A279254 -id:A279254 - cp's OEIS frontend

A302249 a(n) is the number of divisors of A279254(n) in Gaussian integers.

1, 3, 5, 6, 7, 12, 20, 24, 28, 40, 45, 56, 60, 63, 72, 80, 90, 96, 112, 126, 160, 162, 180, 224, 240, 252, 288, 360, 384, 504, 540, 640, 648, 720, 756, 896, 960, 1008, 1152, 1440, 2016, 2160, 2592, 3024, 3168, 3584, 3888, 4032

Offset: 1

Jianing Song, Apr 04 2018

nonn

The divisors are counted mod associates.

Conjecture: a(14) = 63 is the largest odd term.

			A279254(14) = 200 and 200 has 63 divisors in Gaussian integers (up to association), so a(14) = 63.

Amiram Eldar, Table of n, a(n) for n = 1..394 (calculated from the b-file at A279254)
Index entries for Gaussian integers and primes

Cf. A062327, A279254.

With[{s = Array[DivisorSigma[0, #, GaussianIntegers -> True] &, 10^6]}, Union@ FoldList[Max, s]] (* Michael De Vlieger, Apr 05 2018 *)

b(n)= \\ A062327
{
    my(r=1, f=factor(n));
    for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);
        if(p==2, r*=(2*e+1));
        if(p%4==1, r*=(e+1)^2);
        if(p%4==3, r*=(e+1));
    );
    return(r);
}
{ my(r=0, t); for(n=1, 10^6, t=b(n); if(t>r, r=t; print1(t, ", "))); }
\\ Joerg Arndt, Apr 04 2018

a(n) = A062327(A279254(n)).

A332320 Numbers k that are highly norm-abundant in Gaussian integers, i.e., A103230(m) < A103230(k) for all m < k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 15, 18, 20, 26, 30, 50, 60, 70, 78, 90, 130, 150, 170, 180, 210, 260, 270, 330, 390, 510, 630, 780, 870, 910, 990, 1020, 1050, 1110, 1170, 1530, 1890, 1950, 2210, 2340, 2550, 2730, 3510, 4290, 4590, 5070, 5460, 5610, 5850, 6630, 8190, 10530

Offset: 1

Amiram Eldar, Feb 09 2020

nonn

Analogous to highly abundant numbers (A002093), with the norm of the sum of divisors function generalized for Gaussian integers (A103230) instead of the sum of divisors function (A000203).

			The first 6 terms of A103230 are 1, 13, 16, 41, 80, 208, 64, 113, 169, 1040. The record values occur at n = 1, 2, 3, 4, 5, 6, 10, the first 7 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..375
Robert Spira, The Complex Sum of Divisors, The American Mathematical Monthly, Vol. 68, No. 2 (1961), pp. 120-124.

Cf. A000203, A002093, A103228, A103229, A103230, A279254, A332321.

s[n_] := Abs[DivisorSigma[1, n, GaussianIntegers -> True]]^2; sm = 0; seq = {}; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 1,10^4}]; seq

A332321 Numbers k that are norm-superabundant in Gaussian integers, i.e., A103230(m)/m^2 < A103230(k)/k^2 for all m < k.

Original entry on oeis.org

1, 2, 6, 10, 30, 90, 130, 210, 390, 1170, 2730, 5850, 6630, 19890, 46410, 99450, 139230, 192270, 576810, 1345890, 2884050, 4037670, 7883070, 12113010, 20188350, 23649210, 44414370, 49797930, 55181490, 118246050, 149393790, 165544470, 496633410, 746968950, 827722350

Offset: 1

Amiram Eldar, Feb 09 2020

nonn

Analogous to superabundant numbers (A004394), with the magnitude of the sum of divisors function generalized for Gaussian integers (sqrt(A103230)) instead of the sum of divisors function (A000203).

			The first 6 terms of A103230 are 1, 13, 16, 41, 80, 208. The corresponding values of A103230(n)/n^2 are 1, 3.25, 1.777..., 2.5625, 3.2, 5.777... and the record values occur at n = 1, 2, 6, the first 3 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..40
Robert Spira, The Complex Sum of Divisors, The American Mathematical Monthly, Vol. 68, No. 2 (1961), pp. 120-124.

