A036896 -id:A036896 - cp's OEIS frontend

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

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).

A120322 RF(11): refactorable numbers with 11 as smallest prime factor.

Original entry on oeis.org

25937424601, 3575694237941010577249, 52289689788971837545849, 159024068785448665562401, 604292326212030787555081, 1074497011086501939579049, 10912062142819279835644801

Offset: 1

Walter Kehowski, Jun 21 2006

nonn

See A033950 for references. For any prime p, p^(p-1) is the smallest element of RF(p), the refactorable numbers whose smallest prime factor is p. Thus 11^(11-1)=25937424601 is the first element. Other elements would also be 11^10*17^10 or 11^16*17^10. Here are the prime factorizations for the first 48 elements of RF11: (11^10), (11^10)*(13^10), (11^10)*(17^10), (11^10)*(19^10), (11^10)*(13^12), (11^10)*(23^10), (11^10)*(29^10), (11^10)*(31^10), (11^10)*(37^10), (11^10)*(41^10), (11^10)*(43^10), (11^10)*(47^10), (11^10)*(53^10), (11^10)*(59^10), (11^10)*(61^10), (11^10)*(67^10), (11^10)*(71^10), (11^10)*(73^10), (11^10)*(79^10), (11^10)*(83^10), (11^10)*(89^10), (11^10)*(17^16), (11^10)*(97^10), (11^10)*(101^10), (11^10)*(103^10), (11^10)*(107^10), (11^10)*(109^10), (11^10)*(113^10), (11^10)*(127^10), (11^10)*(131^10), (11^10)*(137^10), (11^10)*(139^10), (11^10)*(149^10), (11^10)*(151^10), (11^10)*(157^10), (11^10)*(163^10), (11^10)*(167^10), (11^10)*(173^10), (11^10)*(179^10), (11^10)*(181^10), (11^10)*(191^10), (11^10)*(193^10), (11^10)*(197^10), (11^10)*(199^10), (11^10)*(211^10), (11^10)*(13^10)*(17^10), (11^10)*(2 23^10), (11^10)*(227^10).

			a(1)=11^(11-1)=25937424601.

Cf. A033950, A036896, A036897.

with(numtheory); p:=11: a:=p^(p-1): RF11:=[a]: P:=[seq(ithprime(i),i=2..pi(p)-1)]; for w to 1 do for j from 1 to 12^3 do k:=2*j+1; if andmap(z -> k mod z <> 0, P) then for s from 2 to p-1 by 2 do #accelerate creation n:=a*k^s; t:=tau(n); if not n in RF11 and (n mod t = 0) then RF11:=[op(RF11),n]; print(ifactor(n)); fi; od; fi; od od; RF11:=sort(RF11);

a(n) = odd square, 11 is the smallest prime factor and refactorable.

A360779 Refactorable numbers gaps: differences between consecutive refactorable numbers.

Original entry on oeis.org

1, 6, 1, 3, 6, 6, 12, 4, 16, 4, 12, 8, 4, 4, 8, 8, 4, 20, 4, 4, 16, 4, 24, 4, 20, 21, 3, 4, 8, 8, 4, 24, 12, 8, 32, 16, 4, 12, 12, 4, 8, 12, 28, 17, 3, 4, 2, 18, 4, 8, 8, 4, 12, 12, 20, 24, 4, 4, 16, 16, 12, 13, 7, 4, 4, 24, 8, 12, 24, 4, 8, 12, 44, 16, 12, 4, 16, 4, 24

Offset: 1

Ctibor O. Zizka, Feb 20 2023

nonn

Empirically it looks as though the consecutive refactorable numbers >= 8 with odd gaps between them always occur in triples: [8, 9, 12], [204, 225, 228], [424, 441, 444], [612, 625, 632], [1068, 1089, 1096], [1520, 1521, 1524], and so on. The sum of the gaps in the triple is divisible by 4. The middle term of a triple is an odd refactorable number, see A036896.

			a(1) = 2 - 1 = 1;
a(2) = 8 - 2 = 6;
a(3) = 9 - 8 = 1;
and so on.

Cf. A033950, A036896, A036898, A114617.

Differences[Select[Range[1000], Divisible[#, DivisorSigma[0, #]] &]] (* Amiram Eldar, Feb 20 2023 *)

lista(nn) = my(v=select(x->!(x % numdiv(x)), [1..nn])); vector(#v-1, k, v[k+1]-v[k]); \\ Michel Marcus, Feb 20 2023

a(n) = A033950(n + 1) - A033950(n).

cp's OEIS Frontend

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

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A120322 RF(11): refactorable numbers with 11 as smallest prime factor.

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A360779 Refactorable numbers gaps: differences between consecutive refactorable numbers.

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