A057265 -id:A057265 - cp's OEIS frontend

A036436 Numbers whose number of divisors is a square.

1, 6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 120, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 168, 177, 178, 183

Offset: 1

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

Invented by the HR (named after Hardy and Ramanujan) concept formation program.
Numbers in this sequence but not in A036455 are 1, 1260, 1440, 1800, 1980 etc. [From R. J. Mathar, Oct 20 2008]
tau(p^(n^2-1)) = n^2 so numbers of this form are in this sequence, and because tau is multiplicative: if a and b are in this sequence and (a,b)=1 then a*b is also in a(n). - Enrique Pérez Herrero, Jan 22 2013
What is the density of this sequence? It contains A030229 and thus has (lower) density at least 3/Pi^2 = 0.30396...; it does not contain any members of A030059 or A060687, and hence has (upper) density at most 1 - 3/Pi^2 - 6*A179119/Pi^2 = 0.49528.... - Charles R Greathouse IV, Jan 11 2025

			tau(6)=4, which is a square number, so 6 is in this sequence.

S. Colton, Automated Theorem Discovery: A Future Direction for Theorem Provers, 2002.

Amiram Eldar, Table of n, a(n) for n = 1..10000
S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
S. Colton, HR - Automatic Theory Formation in Pure Mathematics (Unfortunately [403 Forbidden])
S. Colton and A. Bundy, Automatic Concept Formation in Pure Mathematics
S. Colton, A. Bundy and T. Walsh, Agent Based Cooperative Theory Formation in Pure Mathematics
S. Colton, R. McCasland and A. Bundy, Automated Theory Formation for Tutoring Tasks in Pure Mathematics, 2002.

Contains A030229 as a subsequence.

Cf. A009087, A033950, A057265, A049439.

Select[Range[200],IntegerQ[Sqrt[DivisorSigma[0,#]]]&]  (* Harvey P. Dale, Apr 20 2011 *)

is(n)=issquare(numdiv(n)) \\ Charles R Greathouse IV, Jan 22 2013

Links corrected and edited by Daniel Forgues, Jun 30 2010

A120349 Refactorable numbers k such that the number of odd divisors r is odd, the number of even divisors s is even and both r and s are divisors of k.

Original entry on oeis.org

36, 3600, 8100, 10000, 22500, 26244, 32400, 90000, 142884, 202500, 396900, 518400, 656100, 810000, 980100, 1285956, 1368900, 1587600, 1679616, 2286144, 2340900, 2624400, 2924100

Offset: 1

Walter Kehowski, Jun 24 2006

nonn

In general, since n is even, r is always a multiple of s and even if both r and s are divisors of n, the sum t=r+s may not be. For example, if n=144, then r=3, s=12 and t=r+s=15.

			a(1)=36 since r=3(odd), s=6(even) and t=r+s=9 are all divisors.

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

Cf. A033950, A049439, A057265.

with(numtheory); T := proc(n::posint) local x, y, S; S:=divisors(n); x:=nops( select(z->type(z,odd),S) ); y:=nops( select(z->type(z,even),S) ); return [x,y] end; RF:=[]: N:=12^6/2: CNT:=12^4: for w to 1 do for k from 1 to N do n:=2*k; if k mod CNT = 0 then print((N-k)/CNT) fi; r:=T(n)[1]; s:=T(n)[2]; t:=r+s; if type(s,even) and type(r,odd) and andmap(z -> n mod z = 0, [r,s,t]) then RF:=[op(RF),n]; print(n,r,s,t); fi; od od; RF;

a(n) = n-th number such that n is even, r = number of odd divisors of n, s = number of even divisors of n, t = r+s = number of divisors of n, are all divisors of n and r is odd, s is even.

A120361 Even refactorable numbers k such that the number r of odd divisors of k and the number s of even divisors of k are both odd divisors of k.

