A309141 -id:A309141 - cp's OEIS frontend

A309181 Least number with exactly n nonunitary divisors.

1, 4, 8, 16, 24, 36, 48, 256, 72, 1024, 180, 144, 216, 16384, 288, 65536, 360, 576, 3072, 900, 864, 1296, 720, 2304, 1080, 67108864, 2592, 268435456, 1440, 9216, 196608, 5184, 2160, 17179869184, 2880, 36864, 10368, 3600, 6300, 1099511627776, 4320, 20736, 6480

Offset: 0

Amiram Eldar, Jul 15 2019

nonn

Amiram Eldar, Table of n, a(n) for n = 0..100

Cf. A005179, A048105, A309141.

nud[n_] := DivisorSigma[0, n] - 2^PrimeNu[n]; m = 24; c = 0; s = Table[0, {m + 1}]; k = 1; While[c < m + 1, n = nud[k]; If[n <= m && s[[n + 1]] == 0, c++; s[[n + 1]] = k]; k++]; s

nnud(n) = numdiv(n) - 2^omega(n); \\ A048105
a(n) = my(k=1); while (nnud(k) != n, k++); k; \\ Michel Marcus, Jul 17 2019

A048105(a(n)) = n and A048105(k) != n for all k < a(n).

a(n) <= 2^(n+1).

A174961 Number of non-unitary divisors of the n-th nonsquarefree number.

Original entry on oeis.org

1, 2, 1, 2, 3, 2, 2, 4, 1, 2, 2, 4, 5, 4, 2, 2, 6, 1, 2, 2, 4, 4, 4, 2, 5, 2, 8, 2, 2, 6, 3, 4, 4, 4, 2, 8, 2, 2, 5, 4, 8, 6, 2, 2, 8, 1, 2, 2, 4, 6, 4, 4, 4, 4, 11, 2, 2, 4, 4, 2, 4, 8, 6, 2, 8, 1, 2, 2, 2, 6, 10, 4, 2, 4, 10, 5, 4, 8, 4, 2, 6, 2, 12, 4, 8, 5, 4, 4, 4, 2, 12, 2, 4, 2

Offset: 1

N. Wu (neil_wu0626(AT)yahoo.com), Apr 02 2010

nonn

The nonzero terms of A048105.

Also number of nonsquarefree divisors of the n-th nonsquarefree number. The terms in A013929 which correspond to records in this sequence are given by A309141(n); n >= 2. - David James Sycamore, Jan 07 2025

			For n = 4, the fourth nonsquarefree number is A013929(4) = 12 which has 2 non-unitary divisors, 2 and 6. Therefore a(4) = 2.
The number of nonsquarefree divisors of 12 is also = 2 (4 and 12). For n = 55, A013929(55) = 144 so by the third formula above a(55) = A000005(144) - A000005(6) = 15 - 4 = 11 = number of nonsquarefree divisors of 144 (4,8,9,12,16,18,24,36,48,72,144). - _David James Sycamore_, Jan 07 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from G. C. Greubel)

Cf. A013929, A048105.
Cf. A001620, A013661, A306016.
Cf. A034444, A309141.

Select[Table[DivisorSigma[0, n] - 2^(PrimeNu[n]), {n, 1, 500}], # > 0 &] (* G. C. Greubel, May 21 2017 *)

lista(kmax) = {my(f); for(k = 1, kmax, f = factor(k); if(!issquarefree(f), print1(numdiv(f) - 2^omega(f), ", ")));} \\ Amiram Eldar, Dec 09 2023

from math import prod, isqrt
from sympy import mobius, factorint
def A174961(n):
    def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return -(1<Chai Wah Wu, Aug 12 2024

From Amiram Eldar, Dec 09 2023: (Start)
a(n) = A048105(A013929(n)).
Sum_{k=1..n} a(k) ~ (n/zeta(2)) * (log(n) + 2*gamma - 1 - 2*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). (End)
a(n) = A000005(A013929(n)) - A000005(A007947(A013929(n))). - David James Sycamore, Jan 07 2025

Edited by Amiram Eldar, Dec 09 2023

A329882 Nonunitary superabundant numbers: numbers m such that nusigma(m)/m > nusigma(k)/k for all k < m, where nusigma(m) is the sum of nonunitary divisors of m (A048146).

Original entry on oeis.org

1, 4, 8, 16, 24, 36, 48, 72, 144, 288, 360, 432, 720, 1440, 1800, 2160, 3600, 7200, 10800, 15120, 21600, 25200, 50400, 75600, 151200, 302400, 453600, 529200, 831600, 1058400, 1663200, 2116800, 3175200, 3326400, 4989600, 5821200, 9979200, 11642400, 21621600

Offset: 1

Amiram Eldar, Nov 23 2019

nonn

The nonunitary version of A004394.

Cf. A034448, A048146, A064597, A309141.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); nusigma[n_] := DivisorSigma[1, n] - usigma[n]; rm = -1; s = {}; Do[r = nusigma[n]/n; If[r > rm, rm = r; AppendTo[s, n]], {n, 1, 10000}]; s

A329883 Nonunitary highly abundant numbers: numbers m such that nusigma(m) > nusigma(k) for all k < m, where s(n) is the sum of nonunitary divisors of n (A048146).

