A114617 -id:A114617 - cp's OEIS frontend

A036898 List of pairs of consecutive refactorable numbers.

1, 2, 8, 9, 1520, 1521, 50624, 50625, 62000, 62001, 103040, 103041, 199808, 199809, 221840, 221841, 269360, 269361, 463760, 463761, 690560, 690561, 848240, 848241, 986048, 986049, 1252160, 1252161, 1418480, 1418481, 2169728, 2169729, 2692880

Offset: 1

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

nonn

Zelinsky (2002, Theorem 59, p. 15) proved that if k > 1, k and k+1 are both refactorable numbers, then k is even. As a result, a(n) == n-1 (mod 2) for n >= 3. See also A114617. - Jianing Song, Apr 01 2021

			8 is refactorable because tau(8)=4 and 4 divides 8.
9 is refactorable because tau(9)=3 and 3 divides 9.

Donovan Johnson, Table of n, a(n) for n = 1..2000
S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
S. Colton, HR - Automatic Theory Formation in Pure Mathematics
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, A114617.

SequencePosition[Table[If[Divisible[n,DivisorSigma[0,n]],1,0],{n,27*10^5}],{1,1}]//Flatten (* Harvey P. Dale, Dec 07 2021 *)

isrefac(n) = ! (n % numdiv(n));
lista(nn) = {for (n = 1, nn, if (isrefac(n) && isrefac(n+1), print1(n, ", ", n+1, ", ")););} \\ Michel Marcus, Aug 31 2013

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.

A309347 Sums of two refactorable numbers whose difference is refactorable.

Original entry on oeis.org

3, 10, 17, 27, 36, 48, 54, 60, 84, 88, 96, 104, 108, 112, 120, 128, 132, 136, 144, 152, 156, 168, 176, 180, 184, 192, 200, 204, 208, 216, 224, 228, 232, 240, 248, 252, 256, 264, 272, 276, 280, 288, 296, 300, 304, 312, 324, 328, 336, 344, 348, 352, 360, 368

Offset: 1

Wesley Ivan Hurt, Jul 24 2019

nonn

Includes k+2 for k in A114617. - Robert Israel, Jul 29 2025

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

Cf. A033950, A114617.

isref:= proc(n) option remember; n mod numtheory:-tau(n) = 0 end proc:
filter:= proc(n)
  ormap(t -> isref(t) and isref(n-t) and isref(n-2*t), [$1 .. (n-1)/2])
end proc:
select(filter, [$1..1000]); # Robert Israel, Jul 29 2025

Flatten[Table[If[Sum[ (1 - Ceiling[(n - 2 i)/DivisorSigma[0, n - 2 i]] + Floor[(n - 2 i)/DivisorSigma[0, n - 2 i]]) (1 - Ceiling[i/DivisorSigma[0, i]] + Floor[i/DivisorSigma[0, i]]) (1 - Ceiling[(n - i)/DivisorSigma[0, n - i]] + Floor[(n - i)/DivisorSigma[0, n - i]]), {i, Floor[(n - 1)/2]}] > 0, n, {}], {n, 400}]]

A341779 Numbers k such that k and k+1 are both anti-tau numbers (A046642).

Original entry on oeis.org

3, 4, 15, 16, 64, 100, 195, 196, 255, 256, 483, 484, 676, 783, 784, 1023, 1024, 1155, 1156, 1295, 1296, 1443, 1444, 1599, 1600, 1936, 2116, 2703, 2704, 3363, 3364, 3844, 4096, 4623, 4624, 4899, 4900, 5183, 5184, 5476, 5776, 6399, 6400, 6723, 6724, 7395, 7396

Offset: 1

Amiram Eldar, Feb 19 2021

nonn

Since the even anti-tau numbers (A268066) are square numbers, all the terms are either of the form 4*k^2 - 1 or 4*k^2.

			3 is a term since 3 and 4 are both anti-tau numbers: gcd(3, tau(3)) = gcd(3, 2) = 1 and gcd(4, tau(4)) = gcd(4, 3) = 1.

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

Subsequence of A046642 and A081350.

Cf. A000010, A009191, A114617, A268066, A341780.

antiTauQ[n_] := CoprimeQ[n, DivisorSigma[0, n]]; s = {}; Do[k = 4*n^2; If[antiTauQ[k], If[antiTauQ[k - 1], AppendTo[s, k - 1]]; If[antiTauQ[k + 1], AppendTo[s, k]]], {n, 1, 50}]; s

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

Original entry on oeis.org

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.

A351852 Numbers k such that k and k+1 are both divisible by the number of their divisors over the Gaussian integers.

Original entry on oeis.org

31328, 173888, 893024, 1734488, 1896128, 4322240, 5405624, 8485568, 8982008, 9345248, 21874328, 38750624, 78588224, 137663288, 139074848, 189035000, 198387224, 270174968, 281333528, 290873024, 315950624, 326271968, 340513208, 357096608, 499656608, 584527328, 693637568

Offset: 1

Amiram Eldar, Feb 22 2022

nonn

Numbers k such that A062327(k) | k and A062327(k+1) | k+1.

All the terms are even numbers of the form k^2 - 1 (A033996).

			31328 is a term since 31328 is divisible by A062327(31328) = 88 and 31329 is divisible by A062327(31329) = 9.

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

Subsequence of A033996 and A351851.

Cf. A062327, A114617, A351854.

q[n_] := Divisible[n, DivisorSigma[0, n, GaussianIntegers -> True]]; Select[Range[1, 3*10^4, 2]^2 - 1, q[#] && q[# + 1] &]

A351854 Numbers k such that k and k+1 are both divisible by the number of their divisors over the Eisenstein integers.