Original entry on oeis.org

2, 18, 72, 450, 882, 1250, 2178, 3042, 4050, 5202, 6498, 9522, 11250, 13122, 15138, 16200, 17298, 24642, 30258, 33282, 39762, 45000, 50562, 52488, 56448, 62658, 64800, 66978, 71442, 80802, 90738, 95922, 101250, 112338, 124002, 142578, 169362, 180000, 183618, 190962, 198450, 206082

Offset: 1

Walter Kehowski, Jun 25 2006

nonn

Note that s is necessarily a multiple of r.

			a(2) = 18 since r = 3, s = 3, t = r+s = 6.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000005, A001227, A033950, A049439, A057265, A183063.

seq[kmax_] := Module[{v = {}, r, s, t}, Do[t = DivisorSigma[0, k]; r = t - DivisorSigma[0, k/2]; s = t - r; If[OddQ[r] && OddQ[s] && Divisible[k, t] && Divisible[k, r] && Divisible[k, s], AppendTo[v, k]], {k, 2, kmax, 2}]; v]; seq[10^6] (* Amiram Eldar, Aug 01 2024 *)

a(36)-a(42) from Amiram Eldar, Aug 01 2024

A120359 Even refactorable numbers k such that the number r of odd divisors is odd, the number s of even divisors is even, both r and s are divisors of k and k is the first number for which the triple (r,s,t) occurs, where t is the number of divisors of k.

Original entry on oeis.org

36, 3600, 8100, 10000, 26244, 32400, 142884, 202500, 396900, 518400, 656100, 810000, 1587600, 1679616, 2286144, 2624400, 3572100, 6350400, 9144576, 9922500, 12960000, 14288400, 20575296, 25401600, 28579716, 32148900, 39690000, 41990400, 48024900, 57153600, 89302500

Offset: 1

Walter Kehowski, Jun 25 2006

nonn

Note that s is necessarily a multiple of r.

			a(1) = 36 since r = 3, s = 6 and t = r+s = 9.

Cf. A000005, A001227, A033950, A049439, A057265, A120358, A120361, A183063.

seq[kmax_] := Module[{triples = {}, v = {}, r, s, t}, Do[t = DivisorSigma[0, k]; r = t - DivisorSigma[0, k/2]; s = t - r; If[OddQ[r] && EvenQ[s] && FreeQ[triples, {r, s, t}] && Divisible[k, t] && Divisible[k, r] && Divisible[k, s], AppendTo[v, k]; AppendTo[triples, {r, s, t}]], {k, 2, kmax, 2}]; v]; seq[10^6] (* Amiram Eldar, Aug 01 2024 *)

a(21)-a(22) inserted and a(24)-a(31) added by Amiram Eldar, Aug 01 2024

A208251 Number of refactorable numbers less than or equal to n.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 1

Wesley Ivan Hurt, Jan 12 2013

A number is refactorable if it is divisible by the number of its divisors.

			a(1) = 1 since 1 is the first refactorable number, a(2) = 2 since there are two refactorable numbers less than or equal to 2, a(3) through a(7) = 2 since the next refactorable number is 8.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Refactorable Number.

Cf. A033950, A000005, A141586, A057265, A036896, A036898, A114617.

Partial sums of A336040.

with(numtheory) a:=n->sum((1 + floor(i/tau(i)) - ceil(i/tau(i))), i=1..n);

Accumulate[Table[If[Divisible[n, DivisorSigma[0, n]], 1, 0], {n, 1,100}]] (* Amiram Eldar, Oct 11 2023 *)

a(n) = sum(i=1, n, q = i/numdiv(i); 1+ floor(q) - ceil(q)); \\ Michel Marcus, Sep 10 2018

a(n) = Sum_{i=1..n} 1 + floor(i/d(i)) - ceiling(i/d(i)), where d(n) is the number of divisors of n.

A120350 Refactorable numbers k such that the number of odd divisors and the number of even divisors of k are both divisors of k.

