A062327 -id:A062327 - cp's OEIS frontend

A279254 Positive integers that have a record number of divisors in Gaussian integers.

1, 2, 4, 6, 8, 10, 20, 30, 40, 60, 100, 120, 180, 200, 240, 260, 300, 390, 520, 600, 780, 1200, 1300, 1560, 2340, 2600, 3120, 3900, 6630, 7800, 11700, 13260, 15600, 22100, 23400, 26520, 39780, 44200, 53040, 66300, 132600, 198900, 265200, 397800, 530400, 663000, 795600, 928200

Offset: 1

J. Lowell, Dec 08 2016

nonn

Indices of records in A062327.

Charles R Greathouse IV, Table of n, a(n) for n = 1..394

Subsequence of A097752.

Cf. A062327, A002182 (factorization over the integers).

With[{s = Array[DivisorSigma[0, #, GaussianIntegers -> True] &, 10^6]}, Map[FirstPosition[s, #][[1]] &, 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(n,", "))); }
\\ Joerg Arndt, Dec 09 2016

Terms a(11) and beyond from Joerg Arndt, Dec 09 2016

A332313 Numbers k such that k, k + 1 and k + 2 have the same number of divisors in Gaussian integers.

Original entry on oeis.org

23824, 38574, 52974, 62224, 71406, 105424, 110574, 191824, 201616, 209424, 240174, 249775, 282896, 285102, 297774, 326574, 340974, 375824, 393424, 407824, 440656, 451024, 496174, 509776, 553774, 587536, 599632, 600174, 606032, 623824, 628974, 631376, 667024, 672174

Offset: 1

Amiram Eldar, Feb 09 2020

nonn

			23824 is a term since 23824, 23825 and 23826 each have 36 divisors in Gaussian integers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005238, A062327, A332312, A332314.

Flatten[Position[Partition[DivisorSigma[0, Range[3*10^5], GaussianIntegers -> True], 3, 1], {x_, x_, x_}]] (* after Harvey P. Dale at A005238 *)

A062711 Number of prime Gaussian integers z=a+bi with |z|<=n.

Original entry on oeis.org

0, 1, 4, 6, 8, 10, 15, 19, 21, 25, 32, 34, 38, 44, 46, 52, 60, 66, 73, 79, 87, 93, 98, 104, 114, 122, 128, 138, 146, 154, 163, 173, 181, 193, 203, 213, 221, 231, 239, 245, 259, 273, 280, 294, 304, 316, 327, 343, 359, 369

Offset: 1

Reiner Martin, Jul 14 2001

Cf. A000328, A062327, A000720.

m = 50;
t = Table[x + y I, {x, -m, m}, {y, -m, m}] // Flatten[#, 1]& // Select[#, PrimeQ[#, GaussianIntegers -> True]& ]& // Sort // DeleteDuplicates[#, Abs[#1] == Abs[#2] && MatchQ[#1 /#2 , 1|-1|I|-I]& ]&;
a[n_] := Select[t, Abs[#] <= n&] // Length;
Array[a, m] (* Jean-François Alcover, Jul 29 2016 *)

Two prime Gaussian integers are not counted separately if they are associated, i.e. if their quotient is a unit (1, i, -1 or -i).
Similar to the ordinary prime number theorem (see A000720) we have the asymptotic expression: a(n) ~ n^2/(2 * log(n)) - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 16 2001
a(1)=0, a(n)=1+A066339(n^2)+A066490(n) for n>0. - T. D. Noe, Feb 20 2007

A332312 Numbers k such that k and k + 1 have the same number of divisors in Gaussian integers.

Original entry on oeis.org

62, 153, 175, 206, 278, 404, 422, 494, 657, 774, 801, 833, 854, 873, 891, 926, 1017, 1070, 1126, 1142, 1233, 1322, 1424, 1502, 1617, 1718, 1737, 1881, 1910, 1953, 2097, 2222, 2302, 2673, 2694, 2793, 2798, 2817, 2825, 2961, 2996, 3014, 3174, 3177, 3266, 3446, 3577

Offset: 1

Amiram Eldar, Feb 09 2020

nonn

			62 is a term since both 62 and 63 have 6 divisors in Gaussian integers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005237, A062327, A332313, A332314.

SequencePosition[DivisorSigma[0, Range[3500], GaussianIntegers -> True], {x_, x_}][[All, 1]] (* after Harvey P. Dale at A005237 *)

A332314 Numbers k such that k, k + 1, k + 2 and k + 3 have the same number of divisors in Gaussian integers.

Original entry on oeis.org

263449773, 334047725, 760228973, 862305773, 1965540624, 2136055725, 2362380525, 2477365422, 2515570575, 2613782223, 2939626925, 3181603023, 3814526223, 3987335022, 4610697039, 4771214574, 4981539822, 5018728272, 5035157775, 5186567824, 6189727725, 6329159823, 6569396973

Offset: 1

Amiram Eldar, Feb 09 2020

nonn

			263449773 is a term since 263449773, 263449774, 263449775 and 263449776 each have 72 divisors in Gaussian integers.

Cf. A006601, A062327, A332312, A332313.

gaussNumDiv[n_] := DivisorSigma[0, n, GaussianIntegers -> True]; m = 4; s = gaussNumDiv /@ Range[m]; seq = {}; n = m + 1; While[Length[seq] < 10, If[Length @ Union[s] == 1, AppendTo[seq, n - m + 1]]; n++; s = Join[Rest[s], {gaussNumDiv[n]}]]; seq

A332476 The number of unitary divisors of n in Gaussian integers.

