A033950 -id:A033950 - cp's OEIS frontend

A036762 The integer values of x/d(x) in order of magnitude of x in A033950.

1, 1, 2, 3, 2, 3, 3, 4, 5, 7, 5, 6, 8, 7, 11, 8, 13, 9, 16, 11, 17, 19, 13, 10, 23, 17, 25, 19, 29, 12, 31, 14, 23, 16, 37, 41, 43, 29, 15, 31, 47, 24, 22, 53, 49, 37, 32, 25, 26, 59, 20, 61, 41, 21, 43, 67, 28, 47, 71, 73, 25, 34, 125, 79, 53, 40, 83, 28, 38, 59

Offset: 1

Labos Elemer

			If n=63, then x=625 and d(x) = 5 divides x. The quotient is 125 = a(63).

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A000005, A033950, A036761, A036763, A039819.

with(numtheory): A033950 := proc(n) option remember: local k: if(n=1)then return 1: else k:=procname(n-1)+1: do if(type(k/tau(k),integer))then return k: fi: k:=k+1: od: fi: end: A036762 := proc(n) return A033950(n)/tau(A033950(n)): end: seq(A036762(n),n=1..70); # Nathaniel Johnston, May 04 2011

Select[Table[n/DivisorSigma[0, n], {n, 708}], IntegerQ] (* Michael De Vlieger, Jul 04 2016 *)

A336040 Characteristic function of refactorable numbers (A033950).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0

Offset: 1

Wesley Ivan Hurt, Jul 07 2020

			a(1) = 1 since d(1) = 1 and 1 divides 1.
a(2) = 1 since d(2) = 2 and 2 divides 2.
a(3) = 0 since d(3) = 2, but 2 does not divide 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Eric Weisstein's World of Mathematics, Refactorable Number
Index entries for characteristic functions

Cf. A000005, A033950, A054008, A336041 (inverse Möbius transform), A335182, A335665, A349322.

a[n_] := Boole @ Divisible[n, DivisorSigma[0, n]]; Array[a, 100] (* Amiram Eldar, Jul 08 2020 *)

a(n) = n%numdiv(n) == 0; \\ Michel Marcus, Jul 07 2020

a(n) = 1 - ceiling(n/d(n)) + floor(n/d(n)), where d(n) is the number of divisors of n (A000005).
a(n) = [A054008(n) == 0], where [ ] is the Iverson bracket. - Antti Karttunen, Nov 24 2021
a(p) = 0 for odd primes p. - Wesley Ivan Hurt, Nov 28 2021

A039819 Number of divisors of n-th refactorable number (A033950(n)).

Original entry on oeis.org

1, 2, 4, 3, 6, 6, 8, 9, 8, 8, 12, 12, 10, 12, 8, 12, 8, 12, 8, 12, 8, 8, 12, 18, 8, 12, 9, 12, 8, 20, 8, 18, 12, 18, 8, 8, 8, 12, 24, 12, 8, 16, 18, 8, 9, 12, 14, 18, 18, 8, 24, 8, 12, 24, 12, 8, 20, 12, 8, 8, 24, 18, 5, 8, 12, 16, 8, 24, 18, 12, 8, 30, 12, 8, 24, 12, 8, 8, 18, 12, 8, 24, 8

Offset: 1

N. J. A. Sloane

nonn

A number n is refactorable if the number of divisors of n divides n.

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

Cf. A033950, A000005, A036762, A281188.

v:=[ n: n in [1..900] | n mod NumberOfDivisors(n) eq 0 ]; [NumberOfDivisors(v[i]): i in [1..#v]]; // Marius A. Burtea, Jul 02 2019

fQ[n_] := Mod[n, DivisorSigma[0, n]] == 0; DivisorSigma[0, # ] & /@ Select[ Range[1000], fQ[ # ] &] (* Robert G. Wilson v *)

A033950(n)/a(n) = A036762(n).

a(n) = A000005(A033950(n)). - Omar E. Pol, Jan 17 2017

More terms from Robert G. Wilson v, Oct 29 2005

Minor edits by Franklin T. Adams-Watters, Jan 17 2017

A047728 Intersection of A046985 and A033950: multiply perfect, refactorable numbers with integer average divisor dividing the number.

Original entry on oeis.org

1, 672, 30240, 23569920, 45532800, 14182439040, 153003540480, 403031236608, 13661860101120, 154345556085770649600, 143573364313605309726720, 352338107624535891640320, 680489641226538823680000, 34384125938411324962897920, 156036748944739017459105792, 3638193973609385308194865152

Offset: 1

Labos Elemer

nonn

Colton proves that perfect numbers (A000396) cannot be refactorable.

			k = 45532800 is a term since s0 = d(k) = 384, s1 = sigma(k) = 571963392, and the four quotients s1/s0 = 474300, (k * s0)/s1 = 96, s1/k = 4 and k/s0 = 118580 are all integers.

Amiram Eldar, Table of n, a(n) for n = 1..305
Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.

