086 (unitary), A063947 (infinitary), A286325 (bi-unitary), A319745 (nonunitary).
Cf. A324706, A335385.

Programs

  • Mathematica
    f1[p_, e_] := If[e == 3 || e == 6, 4, 2]; f2[p_, e_] := If[e == 3, (p^4 - 1)/(p - 1), If[e == 6, (p^8 - 1)/(p^2 - 1), p^e + 1]]; f[p_, e_] := p^e * f1[p, e]/f2[p, e]; tuhQ[1] = True; tuhQ[n_] := IntegerQ[Times @@ (f @@@ FactorInteger[n])]; Select[Range[10^4], tuhQ]
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.