A125057 -id:A125057 - cp's OEIS frontend

A126888 a(n) is the smallest positive integer such that floor(a(n)/d(a(n))) = n, or -1 if no such number exists, where d(m) is the number of positive divisors of m.

1, 5, 7, 28, 11, 13, 44, 17, 19, 63, 23, 51, 55, 29, 31, 49, 69, 37, 77, 41, 43, 91, 47, 147, 153, 53, 111, 115, 59, 61, 125, 129, 67, 207, 71, 73, 296, 155, 79, 121, 83, 680, 261, 89, 183, 185, 284, 97, 399, 101, 103, 209, 107, 109, 221, 113, 459, 235, 237, 363, 247, 249

Offset: 1

Leroy Quet, Dec 30 2006

nonn

Hugo van der Sanden and D. W. Wilson, Table of n, a(n) for n = 1..10000

Cf. A000005, A126889, A078709, A125056, A125057.

f[n_] := Block[{k = 1},While[Floor[k/Length[Divisors[k]]] != n, k++ ];k];Table[f[n], {n, 62}] (* Ray Chandler, Jan 04 2007 *)

Extended by Ray Chandler, Jan 04 2007.

Added escape clause to definition at the suggestion of Hugo van der Sanden. - N. J. A. Sloane, Jul 01 2022

A125056 a(n) is the largest positive integer such that floor(a(n)/d(a(n))) = n, where d(m) is the number of positive divisors of m.

Original entry on oeis.org

6, 12, 30, 48, 60, 72, 120, 96, 144, 180, 140, 240, 216, 252, 360, 336, 420, 224, 312, 480, 504, 540, 378, 720, 600, 840, 660, 672, 352, 364, 756, 780, 1080, 960, 1260, 864, 594, 924, 936, 1440, 1320, 1680, 1050, 1056, 1092, 1120, 1512, 1560

Offset: 1

Hugo van der Sanden, Jan 09 2007

nonn

We know the sequence is well-defined given the limit x/d(x) > 0.5*sqrt(x) from comments in A036763.

Does every positive integer n equal floor(m/d(m)) for some m?

Hugo van der Sanden and D. W. Wilson, Table of n, a(n) for n = 1..10000

Cf. A126888, A000005, A126889, A078709, A125057.

t = Table[ Floor[ n / DivisorSigma[0, n]], {n, 10^5}]; f[n_] := Max@ Flatten@ Position[t, n]; Array[f, 51] (* Robert G. Wilson v, Jan 12 2007 *)

A353960 a(1)=1. Thereafter, if a(n) has occurred k (>1) times in a(j), 1 <= j <= n then a(n+1) = k*a(n). If a(n) is a first occurrence, a(n+1) = A078709(a(n)).

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 5, 2, 4, 8, 2, 6, 1, 6, 12, 2, 8, 16, 3, 6, 18, 3, 9, 3, 12, 24, 3, 15, 3, 18, 36, 4, 12, 36, 72, 6, 24, 48, 4, 16, 32, 5, 10, 2, 10, 20, 3, 21, 5, 15, 30, 3, 24, 72, 144, 9, 18, 54, 6, 30, 60, 5, 20, 40, 5, 25, 8, 24, 96, 8, 32, 64, 9, 27, 6, 36, 108, 9

Offset: 1

David James Sycamore, May 12 2022

nonn

A nonprime term arises either following a multiple (>1) of a prior term, or following a novel term m such that A078709(m) is a nonprime number. A prime term arises either as a multiple of 1 (first occurrences of 2,3,5), or following a novel term m such that A078709(m) is a prime. Let C(m) be the cardinality of m in the sequence (starting 6,5,10,5,8,8,6,13,8,6,8,9,...).

Conjecture: C(m) = A125057(m) + k, where k is the number of divisors d of m such that 1 < d <= C(m/d). The cardinality of m due to occurrences of novel terms is A125057(m), whereas the cardinality due to repeat terms is counted by k, and all of the latter terms occur prior to any of the former. (Note comments in A125057, A126888, A125056, concerning definitions of those sequences.)

			a(8)=1, the 5th occurrence of 1, so the next term a(9)=5.
a(43)=32, a first occurrence, so a(44)=A078709(32)=5.
a(33)=36 (consequence of two occurrences of 18), and 36 has not occurred before, so a(34)=A078709(36)=4.
C(12)=9 because A125057(12)=6, and there are 3 occurrences of 12 which are consequent to repeat terms (2 terms 6, 4 terms 3, and 3 terms 4).
Alternatively, 12 has 5 divisors >1, of which 3 (d=2,3,4) satisfy d <= C(12/d), therefore k=3 and C(12) = A125057(12) + 3 = 6 + 3 = 9.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Cf. A000005, A078709, A126888, A033950, A130290, A125056, A125057.

Block[{a, c, k, s, nn}, nn = 120; c[] = 0; s = {1}; Table[(Set[a[i], #]; c[#]++) &@ s[[i]], {i, Length[s]}]; Do[If[c[#] == 1, Set[k, Floor[#/DivisorSigma[0, #]]], Set[k, c[#] #]] &@ a[i - 1]; a[i] = k; c[k]++, {i, Length[s] + 1, nn}]; Array[a, nn] ] (* _Michael De Vlieger, May 14 2022 *)

f(n) = n\numdiv(n); \\ A078709
lista(nn) = {my(v=vector(nn), k); v[1] = 1; for (n=2, nn, if ((k=#select(x->(x==v[n-1]), Vec(v, n-1))) > 1, v[n] = k*v[n-1], v[n] = f(v[n-1]));); v;} \\ Michel Marcus, May 16 2022

from itertools import islice
from sympy import divisor_count
def A353960_gen(): # generator of terms
    adict, a = {}, 1
    yield a
    while True:
        if a in adict:
            adict[a] += 1
            a *= adict[a]
        else:
            adict[a] = 1
            a //= divisor_count(a)
        yield a
A353960_list = list(islice(A353960_gen(),30)) # Chai Wah Wu, Jun 04 2022

For p prime = 2,3,5, C(p) = A125057(p) + 1; for p > 5, C(p) = A125057(p).

More terms from Michel Marcus, May 16 2022

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A126888 a(n) is the smallest positive integer such that floor(a(n)/d(a(n))) = n, or -1 if no such number exists, where d(m) is the number of positive divisors of m.

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A125056 a(n) is the largest positive integer such that floor(a(n)/d(a(n))) = n, where d(m) is the number of positive divisors of m.

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A353960 a(1)=1. Thereafter, if a(n) has occurred k (>1) times in a(j), 1 <= j <= n then a(n+1) = k*a(n). If a(n) is a first occurrence, a(n+1) = A078709(a(n)).

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