Original entry on oeis.org

1, 2, 80, 3968, 50624, 497024, 505520, 3207680, 6890624, 9150624, 12383360, 12852224, 13549760, 19210688, 20657024, 25250624, 41796224, 41873840, 47900240, 48650624, 79121024, 81450624, 86099840, 132503120, 140920640, 149450624, 174636224, 186732224, 214769024

Offset: 1

Amiram Eldar, Feb 22 2022

nonn

Numbers k such that A319442(k) | k and A319442(k+1) | k+1.

Except for 1 and 2, all the terms are even numbers of the form k^2 - 1 (A033996).

			2 is a term since 2 is divisible by A319442(2) = 2 and 3 is divisible by A319442(3) = 3.
80 is a term since 80 is divisible by A319442(80) = 10 and 81 is divisible by A319442(81) = 9.

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

Subsequence of A351853.

Cf. A033996, A114617, A319442, A351852.

f[p_, e_] := Switch[Mod[p, 3], 0, 2*e + 1, 1, (e + 1)^2, 2, e + 1]; eisNumDiv[1] = 1; eisNumDiv[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := Divisible[n, eisNumDiv[n]]; Join[{1, 2}, Select[Range[3, 15000, 2]^2 - 1, q[#] && q[# + 1] &]]

A209291 Sum of the refactorable numbers less than or equal to n.

Original entry on oeis.org

1, 3, 3, 3, 3, 3, 3, 11, 20, 20, 20, 32, 32, 32, 32, 32, 32, 50, 50, 50, 50, 50, 50, 74, 74, 74, 74, 74, 74, 74, 74, 74, 74, 74, 74, 110, 110, 110, 110, 150, 150, 150, 150, 150, 150, 150, 150, 150, 150, 150, 150, 150, 150, 150, 150, 206, 206, 206, 206, 266

Offset: 1

Wesley Ivan Hurt, Jan 16 2013

nonn

A number is refactorable if it is divisible by the number of its divisors.
The first 8 terms are odd. The next odd term after 11 is a(225) = 2395.
600 out of the first 1000 terms are odd, including every term from a(625) up to and including a(1000). - Harvey P. Dale, Aug 07 2019

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Refactorable Number

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

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

Accumulate[Table[If[Divisible[n,DivisorSigma[0,n]],n,0],{n,60}]] (* Harvey P. Dale, Aug 07 2019 *)

a(n) = Sum_{i=1..n} i*(1 + floor(i/d(i)) - ceiling(i/d(i))) where d(n) is the number of divisors of n (A000005).

A120314 First occurrence of consecutive refactorable numbers a(n)-1, a(n) where the smallest prime factor of a(n) is the n-th prime prime(n).

Original entry on oeis.org

2, 9, 81450625, 6080399213078595601, 38604666779024731098340977806401, 79600343456925208350554324952070658488321, 111999530649584986702170994086297063568244097100801

Offset: 1

Walter Kehowski, Jun 20 2006

nonn

The factorizations are 2, 3^2, (5^4)*(19^4), (7^6)*(193^6), (11^10)*(131^10), (13^12)*(197^12), (17^16)*(79^16). Since the squares of all odd primes greater than 3 are 1 mod 12, the refactorable a(n)-1 is divisible by 12 whenever n>=3.

			Simplest: a(2)=9=3^2 since tau(9)=3; 9-1=8=2^3 so tau(8)=4.

Cf. A033950, A036898, A114617.

with(numtheory); RFC:=[[1,2,2]]: for w to 1 do for i from 2 to 12 do p:=ithprime(i); P:=[seq(ithprime(j),j=1..i-1)]; for k from 1 to 12^6 do if andmap(z -> k mod z <>0, P) then n:=(p*k)^(p-1); t:=tau(n); n1:=n-1; t1:=tau(n-1); if (n mod t = 0) and (n1 mod t1 = 0) then RFC:=[op(RFC),[k,p,n]]; print(ifactor(n)); break; fi fi od od od;

a(n) is the first integer of the form (k*p)^(p-1) for some k such that a(n)-1 and a(n) are refactorable and the smallest prime divisor of a(n) is prime(n).

A120316 Consecutive refactorable numbers a(n)-1, a(n) in which 5 is the smallest prime divisor of a(n).

Original entry on oeis.org

81450625, 12594450625, 29661450625, 81924750625, 195562950625, 233313150625, 308052750625, 379733250625, 764269350625, 1490902050625, 4606281750625, 4995336750625, 5551796250625, 7502257950625, 8450794350625, 25857479250625

Offset: 1

Walter Kehowski, Jun 20 2006

nonn

Cf. A033950, A036898, A114617.

with(numtheory); RFC5:=[]: p:=ithprime(3): P:=[seq(ithprime(i),i=1..2)]; for w to 1 do for k from 3 to 12^4 by 2 do if andmap(z -> k mod z <> 0, P) then m:=p*k; n:=m^(p-1); t:=tau(n); n1:=n-1; t1:=tau(n1); if (n mod t = 0) and (n1 mod t1 = 0) then RFC5:=[op(RFC5),n]; print(ifactor(n)); fi fi; od od;

a(n) is the first integer of the form (5*k)^(5-1) such that both a(n) and a(n)-1 is refactorable and 5 is the smallest prime divisor of a(n).

cp's OEIS Frontend

A036898 List of pairs of consecutive refactorable numbers.

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

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A309347 Sums of two refactorable numbers whose difference is refactorable.

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Maple

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A341779 Numbers k such that k and k+1 are both anti-tau numbers (A046642).

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

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A351852 Numbers k such that k and k+1 are both divisible by the number of their divisors over the Gaussian integers.

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A351854 Numbers k such that k and k+1 are both divisible by the number of their divisors over the Eisenstein integers.

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A209291 Sum of the refactorable numbers less than or equal to n.

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