Cf. A000005, A000203, A000396, A046985, A001599, A007691.

q[n_] := Module[{d = DivisorSigma[0, n], s = DivisorSigma[1, n]}, Divisible[s, n] && Divisible[n * d, s] && Divisible[s, d] && Divisible[n, d]]; Select[Range[31000], q] (* Amiram Eldar, May 09 2024 *)

is(k) = {my(f = factor(k), s = sigma(f), d = numdiv(f)); !(s % k) && !((k * d) % s) && !(s % d) && !(k % d);} \\ Amiram Eldar, May 09 2024

Let s1 = sigma(k) = A000203(k) be the sum of divisors of k and s0 = d(k) = A000005(k) be the number of divisors of k. Then, k is a term if s1/s0, (k * s0)/s1, s1/k, and k/s0 are all integers.

a(7)-a(13) from Donovan Johnson, Apr 09 2010

Edited and a(14)-a(16) added by Amiram Eldar, May 09 2024

A172398 Number of partitions of n into the sum of two refactorable numbers (A033950).

Original entry on oeis.org

0, 1, 1, 1, 0, 0, 0, 0, 1, 2, 1, 0, 1, 1, 0, 1, 1, 1, 1, 2, 1, 0, 0, 1, 1, 2, 1, 0, 0, 1, 0, 1, 1, 0, 0, 2, 1, 1, 0, 0, 1, 2, 0, 1, 1, 0, 0, 3, 1, 0, 0, 1, 0, 1, 0, 0, 1, 2, 0, 1, 1, 1, 0, 2, 1, 0, 0

Offset: 1

Juri-Stepan Gerasimov, Nov 20 2010

nonn

			a(10)=2 because 10 = 1(refactorable) + 9(refactorable) = 2(refactorable) + 8(refactorable).

R. J. Mathar, Table of n, a(n) for n = 1..1000

Cf. A033950.

with(numtheory);
a:=n-> sum( ((1 + floor(i/tau(i)) - ceil(i/tau(i))) * (1 + floor((n-i)/tau(n-i)) - ceil((n-i)/tau(n-i))) ), i=1..floor(n/2));
# alternative
isA033950 := proc(n)
    if modp(n,numtheory[tau](n)) = 0 then
        true;
    else
        false;
    end if;
end proc:
A172398 := proc(n)
    local a;
    a := 0 ;
    for i from 1 to n/2 do
        if isA033950(i) and isA033950(n-i) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Jul 21 2015

a[n_] := IntegerPartitions[n, {2}, Select[Range[n], Divisible[#, DivisorSigma[0, #]]&]] // Length;
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 04 2023 *)

a(n) = Sum_{i=1..floor(n/2)} ((1+floor(i/d(i)) - ceiling(i/d(i))) * (1 + floor((n-i)/d(n-i)) - ceiling((n-i)/d(n-i)))). - Wesley Ivan Hurt, Jan 12 2013

Corrected by D. S. McNeil, Nov 20 2010

A036761 Number of refactorable integers (A033950) of binary order (A029837) n.

Original entry on oeis.org

1, 1, 0, 1, 2, 2, 4, 8, 13, 22, 39, 77, 137, 254, 459, 889, 1665, 3175, 6041, 11619, 22319, 42979, 83123, 160649, 311826, 605225, 1176998, 2291702, 4466923, 8716126, 17023771, 33279942, 65109458, 127484313, 249783733, 489738130, 960801221, 1886039740

Offset: 0

Labos Elemer

nonn

Since for any epsilon d(n) <= n^epsilon if n is large enough, a(n) does not grow very quickly.

			{1} has binary order 0, {2} has binary order 1, no term has binary order 2, {8} has binary order 3, {9,12} have binary order 4, {18,24} have binary order 5, ...
The 8 numbers, between 65 and 128 (with binary order 7) which are divided by d(x) (A000005) are 72,80,84,88,96,104,108,128, so a(7)=8.

Cf. A000005, A029837, A033950.

with(numtheory): A036761 := proc(n) local ct,k,lim: if(n=0)then return 1: else ct:=0: lim:=2^n: for k from 2^(n-1)+1 to lim do if(k mod tau(k) = 0)then ct:=ct+1: fi: od: return ct: fi: end: seq(A036761(n),n=0..10); # Nathaniel Johnston, May 04 2011

Table[Count[Range[2^(n - 1) + 1, 2^(n)], k_ /; Divisible[k, DivisorSigma[0, k]]] + Boole[n == 0], {n, 0, 22}] (* Michael De Vlieger, May 20 2017 *)

a(22)-a(37) from Donovan Johnson, Aug 29 2012

A047727 Average divisor is an integer (A003601) and the number is refactorable (A033950).