Cf. A000203, A004394, A103228, A103229, A103230, A279254, A332320.

r[n_] := Abs[DivisorSigma[1, n, GaussianIntegers -> True]]^2/n^2; rm = 0; seq = {}; Do[r1 = r[n]; If[r1 > rm, rm = r1; AppendTo[seq, n]], {n, 1, 6*10^5}]; seq

A323392 Positive integers that have a record number of divisors in Eisenstein integers.

Original entry on oeis.org

1, 2, 3, 6, 12, 18, 21, 36, 42, 84, 126, 168, 252, 420, 504, 546, 1008, 1092, 1638, 2184, 3276, 5460, 6552, 7644, 9828, 10374, 13104, 15288, 16380, 20748, 31122, 38220, 41496, 62244, 103740, 124488, 145236, 186732, 207480, 248976, 290472, 311220, 435708, 622440, 726180, 871416

Offset: 1

Jianing Song, Jan 13 2019

nonn

Indices of records in A319442.
Analog of A002182 and A279254, which list the positive integers that have a record number of divisors in rational integers and Gaussian integers respectively.
It seems that 21 is the largest odd term.

			252 has 60 divisors up to association in Eisenstein integers, more than any previous positive integers, so 252 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..100
Wikipedia, Eisenstein integer

Cf. A002182, A279254, A319442.

For the number of divisors of a(n) see A323393.

vmax:= 0: recinds:= NULL:
for n from 1 to 100000 do
    v := A319442(n);
    if v > vmax then vmax:= v; recinds:= recinds, n fi
od:
recinds; # Peter Luschny, Jan 19 2019

f[p_, e_] := Switch[Mod[p, 3], 0, 2*e + 1, 1, (e + 1)^2, 2, e + 1]; eisNumDiv[1] = 1; eisNumDiv[n_] := Times @@ f @@@ FactorInteger[n]; seq = {}; emax = 0; Do[eis = eisNumDiv[n]; If[eis > emax, emax = eis; AppendTo[seq, n]], {n, 1, 10^6}]; seq (* Amiram Eldar, Mar 02 2020 *)

my(r=0, t); for(n=1, 10^6, t=A319442(n); if(t>r, r=t; print1(n, ", ")));

A335853 Numbers that are highly powerful in Gaussian integers.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 100, 200, 400, 500, 800, 1000, 2000, 4000, 5000, 8000, 10000, 18000, 20000, 27000, 36000, 40000, 50000, 54000, 80000, 90000, 108000, 135000, 180000, 216000, 270000, 450000, 540000, 810000, 1080000, 1350000, 1620000, 2160000, 2700000

Offset: 1

Amiram Eldar, Jun 26 2020

nonn

Numbers with a record value of the product of the exponents in the prime factorization in Gaussian integers (A335852). Equivalently, numbers with a record number of powerful divisors in Gaussian integers.

The corresponding record values are 1, 2, 4, 6, 8, 10, 12, 16, 24, 32, 36, 40, 54, 72, 90, 96, ... (see the link for more values).

			The factorization of 1, 2, 3 and 4 in Gaussian integers are 1, -i*(1+i)^2, 3 and -(1+i)^4, and the corresponding products of the exponents are 1, 2, 1 and 4. The record values, 1, 2 and 4, occur at 1, 2 and 4 that are the first 3 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..81
Amiram Eldar, Table of n, a(n), A335852(a(n)) for n = 1..81

Cf. A005934, A279254, A335851, A335852.

With[{s = Array[Times @@ FactorInteger[#, GaussianIntegers -> True][[All, -1]] &, 10^5]}, Map[FirstPosition[s, #][[1]] &, Union@FoldList[Max, s]]] (* after Michael De Vlieger at A005934 *)

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A302249 a(n) is the number of divisors of A279254(n) in Gaussian integers.

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A332320 Numbers k that are highly norm-abundant in Gaussian integers, i.e., A103230(m) < A103230(k) for all m < k.

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A332321 Numbers k that are norm-superabundant in Gaussian integers, i.e., A103230(m)/m^2 < A103230(k)/k^2 for all m < k.

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A323392 Positive integers that have a record number of divisors in Eisenstein integers.

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A335853 Numbers that are highly powerful in Gaussian integers.

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