Original entry on oeis.org

1, 2, 2, 2, 4, 4, 2, 2, 2, 8, 2, 4, 4, 4, 8, 2, 4, 4, 2, 8, 4, 4, 2, 4, 4, 8, 2, 4, 4, 16, 2, 2, 4, 8, 8, 4, 4, 4, 8, 8, 4, 8, 2, 4, 8, 4, 2, 4, 2, 8, 8, 8, 4, 4, 8, 4, 4, 8, 2, 16, 4, 4, 4, 2, 16, 8, 2, 8, 4, 16, 2, 4, 4, 8, 8, 4, 4, 16, 2, 8, 2, 8, 2, 8, 16

Offset: 1

Amiram Eldar, Feb 13 2020

			a(2) = 2 since 2 = -i * (1 + i)^2, so it has 2 unitary divisors (up to associates): 1 and (1 + i)^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A034444, A062327, A086275, A332472, A332473, A332474.

f[p_, e_] := If[Abs[p] == 1, 1, 2]; a[n_] := Times @@ f @@@ FactorInteger[n, GaussianIntegers -> True]; Array[a, 100]

Multiplicative with a(p^e) = 4 if p == 1 (mod 4) and 2 otherwise.

a(n) = 2^A086275(n).

A335852 Product of the exponents in the prime factorization of n in the ring of Gaussian integers.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 26 2020

a(n) is also the number of divisors of n in Gaussian integers that are powerful (A335851).

			a(2) = 2 since 2 = -i * (1 + i)^2 and the Gaussian prime (1 + i) has an exponent 2.
a(100) = 16 since 100 = (1 + i)^4 * (1 + 2*i)^2 * (2 + i)^2 and 4*2*2 = 16.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Gaussian Integer.
Wikipedia, Gaussian integer.

Cf. A005361, A062327, A335851.

a[n_] := Times @@ FactorInteger[n, GaussianIntegers -> True][[All, 2]]; Array[a, 100]

a(n) = my (f = factor(n*I)); f[1,1] /= I; prod(k=1, #f~, f[k,2]); \\ Michel Marcus, Jun 28 2020

Multiplicative with a(p^e) = 2*e if p = 2, e if p == 3 (mod 4) and e^2 if p == 1 (mod 4).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (5/2) * Product_{p prime == 3 (mod 4)} (p^2 - p + 1)/(p*(p-1)) * Product_{p prime == 3 (mod 1)} (p^4 - 3*p^3 + 6*p^2 - 5*p + 1)/(p*(p-1)^3) = 3.73805905189... . - Amiram Eldar, Oct 15 2022

A351851 Numbers that are divisible by the number of their divisors over the Gaussian integers.

Original entry on oeis.org

1, 6, 9, 18, 20, 56, 126, 168, 180, 198, 280, 342, 352, 414, 432, 441, 486, 504, 558, 616, 625, 728, 774, 832, 846, 952, 1056, 1062, 1064, 1089, 1176, 1206, 1278, 1288, 1422, 1494, 1512, 1624, 1736, 1760, 1848, 1854, 1920, 1926, 2025, 2072, 2160, 2286, 2296, 2358

Offset: 1

Amiram Eldar, Feb 22 2022

nonn

Numbers k such that A062327(k) | k.

All the odd terms are squares.

			6 is a term since it is divisible by A062327(6) = 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A033950, A048166, A062327, A351853.

Select[Range[2400], Divisible[#, DivisorSigma[0, #, GaussianIntegers -> True]] &]

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

Original entry on oeis.org

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)).

A302252 Smallest number with exactly n divisors in Gaussian integers.

Original entry on oeis.org

1, 3, 2, 5, 4, 6, 8, 15, 16, 12, 32, 10, 64, 24, 36, 65, 256, 48, 512, 20, 72, 96, 2048, 30, 324, 192, 50, 40, 16384, 252, 32768, 195, 288, 768, 648, 80, 262144, 1536, 576, 60, 1048576, 504, 2097152, 160, 100, 6144, 8388608, 130, 5832, 1875, 2304, 320

Offset: 1

Jianing Song, Apr 04 2018

nonn

The divisors are counted up to association.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Index entries for Gaussian integers and primes.

Cf. A005179, A062327.

a[n_] := If[n > 2 && PrimeQ[n], 2^((n-1)/2), Block[{k=1}, While[ DivisorSigma[0, k, GaussianIntegers -> True] != n, k++]; k]]; Array[a, 52] (* Giovanni Resta, Apr 04 2018 *)

nbd(n) = {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);}  \\ A062327
a(n) = {my(k=1); while (nbd(k) != n, k++); k;} \\ Michel Marcus, Apr 26 2018

For prime p > 2, a(p) = 2^((p-1)/2) = sqrt(A005179(p)).

More terms from Giovanni Resta, Apr 04 2018

cp's OEIS Frontend

A279254 Positive integers that have a record number of divisors in Gaussian integers.

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A332313 Numbers k such that k, k + 1 and k + 2 have the same number of divisors in Gaussian integers.

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A062711 Number of prime Gaussian integers z=a+bi with |z|<=n.

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A332312 Numbers k such that k and k + 1 have the same number of divisors in Gaussian integers.

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A332314 Numbers k such that k, k + 1, k + 2 and k + 3 have the same number of divisors in Gaussian integers.

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A332476 The number of unitary divisors of n in Gaussian integers.

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A335852 Product of the exponents in the prime factorization of n in the ring of Gaussian integers.

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A351851 Numbers that are divisible by the number of their divisors over the Gaussian integers.

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