Original entry on oeis.org

1, 56, 60, 96, 132, 184, 204, 248, 276, 348, 376, 480, 492, 504, 564, 568, 612, 632, 636, 672, 708, 824, 852, 864, 996, 1016, 1056, 1068, 1208, 1212, 1248, 1284, 1336, 1356, 1520, 1528, 1572, 1592, 1632, 1644, 1656, 1784, 1788, 1824, 1908, 1912, 1980

Offset: 1

Labos Elemer

nonn

			x = 56, sigma(x) = 120, number of divisors of x = 8. 120/8 and 56/8 are integers.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A007691, A001599, A046985.

adiQ[n_]:=Module[{ds1=DivisorSigma[1,n],ds0=DivisorSigma[0,n]} ,Divisible[ ds1,ds0]&&Divisible[n,ds0]]; Select[Range[2000],adiQ] (* Harvey P. Dale, Apr 27 2012 *)

isok(n) = my(d = numdiv(n)); !(n % d) && !(sigma(n) % d); \\ Michel Marcus, Oct 15 2016

Both sigma_1(x)/sigma_0(x) and x/sigma_0(x) are integers. - clarified by Harvey P. Dale, Apr 27 2012

A359964 Refactorable numbers (A033950) having more divisors than all smaller refactorable numbers.

Original entry on oeis.org

1, 2, 8, 12, 24, 36, 60, 180, 240, 360, 720, 1260, 1680, 3360, 5040, 10080, 15120, 20160, 25200, 30240, 55440, 100800, 110880, 221760, 277200, 443520, 665280, 720720, 1108800, 1441440, 2494800, 2882880, 3603600, 5765760, 8648640, 12972960, 14414400, 25945920, 28828800

Offset: 1

Amiram Eldar, Jan 20 2023

nonn

The corresponding numbers of divisors are 1, 2, 4, 6, 8, 9, 12, 18, 20, 24, ... .

This sequence if infinite since there are refactorable numbers with arbitrarily large number of divisors. E.g., for any prime p, p^(p-1) is a refactorable number with p divisors.

Amiram Eldar, Table of n, a(n) for n = 1..60

Subsequence of A033950.
Cf. A000005, A073904, A110821.
Similar sequences: A002182, A335317, A356078, A359963.

seq[nmax_] := Module[{s = {}, dm = 0, d}, Do[d = DivisorSigma[0, n]; If[d > dm && Divisible[n, d], dm = d; AppendTo[s, n]], {n, 1, nmax}]; s]; seq[10^6]

lista(nmax) = {my(dm = 0, d); for(n = 1, nmax, d = numdiv(n); if(d > dm && n%d == 0, dm = d; print1(n, ", "))); }

A235176 Numbers k such that the sum of the first k refactorable numbers (A033950) is also a refactorable number.

Original entry on oeis.org

1, 47, 125, 131, 185, 187, 189, 191, 198, 201, 204, 206, 256, 257, 262, 264, 268, 276, 283, 285, 294, 800, 809, 812, 818, 822, 824, 827, 829, 840, 844, 848, 1076, 1080, 1118, 1119, 1133, 1135, 1151, 1153, 1164, 1171, 1175, 1183, 1186, 1189, 1195, 1208, 1210

Offset: 1

Ivan N. Ianakiev, Jan 04 2014

nonn

			The sum of the first 47 terms of A033950 equals 9120, which is A033950(616).

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A033950, A235177.

a(8)-a(49) from Giovanni Resta, Jan 04 2014

A341781 Refactorable numbers (or tau numbers, A033950) k such that k/tau(k) is even, where tau(k) = A000005(k).

Original entry on oeis.org

8, 12, 36, 72, 80, 96, 128, 180, 240, 252, 288, 384, 396, 448, 468, 480, 560, 612, 640, 672, 684, 720, 828, 864, 880, 896, 972, 1040, 1044, 1056, 1116, 1152, 1200, 1248, 1332, 1344, 1360, 1408, 1440, 1476, 1520, 1548, 1620, 1632, 1664, 1680, 1692, 1800, 1824

Offset: 1

Amiram Eldar, Feb 19 2021

nonn

Zelinsky (2002) called these numbers p-generators. He proved that these are the tau numbers k such that for any prime p, if p does not divide k then p*k is also a tau number. He used these numbers to prove that the number of tau numbers not exceeding m is > pi(m)/2 for all m > 7.42*10^13, where pi(m) = A000720(m).

			8 is a term since 8/tau(8) = 8/4 = 2 is even.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.

Cf. A000005, A000720, A033950, A036762, A039819, A281188.

q[n_] := Divisible[n, (d = DivisorSigma[0, n])] && EvenQ[n/d]; Select[Range[2000], q]

isok(k) = my(q=k/numdiv(k)); (denominator(q)==1) && ((q%2) == 0); \\ Michel Marcus, Feb 20 2021

cp's OEIS Frontend

A036762 The integer values of x/d(x) in order of magnitude of x in A033950.

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A336040 Characteristic function of refactorable numbers (A033950).

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A039819 Number of divisors of n-th refactorable number (A033950(n)).

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A047728 Intersection of A046985 and A033950: multiply perfect, refactorable numbers with integer average divisor dividing the number.

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A172398 Number of partitions of n into the sum of two refactorable numbers (A033950).

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A036761 Number of refactorable integers (A033950) of binary order (A029837) n.

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Maple

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A047727 Average divisor is an integer (A003601) and the number is refactorable (A033950).

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