A033950 -id:A033950 - cp's OEIS frontend

A342969 Numbers m such that both m^2-1 and m^2 are refactorable numbers (A033950).

3, 39, 225, 249, 321, 447, 471, 519, 681, 831, 921, 993, 1119, 1191, 1473, 1641, 1671, 1857, 1929, 1983, 2361, 2391, 2463, 2625, 2631, 2913, 3321, 3369, 3561, 3591, 3777, 3807, 3831, 3903, 4119, 4281, 4287, 4359, 4545, 4569, 4791, 5001, 5025, 5079, 5241, 5481

Offset: 1

Jianing Song, Apr 01 2021

Numbers m such that m^2-1 is divisible by d(m^2-1) and m^2 is divisible by d(m^2), d = A000005.
Zelinsky (2002, Theorem 59, p. 15) proved that if k > 1, k and k+1 are both refactorable numbers, then k is even. Such k must be of the form m^2-1 for some odd m.
The smallest term not divisible by 3 is a(66) = 9025.
For the first terms we have d(a(n)^2-1) > d(a(n)^2). But this is not always the case. The smallest counterexample is a(30) = 3591, where d(3591^2-1) = 40 and d(3591^2) = 63. The terms m such that d(m^2-1) < d(m^2) are listed in A342970. [Note that d(m^2-1) = d(m^2) is impossible since d(m^2-1) is even and d(m^2) is odd. - Jianing Song, Nov 21 2021]

			39 is a term since 39^2-1 = 1520 is divisible by d(1520) = 20 and 39^2 = 1521 is divisible by d(1521) = 9.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..3110 from Jianing Song)
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, A033950, A036896, A036898, A114617, A342970.

refQ[n_] := Divisible[n, DivisorSigma[0, n]]; Select[Range[6000], And @@ refQ /@ (#^2 - {1, 0}) &] (* Amiram Eldar, Feb 03 2025 *)

isrefac(n) = ! (n % numdiv(n));
isA342969(n) = (n>1) && isrefac(n^2-1) && isrefac(n^2)

A036898(2*n+1) = A114617(n+1) = a(n)^2 - 1; A036898(2*n+2) = A114617(n+1) + 1 = a(n)^2.

A342970 Numbers m such that both m^2-1 and m^2 are refactorable numbers (A033950) and that m^2 has more divisors than m^2-1.

Original entry on oeis.org

3591, 4545, 5481, 6975, 8415, 9639, 11319, 11583, 11745, 12225, 12735, 16065, 18711, 24255, 24759, 30015, 31671, 39105, 40257, 41535, 41769, 44631, 44865, 52065, 52569, 53055, 54975, 56511, 60255, 60705, 64071, 64575, 69825, 72009, 73665, 76095, 81081, 81855, 87129

Offset: 1

Jianing Song, Apr 01 2021

nonn

Numbers m such that m^2-1 is divisible by d(m^2-1), m^2 is divisible by d(m^2) and d(m^2) > d(m^2-1), d = A000005.

The smallest term not divisible by 3 is a(1048) = 2907025.

			5481 is a term since 5481^2-1 is divisible by d(5481^2-1) = 40, 5481^2 is divisible by d(5481^2) = 63, and 63 > 40.
2907025 is a term since 2907025^2-1 is divisible by d(2907025^2-1) = 96, 2907025^2 is divisible by d(2907025^2) = 125, and 125 > 96.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1500 from Jianing Song)

Cf. A000005, A033950, A342969.

q[n_] := Module[{d = DivisorSigma[0, n^2 - {1, 0}]}, d[[1]] < d[[2]] && Divisible[n^2-1, d[[1]]] && Divisible[n^2, d[[2]]]]; Select[Range[10^5], q] (* Amiram Eldar, Feb 03 2025 *)

isA342970(n) = if(n>1, my(d1 = numdiv(n^2-1), d2 = numdiv(n^2)); !((n^2-1) % d1) && !(n^2 % d2) && d2 > d1, 0)

A382065 Exponentially refactorable numbers: numbers whose exponents in their canonical prime factorization are all refactorable numbers (A033950).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Amiram Eldar, Mar 14 2025

First differs from A377019 at n = 55: A377019(55) = 64 is not a term of this sequence.
First differs from A344742 at n = 62: A344742(62) = 72 is not a term of this sequence.
All the cubefree numbers (A004709) are terms. The least term that is not cubefree is a(215) = 256 = 2^8.
Subsequence of A382063 and first differs from it at n = 362: A382063(362) = 432 = 2^4 * 3^3 is not a term of this sequence.
The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^3 + (1 - 1/p) * (Sum_{k>=3} 1/p^A033950(k))) = 0.83493143539605138255... .
The relative density of this sequence within A382063 is the ratio between the densities of the two sequences: 0.997553... .

