A036434 -id:A036434 - cp's OEIS frontend

A036438 Integers which can be written as m*tau(m) for some m, where tau = A000005.

1, 4, 6, 10, 12, 14, 22, 24, 26, 27, 32, 34, 38, 40, 46, 56, 58, 60, 62, 72, 74, 75, 80, 82, 84, 86, 88, 94, 104, 106, 108, 118, 120, 122, 132, 134, 136, 140, 142, 146, 147, 152, 156, 158, 166, 168, 178, 184, 192, 194, 202, 204, 206, 214, 218, 220, 226, 228, 232

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk)

Invented by the HR concept formation program.

			10 = 5 * tau(5).

Antti Karttunen, Table of n, a(n) for n = 1..11329
Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.
Simon Colton, HR - Automatic Theory Formation in Pure Mathematics, 1998-1999. [Wayback Machine link]

Cf. A000005, A001747, A036434.

Range of A038040.

q[k_] := AnyTrue[Divisors[k], # * DivisorSigma[0, #] == k &]; Select[Range[250], q] (* Amiram Eldar, Feb 01 2025 *)

isok(n) = {for (k=1, n, if (k*numdiv(k) == n, return (1));); return (0);} \\ Michel Marcus, Dec 09 2014

up_to = 65536;
A036438list(up_to) = { my(v=vector(up_to), m = Map()); for(n=1,#v,mapput(m,n*numdiv(n),n)); my(k=0,u=0); while((k<#v)&&(u<#v), u++; if(mapisdefined(m,u), k++; v[k] = u)); vector(k,i,v[i]); };
v036438 = A036438list(up_to);
A036438(n) = v036438[n]; \\ Antti Karttunen, Jul 18 2020

A096335 Number of iterations of n -> n + tau(n) needed for the trajectory of n to join the trajectory of A064491, or -1 if the two trajectories never merge.

Original entry on oeis.org

0, 0, 2, 0, 1, 3, 0, 1, 0, 2, 8, 0, 7, 1, 6, 5, 6, 0, 5, 3, 4, 3, 4, 0, 3, 2, 13, 2, 13, 1, 12, 0, 11, 1, 10, 8, 10, 0, 9, 7, 9, 0, 8, 1, 7, 1, 8, 6, 7, 0, 6, 6, 6, 5, 5, 0, 4, 5, 4, 26, 3, 4, 2, 0, 2, 3, 2, 3, 1, 2, 0, 25, 0, 2, 0, 2, 1, 1, 1, 1, 0, 1, 39, 24, 38

Offset: 1

Jason Earls, Jun 28 2004

nonn

Conjecture: For any positive integer starting value n, iterations of n -> n + tau(n) will eventually join A064491 (verified for all n up to 50000).

The graph looks like a forest of stalks. The tops of the stalks form A036434. - N. J. A. Sloane, Jan 17 2013

			a(6)=3 because the trajectory for 1 (sequence A064491) starts
1->2->4->7->9->12->18->24->32->38->42...
and the trajectory for 6 starts
6->10->14->18->24->32->38->42->50->56...
so the sequence beginning with 6 joins A064491 after 3 steps.

Claudia Spiro, Problem proposed at West Coast Number Theory Meeting, 1977. - From N. J. A. Sloane, Jan 11 2013

T. D. Noe, Table of n, a(n) for n = 1..11000
T. D. Noe, Logarithmic plot of 10^6 terms

Cf. A000005, A036434, A064491, A096335.

s = 1; t = Join[{s}, Table[s = s + DivisorSigma[0, s], {n, 2, 1000}]]; mx = Max[t]; Table[r = n; gen = 0; While[r < mx && ! MemberQ[t, r], gen++; r = r + DivisorSigma[0, r]]; If[r >= mx, gen = -1]; gen, {n, 100}] (* T. D. Noe, Jan 13 2013 *)

Escape clause added to definition by N. J. A. Sloane, Nov 09 2020

A348093 Numbers k >= 1 such that there is no pair (x,y) such that x - d(x) = k or y + d(y) = k, where d = A000005 = number of divisors.

Original entry on oeis.org

8, 20, 36, 40, 67, 68, 79, 88, 100, 116, 117, 131, 132, 134, 140, 156, 164, 167, 180, 185, 196, 204, 228, 244, 252, 268, 276, 284, 300, 308, 312, 321, 324, 341, 348, 370, 372, 379, 388, 401, 405, 408, 420, 425, 436, 439, 453, 460, 476, 479

Offset: 1

Ctibor O. Zizka, Sep 29 2021

nonn

Numbers k >= 1 such that A060990(k) + A036431(k) = 0.

			k = 8 is a term: there are no x,y such that x - d(x) = 8, y + d(y) = 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A036431, A049820, A060990, A062249.

Intersection of A036434 and A045765.

With[{max = 480}, Complement[Range[max], Select[Union[Flatten[Table[n + DivisorSigma[0, n]*{-1, 1}, {n, 1, max + 2 + 2*Ceiling[Sqrt[2*max+4]]}]]], # <= max &]]] (* Amiram Eldar, Mar 04 2023 *)

okp(k) = sum(i=1, k, i+numdiv(i) == k) == 0;
okm(k) = sum(i=1, 2*k+2, i-numdiv(i) == k) == 0;
isok(k) = okp(k) && okm(k); \\ Michel Marcus, Oct 01 2021

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A036438 Integers which can be written as m*tau(m) for some m, where tau = A000005.

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A096335 Number of iterations of n -> n + tau(n) needed for the trajectory of n to join the trajectory of A064491, or -1 if the two trajectories never merge.

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A348093 Numbers k >= 1 such that there is no pair (x,y) such that x - d(x) = k or y + d(y) = k, where d = A000005 = number of divisors.

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