A072437 -id:A072437 - cp's OEIS frontend

A378194 Rectangular array, read by descending antidiagonals: row n shows the integers m such that the number of primes of the form 4k+3 (including multiplicities) that divide m is n-1.

1, 2, 3, 4, 6, 9, 5, 7, 18, 27, 8, 11, 21, 54, 81, 10, 12, 33, 63, 162, 243, 13, 14, 36, 99, 189, 486, 729, 16, 15, 42, 108, 297, 567, 1458, 2187, 17, 19, 45, 126, 324, 891, 1701, 4374, 6561, 20, 22, 49, 135, 378, 972, 2673, 5103, 13122, 19683, 25, 23, 57, 147, 405, 1134, 2916, 8019, 15309, 39366, 59049, 26, 24, 66, 171, 441, 1215, 3402, 8748, 24057, 45927, 118098, 177147

Offset: 1

Clark Kimberling, Jan 14 2025

Every positive integer occurs exactly once.

			Corner:
      1     2     4     5     8     10     13     16      17
      3     6     7    11    12     14     15     19      22
      9    18    21    33    36     42     45     49      57
     27    54    63    99   108    126    135    147     171
     81   162   189   297   324    378    405    441     513
    243   486   567   891   972   1134   1215   1323    1539
    729  1458  1701  2673  2916   3402   3645   3969    4617
   2187  4374  5103  8019  8748  10206  10935  11907   13851

Cf. A065339, A002144, A002145, A376961, A378193, A072437 (row 1), A000244 (column 0), A025192 (column 1).

A378194 := proc(n, k)
    option remember;
    local a;
    if k = 0 then
        0;
    else
        for a from procname(n, k-1)+1 do
            if A065339(a) = n-1 then
                return a;
            end if;
        end do;
    end if;
end proc:
seq(seq( A378194(n, d-n), n=1..d-1), d=2..10) ; # R. J. Mathar, Jan 28 2025

u = Map[Map[#[[1]] &, #] &, GatherBy[
    SortBy[Map[{#, 1 + Count[Map[IntegerQ[(# - 3)/4] && PrimeQ[#] &,
             Flatten[Map[ConstantArray[#[[1]], #[[2]]] &,
             FactorInteger[#]]]], True]} &,
      Range[24000]], #[[2]] &], #[[2]] &]];
r[m_] := Take[u[[m]], 10];
w[m_, n_] := r[m][[n]];
Grid[Table[w[m, n], {m, 1, 8}, {n, 1, 9}]]   (* array *)
Table[w[n - k + 1, k], {n, 8}, {k, n, 1, -1}] // Flatten  (* sequence *)
(* Peter J. C. Moses, Nov 19 2024 *)

Definition corrected. - R. J. Mathar, Jan 28 2025

A371014 The number of divisors of n that are the sum of 2 squares.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 1, 4, 2, 4, 1, 3, 2, 2, 2, 5, 2, 4, 1, 6, 1, 2, 1, 4, 3, 4, 2, 3, 2, 4, 1, 6, 1, 4, 2, 6, 2, 2, 2, 8, 2, 2, 1, 3, 4, 2, 1, 5, 2, 6, 2, 6, 2, 4, 2, 4, 1, 4, 1, 6, 2, 2, 2, 7, 4, 2, 1, 6, 1, 4, 1, 8, 2, 4, 3, 3, 1, 4, 1, 10, 3, 4, 1, 3, 4, 2, 2

Offset: 1

Amiram Eldar, Mar 08 2024

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

Cf. A001481, A004614, A046951, A072437, A167181, A371015.

f[p_, e_] := If[Mod[p, 4] == 3, Floor[e/2] + 1, e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1]%4 == 3, f[i, 2]\2 + 1, f[i, 2] + 1));}

Multiplicative with a(p^e) = floor(e/2) + 1 if p == 3 (mod 4), and e+1 otherwise.
a(n) = A000005(n) if and only if n is in A072437.
a(n) = A046951(n) if and only if n is in A004614.
a(n) = 1 if and only if n is in A167181.

A382092 Values taken by gcd(a^2 + b^2 + c^2, abc), where a, b, c are positive integers.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 27, 29, 32, 34, 36, 37, 40, 41, 45, 49, 50, 52, 53, 54, 58, 61, 64, 65, 68, 72, 73, 74, 80, 81, 82, 85, 89, 90, 97, 98, 100, 101, 104, 106, 108, 109, 113, 116, 117, 121, 122, 125, 128, 130, 135, 136, 137

Offset: 1

Yifan Xie, Mar 29 2025

Numbers k such that for each prime p == 3 (mod 4) dividing k, v_p(k) > 1, where v_p(k) is the p-valuation of k.

			3 is not a term because triples (a, b, c) of positive integers such that gcd(a^2 + b^2 + c^2, a*b*c) = 3 do not exist.
9 is a term because gcd(3^2 + 3^2 + 9^2, 3*3*9) = gcd(99, 81) = 9.

Harvard-MIT Math Tournament, HMMT February 2025 Team Round Solutions, Problem 10.

Cf. A002145.

A001481 and A072437 are subsequences.

isok(n) = {
    my(f = factor(n));
    for(i=1, #f[,1], if(f[i, 1] % 4 == 3 && f[i, 2] <= 1, return(0)));
    return(1);
}

cp's OEIS Frontend

A378194 Rectangular array, read by descending antidiagonals: row n shows the integers m such that the number of primes of the form 4k+3 (including multiplicities) that divide m is n-1.

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A371014 The number of divisors of n that are the sum of 2 squares.

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PARI

Formula

A382092 Values taken by gcd(a^2 + b^2 + c^2, abc), where a, b, c are positive integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

cp's OEIS Frontend

A378194 Rectangular array, read by descending antidiagonals: row n shows the integers m such that the number of primes of the form 4k+3 (including multiplicities) that divide m is n-1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions

A371014 The number of divisors of n that are the sum of 2 squares.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A382092 Values taken by gcd(a^2 + b^2 + c^2, a*b*c), where a, b, c are positive integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A382092 Values taken by gcd(a^2 + b^2 + c^2, abc), where a, b, c are positive integers.