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

Cf. A033950, A344742.
Subsequence of A382063.
Subsequence: A004709.
Similar sequences: A197680, A209061, A138302, A268335, A361177, A377019.

refQ[k_] := Divisible[k, DivisorSigma[0, k]]; q[k_] := AllTrue[FactorInteger[k][[;; , 2]], refQ]; Select[Range[100], q]

isref(n) = !(n % numdiv(n));
isok(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!isref(e[i]), return(0))); 1; }

A046525 Numbers common to A033950 and A034884.

Original entry on oeis.org

2, 8, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 96, 108, 132, 180, 240, 252, 288, 360, 480, 504, 720, 1260

Offset: 1

Labos Elemer

Cf. A035033, A036763, A046526.

Select[Range[1265], Mod[#, x = DivisorSigma[0, #]] == 0 && # < x^2 &] (* Jayanta Basu, Jun 27 2013 *)

A046526 Numbers common to A033950 and A035033.

Original entry on oeis.org

1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 96, 108, 132, 180, 240, 252, 288, 360, 480, 504, 720, 1260

Offset: 1

Labos Elemer

Cf. A035033, A036763, A046525.

A071227 Number of solutions 1<=x<=m to gcd(m,x) = tau(m) where m = A033950(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 4, 6, 4, 2, 4, 6, 10, 4, 12, 6, 8, 10, 16, 18, 12, 4, 22, 16, 20, 18, 28, 4, 30, 6, 22, 8, 36, 40, 42, 28, 8, 30, 46, 8, 10, 52, 42, 36, 16, 20, 12, 58, 8, 60, 40, 12, 42, 66, 12, 46, 70, 72, 20, 16, 100, 78, 52, 16, 82, 12, 18, 58, 88, 8, 60, 96, 20, 66, 100

Offset: 1

Benoit Cloitre, Jun 10 2002

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

Cf. A000005 (tau), A033950.

s[n_] := Module[{d = DivisorSigma[0, n]}, If[Divisible[n, d], Count[Range[n], ?(GCD[n, #] == d &)], Nothing]]; Array[s, 1000] (* _Amiram Eldar, Apr 29 2025 *)

for(n=1,1000,if(sum(i=1,n,if(gcd(n,i)-numdiv(n),0,1))>0,print1(sum(i=1,n,if(gcd(n,i)-numdiv(n),0,1)),",")))

Name corrected by Sean A. Irvine, Jul 05 2024

A323326 a(n) = 2*T(n) - pi(n), where T(n) (A208251) is the number of refactorable/tau numbers (A033950) <= n and pi(n) (A000720) is the number of primes <= n.

Original entry on oeis.org

2, 3, 2, 2, 1, 1, 0, 2, 4, 4, 3, 5, 4, 4, 4, 4, 3, 5, 4, 4, 4, 4, 3, 5, 5, 5, 5, 5, 4, 4, 3, 3, 3, 3, 3, 5, 4, 4, 4, 6, 5, 5, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 2, 2, 2, 4, 4, 4, 3, 5, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 2, 4, 3, 3, 3, 3, 3, 3, 2, 4, 4, 4, 3, 5, 5, 5, 5, 7, 6, 6, 6, 6, 6, 6, 6, 8, 7, 7, 7, 7, 6, 6, 5, 7, 7, 7, 6, 8, 7, 7, 7, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 1

Jud McCranie, Jan 11 2019

nonn

Colton conjectured that T(n) >= pi(n)/2 for all n, i.e., this sequence is nonnegative. Zelinsky proved it for n > 7.42*10^13 (see the Zelinsky reference). This calculation went to 7.44*10^13, proving the conjecture.

			For n=6, pi(6)=3, T(6)=2, so a(6) = 2*2 - 3 = 1.

Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999.
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. A033950, A208251, A000720.

A331666 Refactorable numbers (A033950) that are simultaneously arithmetic (A003601) and harmonic (A001599).