Original entry on oeis.org

1, 4, 8, 12, 16, 24, 32, 36, 48, 64, 72, 96, 108, 120, 144, 180, 192, 216, 288, 360, 432, 504, 576, 648, 720, 864, 1008, 1080, 1296, 1440, 1728, 1800, 2016, 2160, 2520, 2880, 3024, 3240, 3456, 3528, 3600, 4320, 5040, 5400, 5760, 6048, 6480, 7056, 7200, 8640

Offset: 1

Amiram Eldar, Nov 23 2019

nonn

The corresponding record values are 0, 2, 6, 8, 14, 24, 30, 41, 56, 62, 105, 120, 140, 144, 233, 246, 248, 348, 489, 630, 764, 840, ...

Amiram Eldar, Table of n, a(n) for n = 1..500

The nonunitary version of A002093.

Cf. A048146, A064597, A309141, A329882.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); nusigma[n_] := DivisorSigma[1, n] - usigma[n]; num = -1; s = {}; Do[nu = nusigma[n]; If[nu > num, num = nu; AppendTo[s, n]], {n, 1, 10^4}]; s

A335386 Tri-unitary highly composite numbers: where the number of tri-unitary divisors (A335385) increases to a record.

Original entry on oeis.org

1, 2, 6, 24, 120, 840, 7560, 83160, 1081080, 18378360, 349188840, 8031343320, 200783583000, 5822723907000, 180504441117000, 6678664321329000, 273825237174489000, 11774485198503027000, 553400804329642269000, 27116639412152471181000, 1437181888844080972593000

Offset: 1

Amiram Eldar, Jun 04 2020

nonn

Amiram Eldar, Table of n, a(n) for n = 1..32

Analogous sequences: A002182 (highly composite), A002110 (unitary), A037992 (infinitary), A293185 (bi-unitary), A318278 (exponential), A306736 (exponential infinitary), A307845 (exponential unitary), A309141 (nonunitary), A322484 (semi-unitary).

Cf. A335385.

f[p_, e_] := If[e == 3 || e == 6, 4, 2]; d[1] = 1; d[n_] := Times @@ (f @@@ FactorInteger[n]); dm = 0; s = {}; Do[If[(d1 = d[n]) > dm, dm = d1; AppendTo[s, n]], {n, 1, 1100000}]; s

A335385(a(n)) = 2^(n-1).

A333931 Recursive highly composite numbers: numbers with a record number of recursive divisors (A282446).

Original entry on oeis.org

1, 2, 4, 6, 12, 30, 36, 60, 180, 420, 900, 1260, 4620, 6300, 13860, 44100, 55440, 69300, 180180, 485100, 720720, 900900, 2882880, 3063060, 6306300, 12252240, 15315300, 49008960, 58198140, 107207100, 232792560, 290990700, 931170240, 1163962800, 2036934900, 4655851200

Offset: 1

Amiram Eldar, Apr 10 2020

nonn

The corresponding record values are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 48, 54, ...

Amiram Eldar, Table of n, a(n) for n = 1..200

Subsequence of A025487.
Cf. A282446.
Analogous sequences: A002182 (highly composite), A002110 (unitary), A037992 (infinitary), A293185 (bi-unitary), A309141 (nonunitary), A318278 (exponential).

recDivNum[1] = 1; recDivNum[n_] := recDivNum[n] = Times @@ (1 + recDivNum/@ (Last /@ FactorInteger[n])); rm = 0; s = {}; Do[r = recDivNum[n]; If[r > rm, rm = r; AppendTo[s, n]], {n, 1, 10^4}]; s

A358253 Numbers with a record number of non-unitary square divisors.

Original entry on oeis.org

1, 8, 32, 128, 288, 864, 1152, 2592, 4608, 10368, 20736, 28800, 41472, 64800, 115200, 259200, 518400, 1036800, 2073600, 4147200, 8294400, 9331200, 12700800, 25401600, 50803200, 101606400, 203212800, 406425600, 457228800, 635040000, 812851200, 914457600, 1270080000

Offset: 1

Amiram Eldar, Nov 05 2022

nonn

Numbers m such that A056626(m) > A056626(k) for all k < m.

The corresponding record values are 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 20, 22, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..283
Amiram Eldar, Table of n, a(n), A056626(a(n)) for n = 1..283

Cf. A056626, A358252.
Subsequence of A025487.
Similar sequences: A002182 (all divisors), A002110 (unitary), A037992 (infinitary), A046952 (square divisors), A053624 (odd divisors), A293185 (bi-unitary), A309141 (non-unitary), A318278 (exponential).

f1[p_, e_] := 1 + Floor[e/2]; f2[p_, e_] := 2^(1 - Mod[e, 2]); f[1] = 0; f[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; s = {}; fmax = -1; Do[If[(fn = f[n]) > fmax, fmax = fn; AppendTo[s, n]], {n, 1, 10^5}]; s

s(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + floor(f[i,2]/2)) - 2^sum(i = 1, #f~, 1 - f[i,2]%2);}
lista(nmax) = {my(smax = -1, sn); for(n = 1, nmax, sn = s(n); if(sn > smax, smax = sn; print1(n, ", "))); }

A348342 Noninfinitary highly composite numbers: where the number of noninfinitary divisors (A348341) increases to a record.

