A036438 -id:A036438 - 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

A338382 Numbers m such that the equation m = k*tau(k) has more than one solution, where tau(k) is the number of divisors of k.

Original entry on oeis.org

108, 192, 448, 1080, 1512, 1920, 2376, 2688, 2808, 3672, 4104, 4224, 4480, 4968, 4992, 6000, 6264, 6528, 6696, 7296, 7992, 8100, 8640, 8832, 8856, 9288, 9856, 10152, 11136, 11448, 11648, 11904, 12096, 12744, 12960, 13176, 14208, 14400, 14472, 15120, 15232, 15336

Offset: 1

Bernard Schott, Oct 23 2020

nonn

The map k -> k*tau(k) = m is not injective (A038040), this sequence lists in increasing order the integers m that have several preimages.

There are primitive terms that generate an infinity of terms because of the multiplicativity of tau(k); for example, a(1) = 108 and with t such that gcd(t,6) = 1, every m = 108*(t*tau(t)) is another term; in particular, with p prime > 3, every m = 216*p is another term: 1080, 1512, 2376, ...

			a(1) = 108 because 18 * tau(18) = 27 * tau(27) = 108.
a(2) = 192 because 24 * tau(24) = 32 * tau(32) = 192.
a(3) = 448 because 56 * tau(56) = 64 * tau(64) = 448.
a(8) = 2688 is the smallest term with 3 preimages because 168 * tau(168) = 192 * tau(192) = 224 * tau(224) = 2688.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B12, p. 102-103.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, entry 168, page 127.

Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.

Cf. A000005, A038040, A327166, A338381, A338383, A338384, A338385.
Cf. A337873 (similar for k*sigma(k)).
Subsequence of A036438.

solNum[n_] := DivisorSum[n, 1 &, # * DivisorSigma[0, #] == n &]; Select[Range[16000], solNum[#] > 1 &] (* Amiram Eldar, Oct 23 2020 *)

isok(m) = {my(nb=0); fordiv(m, d, if (d*numdiv(d) == m, nb++; if (nb>1, return(1))); ); return (0); } \\ Michel Marcus, Oct 24 2020

More terms from Amiram Eldar, Oct 23 2020

A383276 Numbers of the form A034444(k) * k.

Original entry on oeis.org

1, 4, 6, 8, 10, 14, 16, 18, 22, 24, 26, 32, 34, 38, 40, 46, 48, 50, 54, 56, 58, 60, 62, 64, 72, 74, 80, 82, 84, 86, 88, 94, 96, 98, 104, 106, 112, 118, 122, 128, 132, 134, 136, 140, 142, 144, 146, 152, 156, 158, 160, 162, 166, 176, 178, 180, 184, 192, 194, 200

Offset: 1

Amiram Eldar, Apr 21 2025

The sorted values of {abs(A298473(n))}.
Numbers m that have a divisor d such that A034444(d) * d = m.
All the terms above 1 are even since A034444(k) is even for k >= 2.
A number m is a term if and only if either A007814(m) = A005087(m) or A007814(m) > A005087(m) + 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
H. L. Abbott and M. V. Subbarao, On the Distribution of the Sequence {nd*(n)}, Canadian Mathematical Bulletin, Vol. 32, No. 1 (1989), pp. 105-108.

The unitary analog of A036438.
Subsequences: A100484, A138929 \ {2}, A151821.
Cf. A005087, A007814, A034444, A298473, A383277 (characteristic function), A383278 (number of terms not exceeding n), A383279.

q[k_] := AnyTrue[Divisors[k], 2^PrimeNu[#] * # == k &]; Select[Range[200], q]
(* second program: *)
q[k_] := Module[{e = IntegerExponent[k, 2], w}, w = PrimeNu[k/2^e]; e > w + 1 || e == w]; Select[Range[200], q]

isok(k) = fordiv(k, d, if((1 << omega(d)) * d == k, return(1))); 0;

isok(k) = {my(e = valuation(k, 2), w = omega(k >> e)); e > w + 1 || e == w;}

a(n) = A383279(n) * A034444(A383279(n)).

A364146 Numbers k such that k! belongs to A038040.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 10, 11, 12, 13, 14, 15, 16, 17, 21, 22, 25, 26, 28, 29, 31, 32, 35, 36, 37, 38, 39, 40, 41, 49, 50, 52, 53, 54, 55, 57, 58, 59, 64, 66, 67, 70, 71, 76, 77, 78, 79, 80, 81, 82, 83, 85, 90, 91, 92, 95, 96, 97, 98, 99, 101, 103, 106, 108, 115, 121, 122, 123, 124, 125, 126, 127

Offset: 1

Max Alekseyev, Jul 10 2023

nonn

Numbers k such that k! = m * number of divisors of m, for some integer m.

			For k = 10, we have k! = m * A000005(m) for m = 37800 or m = 43200. Hence, 10 is a term.

Max Alekseyev, Table of n, a(n) for n = 1..500

Cf. A000005, A000142, A036438, A038040.

A073068 Minimum value of abs(n-x*tau(x)) x>0.

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 2, 3, 4, 3, 2, 1, 0, 1, 0, 1, 0, 0, 1, 2, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 0, 0, 1, 2, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1

Offset: 1

Benoit Cloitre, Aug 17 2002

nonn

a(n)=vecmin(vector(n,k,abs(n-k*numdiv(k))))

a(A036438(n))=0 . It seems that sum(k=1, n, a(k)) is asymptotic to C*n with 1.5

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

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A338382 Numbers m such that the equation m = k*tau(k) has more than one solution, where tau(k) is the number of divisors of k.

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A383276 Numbers of the form A034444(k) * k.

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A364146 Numbers k such that k! belongs to A038040.

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A073068 Minimum value of abs(n-x*tau(x)) x>0.

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