A078709 -id:A078709 - cp's OEIS frontend

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)).

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

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.

Original entry on oeis.org

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 *)

A125057 a(n) is the number of positive integers m such that floor(m/d(m)) = n, where d(m) is the number of positive divisors of m.

Original entry on oeis.org

5, 4, 9, 3, 7, 5, 6, 11, 7, 4, 8, 6, 9, 5, 4, 16, 7, 4, 8, 7, 11, 5, 10, 7, 7, 8, 7, 12, 9, 6, 10, 8, 8, 8, 10, 6, 4, 7, 7, 15, 8, 4, 11, 11, 8, 12, 7, 11, 7, 9, 8, 8, 12, 14, 8, 12, 8, 8, 11, 5, 14, 7, 7, 9, 5, 8, 4, 13, 7, 8, 12, 10, 6, 9, 14, 11, 9, 8, 9, 12, 13, 8, 8, 9, 9, 10, 7, 11, 14, 3, 10

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?
First occurrence of k>2: 4, 2, 1, 7,5 , 11, 3, 23, 8, 28, 68, 54, 40, 16, 251, 572, 141, ???, ???, ???, 529, ..., (630). - Robert G. Wilson v, Jan 11 2007

D. W. Wilson, Table of n, a(n) for n = 1..10000

Cf. A126888, A000005, A126889, A078709, A125056.

t = Table[Floor[n/DivisorSigma[0, n]], {n, 10^5}]; f[n_] := Length@Select[t, # == n &]; Array[f, 91] (* Robert G. Wilson v, Jan 11 2007 *)

Edited by Robert G. Wilson v, Jan 11 2007

A334762 a(n) = ceiling (n / A000005(n)).

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 2, 3, 3, 6, 2, 7, 4, 4, 4, 9, 3, 10, 4, 6, 6, 12, 3, 9, 7, 7, 5, 15, 4, 16, 6, 9, 9, 9, 4, 19, 10, 10, 5, 21, 6, 22, 8, 8, 12, 24, 5, 17, 9, 13, 9, 27, 7, 14, 7, 15, 15

Offset: 1

Ctibor O. Zizka, May 10 2020

ceiling (n / A000005(n)) - floor (n / A000005(n)) = 0 for A033950(n).

			a(1) = ceiling (1 / 1) = 1;
a(5) = ceiling (5 / 2) = 3;

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A033950, A078709.

seq(ceil(n/numtheory:-tau(n)), n=1..100); # Robert Israel, May 13 2020

a[n_] := Ceiling[n / DivisorSigma[0, n]]; Array[a, 60] (* Amiram Eldar, May 10 2020 *)

from sympy import divisor_count
def A334762(n): return (a := divmod(n,divisor_count(n)))[0] + int((a[1] > 0) == True) # Chai Wah Wu, Jun 20 2022

a(n) = ceiling (n / A000005(n)).

A078710 Integer part of the subinterval sample variance in the partition of [0,n] by the divisors of n.

Original entry on oeis.org

0, 0, 0, 4, 1, 12, 2, 7, 3, 40, 4, 60, 9, 17, 8, 112, 9, 144, 13, 35, 27, 220, 14, 104, 41, 60, 25, 364, 23, 420, 34, 93, 75, 166, 30, 612, 97, 133, 45, 760, 47, 840, 67, 124, 147, 1012, 50, 500, 93, 235, 96, 1300, 78, 410, 91, 297, 243, 1624, 69, 1740, 281, 254, 130, 576

Offset: 2

Joseph L. Pe, Dec 19 2002

nonn

			The divisors of 9 partition the closed interval [0,9] into subintervals [0,1), [1,3), [3,9], with lengths 1, 2, 6, respectively. The sample variance of these lengths has integer part = 7. Hence a(9) = 7. [Corrected by _Sean A. Irvine_, Jul 12 2025]

Cf. A078709.

f[n_] := Module[{d, l, a, i}, d = Divisors[n]; l = Length[d]; a = {1}; For[i = 1, i <= l - 1, i++, a = Append[a, d[[i + 1]] - d[[i]]]]; a]; Table[Floor[Variance[f[i]]], {i, 2, 100}]

A355140 n/d(n) rounded to the nearest integer, where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 2, 3, 3, 6, 2, 7, 4, 4, 3, 9, 3, 10, 3, 5, 6, 12, 3, 8, 7, 7, 5, 15, 4, 16, 5, 8, 9, 9, 4, 19, 10, 10, 5, 21, 5, 22, 7, 8, 12, 24, 5, 16, 8, 13, 9, 27, 7, 14, 7, 14, 15, 30, 5, 31, 16, 11, 9, 16, 8, 34, 11, 17, 9, 36, 6, 37, 19, 13, 13, 19

Offset: 1

Sameer Khan, Jun 20 2022

nonn

In the ambiguous case, fractions are rounded up.

			a(1) = round (1 / 1) = 1;
a(4) = round (4 / 3) = 1;
a(5) = round (5 / 2) = 3;

Cf. A000005, A078709 (floor), A334762 (ceiling), A090395 (numerators), A090387 (denominators).

Table[Floor[n/DivisorSigma[0,n]+1/2],{n,100}] (* Harvey P. Dale, Dec 22 2022 *)

from sympy import divisor_count
def A355140(n): return (2*n+(d:=divisor_count(n)))//(2*d) # Chai Wah Wu, Jun 20 2022

a(n) = round (n / A000005(n)).

cp's OEIS Frontend

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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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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A125057 a(n) is the number of positive integers m such that floor(m/d(m)) = n, where d(m) is the number of positive divisors of m.

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A334762 a(n) = ceiling (n / A000005(n)).

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A078710 Integer part of the subinterval sample variance in the partition of [0,n] by the divisors of n.

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Mathematica

A355140 n/d(n) rounded to the nearest integer, where d(n) is the number of divisors of n (A000005).

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