A141586 -id:A141586 - cp's OEIS frontend

A142727 For definition see Comments lines.

1, 2, 3, 4, 6, 6, 8, 8, 9, 12, 12, 14, 15, 16, 16, 18, 20, 20, 22, 24, 24, 24, 25, 27, 30, 30, 30, 32, 32, 32, 36, 36, 39, 40, 42, 42, 44, 45, 46, 48, 48, 48, 50, 50, 52, 52, 55, 59, 60, 60, 60, 60, 60, 64, 64, 66, 66, 67, 69, 70, 71, 72, 75, 76, 76, 78, 80, 81, 84

Offset: 1

David Applegate and N. J. A. Sloane, Oct 03 2008

nonn

S is a sequence of numbers with repetitions, sorted in nondecreasing order.
Initially a(1) = 1 and S = N = {1,2,3,4,5,6,...}.
For each n >= 2, let p = prime(n), let a(n) = (p-1)st term of S and set S = S union a(n)*N. Repeat.
A sequence arising from one possible attempt to analyze A141586.

			The first few stages in the calculation are as follows:
S = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
n = 2, p = 3, so a(2) is the 2nd term of S, which is 2.
Now S = 1 2 2 3 4 4 5 6 6 7 8 8 9 10 10 11 12 12 13 14 14 ...
n = 3, p = 5, so a(3) is the 4th term of S, which is 3.
Now S = 1 2 2 3 3 4 4 5 6 6 6 7 8 8 9 9 10 10 11 12 12 12 13 14 14 ...
n = 4, p = 7, so a(4) is the 6th term of S, which is 4.
Now S = 1 2 2 3 3 4 4 4 5 6 6 6 7 8 8 8 9 9 10 10 11 12 12 12 12 13 14 14 ...
n = 5, p = 11, so a(5) is the 10th term of S, which is 6.
And so on.

David Applegate, Table of n, a(n) for n = 1..1000
David Applegate, Table of n, p(n), a(n), a(n)/n at intervals of 10000 up to 244190000

A plot of the extended sequence suggests that a(n) ~= c n log(log(n)) + d n for constants c and d. For example, run: $ gnuplot> plot [] [1.27:1.35] a142727.txt using 1:4, 1.12+0.076*log(log(x)).

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).

A212169 List of highly composite numbers (A002182) with an exponent in its prime factorization that is at least as great as the number of positive exponents; intersection of A002182 and A212165.

Original entry on oeis.org

1, 2, 4, 12, 24, 36, 48, 120, 240, 360, 720, 1680, 5040, 10080, 15120, 20160, 25200, 45360, 50400, 110880, 221760, 332640, 554400, 665280, 2882880, 8648640, 14414400, 17297280, 43243200, 294053760

Offset: 1

Matthew Vandermast, May 22 2012

Sequence can be used to find the largest highly composite number in subsequences of A212165 (of which there are several in the database).
Ramanujan showed that, in the canonical prime factorization of a highly composite number with largest prime factor prime(n), the largest exponent cannot exceed 2*log_2(prime(n+1)). (See formula 54 on page 15 of the Ramanujan paper.) This limit is less than n for all n >= 9 (and prime(n) >= 23).
1. Direct calculation verifies this for 9 <= n <= 11.
2. Nagura proved that, for any integer m >= 25, there is always a prime between m and 1.2*m.  Let n = 11, at which point prime(11) = 31 and log_2(prime(n+1)) = log 37/log 2 = 5.209453.... Since log 1.2/log 2 is only 0.263034..., it follows that n must increase by at least 3k before 2*log_2(prime(n+1)) can increase by 2k, for all values of k. Therefore, 2*log_2(prime(n+1)) can never catch up to prime(n) for n > 11.
665280 = 2^6*3^3*5*7*11 is the largest highly composite number whose prime factorization contains an exponent that is strictly greater than the number of positive exponents in that factorization (including the implied 1's).

			A002182(62) = 294053760 = 2^7*3^3*5*7*11*13*17 has 7 positive exponents in its prime factorization, including 5 implied 1's. The maximal exponent in its prime factorization is also 7. Therefore, 294053760 is a term of this sequence.

S. Ramanujan, Highly composite numbers, Proc. Lond. Math. Soc. 14 (1915), 347-409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962.

A. Flammenkamp, List of the first 1200 highly composite numbers
J. Nagura, On the interval containing at least one prime number, Proc. Japan Acad., 28 (1952), 177-181.
S. Ramanujan, Highly Composite Numbers (p. 15) (pages 11-12 introduce some of the notation in formula 54)

A212165 also includes all terms in A006939, A066120, A087980, A130091, A138534, A141586, A166475, A181555, A181813-A181814, A181818, A181823-A181825, A182763.

Other finite subsequences of A002182 include A072938, A095921, A106037, A136253, A162935, A166981, A168263.

okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] >= Length[f]]; a = 0; t = {}; Do[b = DivisorSigma[0, n]; If[b > a, a = b; If[okQ[n], AppendTo[t, n]]], {n, 10^6}]; t (* T. D. Noe, May 24 2012 *)

A306693 a(n) is the least strongly refactorable multiple of n if any, or a(n) = -1 otherwise.

Original entry on oeis.org

1, 2, 12, 12, 240, 12, 3360, 24, 36, 240, 5280, 12, 6240, 3360, 240, 240, 8160, 36, 9120, 240, 3360, 5280, 11040, 24, 21600, 6240, 4320, 3360, 13920, 240, 14880, 480, 5280, 8160, 3360, 36, 17760, 9120, 6240, 240, 19680, 3360, 20640, 5280, 720, 11040, 22560

Offset: 1

Rémy Sigrist, Mar 05 2019

nonn

Strongly refactorable numbers correspond to A141586.

Is a(n) > 0 for any n > 0 ?

			For n = 3:
- the divisors of 3 are: 1, 3,
- the corresponding numbers of divisors are: 1, 2,
- 2 does not divide 3,
- the divisors of 2*3 are: 1, 2, 3, 6,
- the corresponding numbers of divisors are: 1, 2, 2, 4,
- 4 does not divide 2*3,
- the divisors of 2*2*3 are: 1, 2, 3, 4, 6, 12,
- the corresponding numbers of divisors are: 1, 2, 2, 3, 4, 6,
- they all divide 2*2*3,
- hence a(3) = 2*2*3 = 12.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of A000005(n))

See A306645 for a similar sequence.

Cf. A000005, A141586.

a(n) = while (1, my (m=n); fordiv (m, d, m=lcm(m, numdiv(d))); if (n==m, return (n), n=m))

a(A141586(n)) = A141586(n) for any n > 0.

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A142727 For definition see Comments lines.

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

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A212169 List of highly composite numbers (A002182) with an exponent in its prime factorization that is at least as great as the number of positive exponents; intersection of A002182 and A212165.

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A306693 a(n) is the least strongly refactorable multiple of n if any, or a(n) = -1 otherwise.

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