Original entry on oeis.org

2, 12, 18, 24, 36, 72, 80, 180, 240, 252, 360, 396, 450, 468, 480, 504, 560, 612, 684, 720, 792, 828, 880, 882, 896, 936, 972, 1040, 1044, 1116, 1200, 1224, 1250, 1332, 1344, 1360, 1368, 1440, 1476, 1520, 1548, 1620, 1656, 1692, 1840, 1908, 1944, 2000

Offset: 1

Walter Kehowski, Jun 24 2006

nonn

Since s = 0 if k is odd, the number k is necessarily even and then the number of even divisors s is always a multiple of the number of odd divisors r. Note that t = r + s may not be a divisor even if both r and s are divisors. For example, if k = 144, then r = 3, s = 12, but t = r + s = 15.

			a(3) = 18 since r = 3, s = 3 and t = r + s = 6 are all divisors.

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

Cf. A033950, A049439, A057265.

with(numtheory); A:=[]: for w to 1 do for k from 1 to 5000 do n:=2*k; S:=divisors(n); r:=nops( select(z->type(z,odd),S) ); s:=nops( select(z->type(z,even),S) ); t:=r+s; if andmap(z -> n mod z = 0, [r,s,t]) then A:=[op(A),n]; print(n,r,s,t); fi; od od; A;

oddtau[n_] := DivisorSigma[0, n/2^IntegerExponent[n, 2]]; seqQ[n_] := Module[{d = DivisorSigma[0, n], o = odd[n]}, Divisible[n, d] && Divisible[n, o] && Divisible[n, d - o]]; Select[Range[2, 2000, 2], seqQ] (* Amiram Eldar, Jan 15 2020 *)

a(n) = k is even, r = number of odd divisors of k, s = number of even divisors of k and t = r + s = number of divisors of k, are all divisors of k.

Offset corrected by Amiram Eldar, Jan 15 2020

A120351 Even numbers k such that the number of odd divisors r and the number of even divisors s are both divisors of k.

Original entry on oeis.org

2, 4, 6, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 34, 36, 38, 44, 46, 48, 52, 58, 62, 68, 72, 74, 76, 80, 82, 86, 90, 92, 94, 106, 112, 116, 118, 120, 122, 124, 126, 134, 142, 144, 146, 148, 150, 158, 160, 164, 166, 168, 172, 176, 178, 180, 188, 192, 194, 198, 202, 206

Offset: 1

Walter Kehowski, Jun 24 2006

nonn

Since s=0 if k is odd, the number k is necessarily even and then s is always a multiple of r. Note that t=r+s may not be a divisor even if both r and s are divisors. For example, if k=144, then r=3, s=12, but t=r+s=15.

			16 is a term since r=1 and s=4 are both divisors.

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

Cf. A033950, A049439, A057265.

with(numtheory); A:=[]: N:=10^4/2: for w to 1 do for k from 2 to N do n:=2*k; S:=divisors(n); r:=nops( select(z->type(z,odd),S) ); s:=nops( select(z->type(z,even),S) ); if andmap(z -> n mod z = 0,[r,s]) then A:=[op(A),n]; print(n,r,s); fi; od od; A;

aQ[n_] := Divisible[n, (ev = DivisorSigma[0, n/2])] && Divisible[n, DivisorSigma[0, n] - ev]; Select[Range[2, 206, 2], aQ] (* Amiram Eldar, Nov 02 2019 *)

a(n) = n is even, r = number of odd divisors of n, s = number of even divisors of n, are all divisors of n.

Term 2 inserted by Amiram Eldar, Nov 02 2019

A120358 Even refactorable numbers k such that the number r of odd divisors and the number s of even divisors are both odd divisors of k and k is the first number for which the triple (r,s,t) occurs, where t is the number of divisors of k.

Original entry on oeis.org

2, 18, 72, 450, 1250, 4050, 16200, 52488, 56448, 64800, 71442, 101250, 198450, 235298, 285768, 328050, 405000, 793800, 1036800, 1312200, 1620000, 1786050, 3175200, 4572288, 4961250, 5248800, 7144200, 12700800, 14289858, 15059072, 16074450, 19845000, 24012450, 25920000, 28576800

Offset: 1

Walter Kehowski, Jun 25 2006

nonn

Note that s is necessarily a multiple of r.

			a(2) = 18 since r = 3, s = 3 and t = r+s = 6.

