A036431 -id:A036431 - cp's OEIS frontend

A036434 Integers which cannot be written as k+tau(k) for some k.

1, 3, 6, 8, 11, 16, 17, 20, 22, 23, 27, 29, 35, 36, 40, 41, 44, 46, 47, 53, 54, 57, 60, 65, 67, 68, 70, 76, 77, 79, 80, 83, 87, 88, 92, 93, 94, 100, 101, 106, 107, 114, 116, 117, 121, 125, 128, 131, 132, 134, 135, 140, 142, 148, 155, 156, 157, 158, 161, 164, 166

Offset: 1

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

nonn

Invented by the HR concept formation program.

Indices of 0's in A036431. - Sean A. Irvine, Nov 01 2020

			None of 1,2,3,4,5,6,7 are such that k+tau(k)=8, so 8 is in the sequence.

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

Cf. A036431, A036438, A062249.

m = 200; Complement[Range[m], Table[n + DivisorSigma[0, n], {n, 1, m}]] (* Amiram Eldar, Nov 10 2021 *)

isok(n) = sum(k=1, n, k+numdiv(k) != n) == n; \\ Michel Marcus, Dec 08 2014

A036432 Sets a record for the number of positive integers which, when added to the number of their divisors, gives n.

Original entry on oeis.org

1, 2, 7, 38, 122, 2766, 64686, 1972296, 5387768, 56208248, 753815160, 130065181784, 5398921735160

Offset: 0

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

Invented by the HR concept formation program.

a(13) > 10^13. - Giovanni Resta, Feb 24 2020

			1 [ first with 0 ].
2=1+tau(1) [ first with 1 ].
7=4+tau(4) and 5+tau(5) [ first with 2 ].
38=30+tau(30) and 32+tau(32) and 34+tau(34) [ first with 3 ].

S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
S. Colton, HR - Automatic Theory Formation in Pure Mathematics

Cf. A036431.

nbi(n) = {nb = 0; for (i = 1, n, if ((i + numdiv(i)) == n, nb++);); nb;}
lista(nn) = {rec = -1; for (n = 1, nn, new = nbi(n); if (new > rec, print1(n, ", "); rec = new;););} \\ Michel Marcus, Aug 31 2013

Sets record for |{b in N : b + tau(b) = n}|.

More terms from Julian Richardson (julianr(AT)dai.ed.ac.uk), who has searched up to 100000000.
a(10)-a(11) from Donovan Johnson, Feb 19 2009
a(12) from Giovanni Resta, Feb 24 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

cp's OEIS Frontend

A036434 Integers which cannot be written as k+tau(k) for some k.

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A036432 Sets a record for the number of positive integers which, when added to the number of their divisors, gives n.

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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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Mathematica

PARI