A322484 -id:A322484 - cp's OEIS frontend

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

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).

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

A358263 Numbers with a record number of noninfinitary square divisors.

Original entry on oeis.org

1, 16, 144, 256, 1296, 2304, 20736, 57600, 331776, 518400, 2822400, 8294400, 12960000, 25401600, 132710400, 207360000, 228614400, 406425600, 635040000, 2057529600, 3073593600, 6502809600, 10160640000, 27662342400, 31116960000, 51438240000, 76839840000, 248961081600

Offset: 1

Amiram Eldar, Nov 06 2022

nonn

Numbers m such that A358261(m) > A358261(k) for all k < m.

The corresponding record values are 0, 1, 2, 3, 5, 6, 11, 12, 13, 22, 24, 26, 37, 44, 46, 47, 48, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..187
Amiram Eldar, Table of n, a(n), A358261(a(n)) for n = 1..187

Cf. A358261, A358262.
Subsequence of A025487.
Similar sequences: A002182, A002110 (unitary), A037992 (infinitary), A293185, A306736, A307845, A309141, A318278, A322484, A335386, A348632, A358253.

f1[p_, e_] := 1 + Floor[e/2]; f2[p_, e_] := 2^DigitCount[If[OddQ[e], e - 1, e], 2, 1]; 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, 6*10^5}]; s

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

A325839 Exponentially-odd coreful highly composite numbers: numbers with record values of the number of exponentially odd coreful divisors (A325837).

Original entry on oeis.org

1, 8, 32, 128, 512, 864, 3456, 7776, 13824, 31104, 124416, 279936, 497664, 1119744, 1990656, 3888000, 10077696, 15552000, 34992000, 62208000, 139968000, 248832000, 388800000, 559872000, 1259712000, 1555200000, 2239488000, 3499200000, 6220800000, 8957952000

Offset: 1

Amiram Eldar, Sep 07 2019

nonn

The corresponding record numbers of exponentially-odd divisors are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 30, 32, 36, 40, 42, 45, 48, 50, 54, 56, 60, 63, 64, ... (see the link for more terms).
The even version of this sequence is A046952 which is the sequence of numbers with record number of square divisors (only even exponents, A046951).
Numbers with record values of the number of exponentially odd divisors are the same as the numbers with record values of the number of semi-unitary divisors (A322484). - Amiram Eldar, Sep 08 2023

Amiram Eldar, Table of n, a(n) for n = 1..290
Amiram Eldar, Table of n, a(n), A325837(a(n)) for n = 1..290

Cf. A033634, A046951, A046952, A268335, A322484, A325837.

Subsequence of A025487.

fun[p_,e_] := Floor[(e+1)/2]; a[n_] := Times@@(fun@@@FactorInteger[n]); am = 0; s={}; Do[a1=a[n]; If[a1>am, am=a1; AppendTo[s, n]], {n, 1, 300000}]; s

Name corrected by Amiram Eldar, Sep 08 2023

A377710 Numbers that have a record number of infinitary divisors that are exponentially odd numbers (A268335).

Original entry on oeis.org

1, 2, 6, 24, 30, 120, 210, 840, 2310, 7560, 9240, 30030, 83160, 120120, 510510, 1081080, 1921920, 2042040, 9699690, 18378360, 32672640, 38798760, 223092870, 349188840, 620780160, 892371480, 5587021440, 6469693230, 8031343320, 14277943680, 25878772920, 128501493120

Offset: 1

Amiram Eldar, Nov 04 2024

nonn

First differs from A322484 at n = 15.
Indices of records in A363825.
The corresponding record values are 1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, 72, 96, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..139
Amiram Eldar, Table of n, a(n), A363825(a(n)) for n = 1..139

Cf. A268335, A363825, A322484.

Subsequence of A025487.

f[p_, e_] := 1 + If[OddQ[e], 2^DigitCount[e-1, 2, 1], 0]; d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n]; v = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]; seq = {}; dm = 0; Do[If[(dk = d[v[[k]]]) > dm, dm = dk; AppendTo[seq, v[[k]]]], {k, 1, Length[v]}]; seq

cp's OEIS Frontend

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

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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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A353899 Indices of records in A353898.

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A358263 Numbers with a record number of noninfinitary square divisors.

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PARI

A325839 Exponentially-odd coreful highly composite numbers: numbers with record values of the number of exponentially odd coreful divisors (A325837).

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A377710 Numbers that have a record number of infinitary divisors that are exponentially odd numbers (A268335).

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