Original entry on oeis.org

1, 4, 12, 16, 36, 48, 144, 240, 576, 720, 1680, 2880, 3600, 5040, 11520, 14400, 15120, 20160, 25200, 45360, 55440, 80640, 100800, 166320, 176400, 226800, 277200, 498960, 720720, 887040, 1108800, 1587600, 1940400, 2494800, 3603600, 6486480, 9979200, 11531520, 14414400

Offset: 1

Amiram Eldar, Oct 13 2021

nonn

The record numbers of noninfinitary divisors are 0, 1, 2, 3, 5, 6, 11, 12, 13, 22, 24, 26, 37, 44, 46, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..468
Amiram Eldar, Table of n, a(n), A348341(a(n)) for n = 1..468

Cf. A348341.
Subsequence of A025487.
Similar sequences: A002182, A002110 (unitary), A037992 (infinitary), A293185, A306736, A307845, A309141, A318278, A322484, A335386.

nid[1] = 0; nid[n_] := DivisorSigma[0, n] - Times @@ Flatten[2^DigitCount[#, 2, 1] & /@ FactorInteger[n][[;; , 2]]]; dm = -1; s = {}; Do[If[(d = nid[n]) > dm, dm = d; AppendTo[s, n]], {n, 1, 10^6}]; s

A348632 Nonexponential highly composite numbers: where the number of nonexponential divisors (A160097) increases to a record.

Original entry on oeis.org

1, 6, 12, 24, 30, 60, 120, 210, 240, 360, 420, 720, 840, 1260, 1680, 2520, 3360, 5040, 7560, 9240, 10080, 15120, 18480, 25200, 27720, 36960, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 480480, 498960, 554400, 665280, 720720, 1081080, 1441440

Offset: 1

Amiram Eldar, Oct 26 2021

nonn

The corresponding record values are 1, 3, 4, 6, 7, 10, 14, 15, 17, 20, 22, 24, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..651
Amiram Eldar, Table of n, a(n), A160097(a(n)) for n = 1..651

Cf. A160097.
Subsequence of A025487.
Similar sequences: A002182, A002110 (unitary), A037992 (infinitary), A293185, A306736, A307845, A309141, A318278, A322484, A335386.

f1[p_, e_] := e + 1; f2[p_, e_] := DivisorSigma[0, e]; ned[1] = 1; ned[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f; dm = -1; s = {}; Do[If[(d = ned[n]) > dm, dm = d; AppendTo[s, n]], {n, 1, 10^6}]; s

A353899 Indices of records in A353898.

Original entry on oeis.org

1, 2, 4, 6, 12, 30, 36, 60, 180, 420, 900, 1260, 4620, 6300, 13860, 44100, 55440, 69300, 180180, 485100, 720720, 900900, 3063060, 6306300, 12252240, 15315300, 58198140, 107207100, 232792560, 290990700, 1163962800, 2036934900, 5354228880, 6692786100, 22406283900

Offset: 1

Amiram Eldar, May 10 2022

nonn

First differs from A333931 at n=23.

The corresponding record values are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 48, 54, 72, 81, 96, 108, 144, 162, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..145
Amiram Eldar, Table of n, a(n), A353898(a(n)) for n = 1..145

Cf. A333931, A353898.
Subsequence of A025487 and A138302.
Similar sequences: A002182, A002110 (unitary), A037992 (infinitary), A293185, A306736, A307845, A309141, A318278, A322484, A335386.

f[p_, e_] := Floor[Log2[e]] + 2; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq = {}; sm = 0; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 1, 10^6}]; seq

cp's OEIS Frontend

A309181 Least number with exactly n nonunitary divisors.

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A174961 Number of non-unitary divisors of the n-th nonsquarefree number.

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A329882 Nonunitary superabundant numbers: numbers m such that nusigma(m)/m > nusigma(k)/k for all k < m, where nusigma(m) is the sum of nonunitary divisors of m (A048146).

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A329883 Nonunitary highly abundant numbers: numbers m such that nusigma(m) > nusigma(k) for all k < m, where s(n) is the sum of nonunitary divisors of n (A048146).

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A335386 Tri-unitary highly composite numbers: where the number of tri-unitary divisors (A335385) increases to a record.

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A333931 Recursive highly composite numbers: numbers with a record number of recursive divisors (A282446).

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A358253 Numbers with a record number of non-unitary square divisors.

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A348342 Noninfinitary highly composite numbers: where the number of noninfinitary divisors (A348341) increases to a record.

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A348632 Nonexponential highly composite numbers: where the number of nonexponential divisors (A160097) increases to a record.

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