Cf. A000005, A001227, A033950, A049439, A057265, A120359, A120361, A183063.

seq[kmax_] := Module[{triples = {}, v = {}, r, s, t}, Do[t = DivisorSigma[0, k]; r = t - DivisorSigma[0, k/2]; s = t - r; If[OddQ[r] && OddQ[s] && FreeQ[triples, {r, s, t}] && Divisible[k, t] && Divisible[k, r] && Divisible[k, s], AppendTo[v, k]; AppendTo[triples, {r, s, t}]], {k, 2, kmax, 2}]; v]; seq[10^6] (* Amiram Eldar, Aug 01 2024 *)

a(27)-a(35) from Amiram Eldar, Aug 01 2024

A120356 Even refactorable numbers n such that the number r of odd divisors and the number s of even divisors are both even divisors of n and n is the first number for which the triple (r,s,t) occurs, where t is the number of divisors of n.

Original entry on oeis.org

12, 24, 80, 180, 240, 360, 480, 720, 896, 1344, 1440, 1620, 2688, 3240, 3360, 4032, 5040, 6720, 6912, 8064, 10080, 13440, 20160, 20412, 24300, 25200, 30000, 30240, 34560, 40320, 40824, 48600, 56320, 56700, 60000, 60480, 62208, 67584, 69120

Offset: 1

Walter Kehowski, Jun 25 2006

nonn

Note that s is necessarily a multiple of r.

			a(1)=12 since r=2, s=4 and r+s=6.

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A001227, A033950, A049439, A057265, A183063.

triples = {}; seq = {}; Do[t = DivisorSigma[0, n]; r =  DivisorSigma[0, 2 n] - t; s = t - r; tri = {r, s, t}; If[AllTrue[tri, EvenQ[#] &&  Divisible[n, #] &] && !MemberQ[triples, tri], AppendTo[seq, n]; AppendTo[triples, tri]], {n, 2, 69120, 2}]; seq (* Amiram Eldar, Jun 13 2020 *)

A120360 Even refactorable numbers k such that the number of odd divisors of k and the number of even divisors of k are both even divisors of k.

Original entry on oeis.org

12, 24, 80, 180, 240, 252, 360, 396, 468, 480, 504, 560, 612, 684, 720, 792, 828, 880, 896, 936, 972, 1040, 1044, 1116, 1200, 1224, 1332, 1344, 1360, 1368, 1440, 1476, 1520, 1548, 1620, 1656, 1692, 1840, 1908, 1944, 2000, 2088, 2124, 2196, 2232, 2320

Offset: 1

Walter Kehowski, Jun 25 2006

nonn

Note that the number of even divisors s is necessarily a multiple of the number of odd divisors r.

			a(1) = 12 since r = 2, s = 4, t = r + s = 6.

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

Cf. A033950, A049439, A057265, A120350.

oddtau[n_] := DivisorSigma[0, n/2^IntegerExponent[n, 2]]; seqQ[n_] := Module[{d = DivisorSigma[0, n], o = odd[n]}, EvenQ[d] && EvenQ[o] && Divisible[n, d] && Divisible[n, o] && Divisible[n, d - o]]; Select[Range[2, 2320, 2], seqQ] (* Amiram Eldar, Jan 15 2020 *)

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A036436 Numbers whose number of divisors is a square.

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A120349 Refactorable numbers k such that the number of odd divisors r is odd, the number of even divisors s is even and both r and s are divisors of k.

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A120361 Even refactorable numbers k such that the number r of odd divisors of k and the number s of even divisors of k are both odd divisors of k.

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A120359 Even refactorable numbers k such that the number r of odd divisors is odd, the number s of even divisors is even, both r and s are divisors of k and k is the first number for which the triple (r,s,t) occurs, where t is the number of divisors of k.

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A208251 Number of refactorable numbers less than or equal to n.

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A120350 Refactorable numbers k such that the number of odd divisors and the number of even divisors of k are both divisors of k.

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Maple

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Extensions

A120351 Even numbers k such that the number of odd divisors r and the number of even divisors s are both divisors of k.

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