Original entry on oeis.org

1, 672, 30240, 23569920, 45532800, 164989440, 447828480, 623397600, 1381161600, 1862023680, 2144862720, 3134799360, 3831421440, 13584130560, 14182439040, 16569653760, 21943595520, 22933532160, 34482792960, 35032757760, 40752391680, 53621568000, 56481384960

Offset: 1

Jaroslav Krizek, Jan 23 2020

nonn

Numbers m such that all values of sigma(m)/tau(m), m/tau(m) and m * tau(m)/sigma(m) are any integers (f, g, and h respectively).
Corresponding values of numbers f, g and h: (1, 84, 1260, 294624, 474300, 1178496, 2946240, 3298400, 5754840, 11784960, ...); (1, 28, 315, 73656, 118575, 257796, 699732, 721525, 1198925, 2909412, 1675674, ...); (1, 8, 24, 80, 96, 140, 152, 189, 240, 158, 260, 266, 220, 380, 384, 296, 392, ...).
Multiply-perfect numbers from this sequence are in A047728.

			For m = 672, f = sigma(m)/tau(m) = 2016/24 = 84; g = m/tau(m) = 672/24 = 28; h = m * tau(m)/sigma(m) = 672*24/2016 = 8.

Amiram Eldar, Table of n, a(n) for n = 1..118 (terms below 10^14)

Intersection of A033950 and A007340.

Cf. A000005, A000203, A001599, A003601, A047728.

[m: m in [1..10^6] | IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and  IsIntegral(m / NumberOfDivisors(m)) and IsIntegral(m * NumberOfDivisors(m) / SumOfDivisors(m))]

Select[Range[3*10^7], Divisible[#, (d = DivisorSigma[0, #])] && Divisible[(s = DivisorSigma[1, #]), d] && Divisible[#*d, s] &] (* Amiram Eldar, Jan 24 2020 *)

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

A374540 a(1) = 0; for n >= 2, a(n) is the number of iterations needed for the map x -> x/A000005(x) to reach a least integer, when starting from x = A033950(n).

Original entry on oeis.org

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

Offset: 1

Ctibor O. Zizka, Jul 11 2024

nonn

The refactorability "depth" for refactorable numbers. Numbers from A159973 have the refactorability "depth" 0. Records reached for A033950(A360806(n)), i.e. the growth of the sequence is very slow.

			n = 2: A033950(2) = 2, 2/A000005(2) = 1, thus a(2) = 1.
n = 3: A033950(3) = 8, 8/A000005(8) = 2 --> 2/A000005(2) = 1, thus a(3) = 2.
n = 13: A033950(13) = 80, 80/A000005(80) = 8 --> 8/A000005(8) = 2 --> 2/A000005(2) = 1, thus a(13) = 3.

Cf. A000005, A033950, A036762, A159973, A360806.

f[n_] := Module[{v = NestWhileList[# / DivisorSigma[0, #] &, n, IntegerQ[#] && # > 1 &], len}, len = Length[v]; If[IntegerQ[v[[2]]], If[v[[-1]] == 1, len - 1, len - 2], Nothing]]; f[1] = 0; Array[f, 1200] (* Amiram Eldar, Jul 11 2024 *)

A062290 Erroneous version of A033950.

Original entry on oeis.org

1, 2, 8, 12, 36, 40, 60, 80, 96, 128, 180, 225, 288, 448, 450, 480, 600, 625, 640, 1200

Offset: 1

dead

cp's OEIS Frontend

A342969 Numbers m such that both m^2-1 and m^2 are refactorable numbers (A033950).

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A342970 Numbers m such that both m^2-1 and m^2 are refactorable numbers (A033950) and that m^2 has more divisors than m^2-1.

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A382065 Exponentially refactorable numbers: numbers whose exponents in their canonical prime factorization are all refactorable numbers (A033950).

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A046525 Numbers common to A033950 and A034884.

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A046526 Numbers common to A033950 and A035033.

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A071227 Number of solutions 1<=x<=m to gcd(m,x) = tau(m) where m = A033950(n).

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A323326 a(n) = 2*T(n) - pi(n), where T(n) (A208251) is the number of refactorable/tau numbers (A033950) <= n and pi(n) (A000720) is the number of primes <= n.

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A331666 Refactorable numbers (A033950) that are simultaneously arithmetic (A003601) and harmonic (A001599).

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A374540 a(1) = 0; for n >= 2, a(n) is the number of iterations needed for the map x -> x/A000005(x) to reach a least integer, when starting from x = A033950(n).

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