A036898 -id:A036898 - cp's OEIS frontend

A114617 Numbers k such that k and k+1 are both refactorable numbers.

1, 8, 1520, 50624, 62000, 103040, 199808, 221840, 269360, 463760, 690560, 848240, 986048, 1252160, 1418480, 2169728, 2692880, 2792240, 3448448, 3721040, 3932288, 5574320, 5716880, 6066368, 6890624, 6922160, 8485568

Offset: 1

Eric W. Weisstein, Dec 16 2005

nonn

It is not possible to have three consecutive refactorable numbers (see the link). The sequence is best viewed in base 12, with X for 10 and E for 11: 1, 8, X68, 25368, 2EX68, 4E768, 97768, X8468, 10EX68, 1X4468, 293768, 34XX68, 3E6768, 504768, 584X68, 887768, X9X468, E27X68, 11X3768, 12E5468, 1397768, 1X49X68, 1XE8468, 2046768, 2383768, 2399X68, 2X12768. After the first two terms all terms are 68, 368, 468, 668, 768, X68 mod 1000 (base 12). - Walter Kehowski, Jun 19 2006
No successive refactorables seem to be of the form odd, odd+1. If such a pair exist, they must be very large. The first pair of successive refactorables not divisible by 3 is (5*19)^4-1, (5*19)^4. - Walter Kehowski, Jun 25 2006
Zelinsky (2002, Theorem 59, p. 15) proved that all the terms above 1 are even. - Amiram Eldar, Feb 20 2021

Jud McCranie, Table of n, a(n) for n = 1..94342 First 1000 terms by Donovan Johnson.
Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.
Eric Weisstein's World of Mathematics, Refactorable Number.

Cf. A033950, A036898.

Filtered([1..10^6],n->n mod Tau(n)=0 and (n+1) mod Tau(n+1)=0 ); # Muniru A Asiru, Dec 21 2018

with(numtheory); RFC:=[]: for w to 1 do for k from 1 to 12^6 do n:=144*k+(6*12+8); if andmap(z-> z mod tau(z) = 0,[n,n+1]) then RFC:=[op(RFC),n]; print(n); fi od od; # it is possible to remove the condition n = (6*12+8) mod 12^2 but you'll get the same sequence. - Walter Kehowski, Jun 19 2006

Select[Join[{1, 8}, 144*Range[10^5] + 80], Mod[#, DivisorSigma[0, #]] == 0 && Mod[#+1, DivisorSigma[0, #+1]] == 0 & ](* Jean-François Alcover, Oct 25 2012, after Walter Kehowski *)

isok(n) = !(n % numdiv(n)) && !((n+1) % numdiv(n+1)); \\ Michel Marcus, Dec 21 2018

a(n) mod tau(a(n)) = 0 and (a(n)+1) mod tau(a(n)+1) = 0 where tau(n) is the number of divisors of n. - Walter Kehowski, Jun 19 2006

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.

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.

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).

A036899 Products of pairs of consecutive refactorable numbers.

Original entry on oeis.org

2, 72, 2311920, 2562840000, 3844062000, 10617344640, 39923436672, 49213207440, 72555078960, 215073801360, 476873804160, 719511945840, 972291644352, 1567905917760, 2012086928880, 4707721763712, 7251605387280

Offset: 1

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

nonn

The product of a pair of consecutive refactorable numbers is itself refactorable.

			8 and 9 are consecutive refactorable numbers, so 8*9 = 72 is in the sequence.

S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
S. Colton, HR - Automatic Theory Formation in Pure Mathematics

Cf. A033950.

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

a(n) = A036898(2n) * A036898(2n+1). - Michel Marcus, Aug 31 2013

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).

A120317 Consecutive refactorable numbers a(n)-1, a(n) in which 7 the smallest prime divisor of a(n).

Original entry on oeis.org

6080399213078595601, 106451203123324908289, 842122675409157900289, 205035001401532317649921, 690310240598397005456401, 1125500133125681400538801, 1241419580861102113344769

Offset: 1

Walter Kehowski, Jun 20 2006

nonn

Cf. A033950, A036898, A114617.

with(numtheory); RFC7:=[]: p:=ithprime(4): P:=[seq(ithprime(i),i=1..3)]; 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 RFC7:=[op(RFC7),n]; print(ifactor(n)); fi fi; od od;

a(n) is the first integer of the form (7*k)^(7-1) such that both a(n) and a(n)-1 is refactorable and 7 is the smallest prime divisor of a(n).

A120318 Consecutive refactorable numbers a(n)-1, a(n) in which 11 is the smallest prime divisor of a(n).

Original entry on oeis.org

38604666779024731098340977806401, 7208577773559712596404976530284801, 695314235787112476661749457231833601, 313468146036745542621075945985861000534849

Offset: 1

Walter Kehowski, Jun 20 2006

nonn

Cf. A033950, A036898, A114617.

with(numtheory); RFC11:=[]: p:=ithprime(5): P:=[seq(ithprime(i),i=1..4)]; 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 RFC11:=[op(RFC11),n]; print(ifactor(n)); fi fi; od od;

a(n) is the first integer of the form (11*k)^(11-1) such that both a(n) and a(n)-1 is refactorable and 11 is the smallest prime divisor of a(n).

A120335 CRF(13): consecutive refactorable numbers (rf, rf-1 are refactorable) such that 13 is the smallest prime divisor of rf.

Original entry on oeis.org

79600343456925208350554324952070658488321, 67727051825754224132985695308485992267126791150081, 17096333467784942360991864487916588941402614691799041

Offset: 1

Walter Kehowski, Jun 22 2006

The sequence has prime factorization (13*197)^12, (13*1093)^12, (13*1733)^12, (13*17*139)^12, (13*7877)^12, (13*16069)^12.

			a(1)=(13*197)^12 is the first number rf such that rf refactorable, 13 is the smallest prime of rf and rf-1 is refactorable.

Cf. A033950, A036898, A114617.

a(n) = is the n-th number rf such that both rf and rf-1 are refactorable and 13 is the smallest prime divisor of rf.

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A114617 Numbers k such that k and k+1 are both refactorable numbers.

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

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

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

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A036899 Products of pairs of consecutive refactorable numbers.

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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).

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

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A120317 Consecutive refactorable numbers a(n)-1, a(n) in which 7 the smallest prime divisor of a(n).