A065768 -id:A065768 - cp's OEIS frontend

A379223 Sum of the divisors of the n-th odd square: a(n) = sigma((2*n-1)^2).

1, 13, 31, 57, 121, 133, 183, 403, 307, 381, 741, 553, 781, 1093, 871, 993, 1729, 1767, 1407, 2379, 1723, 1893, 3751, 2257, 2801, 3991, 2863, 4123, 4953, 3541, 3783, 6897, 5673, 4557, 7189, 5113, 5403, 10153, 7581, 6321, 9841, 6973, 9517, 11323, 8011, 10431, 12909, 11811, 9507, 16093, 10303, 10713, 22971, 11557

Offset: 1

Antti Karttunen, Dec 21 2024

nonn

Sequence contains duplicates. For example, a(314) = a(375) = 658749 and a(1007) = a(1279) = 6540807.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A016754, A065768 (same sequence sorted into ascending order, with duplicates removed), A379224 [= A065621(a(n))].

First row and column of array A379220, first row of array A379221.

Table[DivisorSigma[1,(2n-1)^2],{n,54}] (* James C. McMahon, Dec 22 2024 *)

PARI
```
A379223(n) = sigma((2*n-1)^2);
```

from math import prod
from sympy import factorint
def A379223(n): return prod((p**((e<<1)|1)-1)//(p-1) for p, e in factorint((n<<1)-1).items()) # Chai Wah Wu, Dec 21 2024

a(n) = A000203(A016754(n-1)).

A379221 Square array A(n, k) = A048720(A065621(sigma((2n-1)^2)), sigma((2k-1)^2)), read by falling antidiagonals, (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), etc.

Original entry on oeis.org

1, 13, 21, 31, 233, 35, 57, 403, 439, 73, 121, 845, 961, 805, 137, 133, 1549, 1899, 1831, 1765, 397, 183, 2753, 4011, 4017, 3943, 3025, 475, 403, 2331, 4399, 7665, 7537, 4123, 2159, 695, 307, 7919, 5945, 9709, 16177, 9365, 5737, 7635, 855, 381, 5839, 12501, 10447, 17965, 18389, 10707, 13261, 5299, 901, 741, 4953, 9525, 27083, 24207, 49465, 24339, 27295, 10093, 4537, 1837

Offset: 1

Antti Karttunen, Dec 22 2024

			The top left corner of the array:
   n\k   |    1      2      3      4       5       6       7       8       9
(*2-1)^2 |    1      9     25     49      81     121     169     225     289
---------+-------------------------------------------------------------------
   1   1 |    1,    13,    31,    57,    121,    133,    183,    403,    307,
   2   9 |   21,   233,   403,   845,   1549,   2753,   2331,   7919,   5839,
   3  25 |   35,   439,   961,  1899,   4011,   4399,   5945,  12501,   9525,
   4  49 |   73,   805,  1831,  4017,   7665,   9709,  10447,  27083,  17515,
   5  81 |  137,  1765,  3943,  7537,  16177,  17965,  24207,  50315,  37163,
   6 121 |  397,  3025,  4123,  9365,  18389,  49465,  60243,  86471, 108263,
   7 169 |  475,  2159,  5737, 10707,  24339,  60215,  52817,  76125, 131005,
   8 225 |  695,  7635, 13261, 27295,  51039,  87019,  76565, 245801, 183625,
   9 289 |  855,  5299, 10093, 18047,  37823, 107915, 130229, 183305, 200041,
  10 361 |  901,  4537, 12003, 22365,  46621, 118545,  98539, 162655, 248191,
  11 441 | 1837,  8945, 24187, 43317,  90741, 232729, 201779, 311335, 504583,
  12 529 | 1657, 11349, 18231, 40193,  66369, 205597, 231263, 338075, 449339,
  13 625 | 1301, 14825, 25235, 56909, 105229, 170945, 156187, 508399, 387535,
  14 729 | 3277, 22929, 36059, 81877, 134293, 416121, 464275, 684551, 888103,
  15 841 | 1451, 15967, 28601, 50979, 110051, 181895, 139777, 469709, 346669,
  16 961 | 1057, 13741, 32767, 58137, 125785, 132133, 182871, 425971, 322387,

Cf. A000203, A016754, A048720, A065621, A277320.
Cf. A379223 (row 1), A379224 (column 1).
Cf. A379121, A379122, A379123, A379124, A379125.
Cf. also A065768, A379220.

up_to = 66;
A048720(b, c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1, n+n-1);
A379221sq(x,y) = A048720(A065621(sigma((x+x-1)^2)), sigma((y+y-1)^2));
A379221list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A379221sq(col,(a-(col-1))))); (v); };
v379221 = A379221list(up_to);
A379221(n) = v379221[n];

A(n, k) = A277320(A379223(n), A379223(k)).

A065767 Intersection of A065764 and A065765: n such that x and y exist with sigma[x^2] = n = sigma[2*(y^2)].

Original entry on oeis.org

399, 5187, 12369, 34671, 48279, 73017, 80199, 122493, 152019, 160797, 220647, 259749, 311619, 347529, 396207, 436107, 561393, 687477, 755307, 900543, 949221, 1042587, 1074801, 1142337, 1412859, 1496649, 1509417, 1592409, 1818243

Offset: 1

Labos Elemer, Nov 19 2001

nonn

			n = 399 = sigma[14^2] = sigma[2*(11^2)] = 1+2+4+7+14+28+49+98+196 = 1+2+11+22+121+242; also sigma[42.42] = sigma[2.33.33] = sigma[1764] = sigma[2378] = 5187.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A028982, A000203, A000290, A065764-A065768.

Intersection[Table[DivisorSigma[1, w^2], {w, 1, 10000}], Table[DivisorSigma[1, 2*(w^2)], {w, 1, 10000}]]

cp's OEIS Frontend

A379223 Sum of the divisors of the n-th odd square: a(n) = sigma((2*n-1)^2).

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A379221 Square array A(n, k) = A048720(A065621(sigma((2n-1)^2)), sigma((2k-1)^2)), read by falling antidiagonals, (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), etc.

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Author

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Examples

Links

Crossrefs

Programs

PARI

Formula

A065767 Intersection of A065764 and A065765: n such that x and y exist with sigma[x^2] = n = sigma[2*(y^2)].

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Author

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Examples

Links

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Programs

Mathematica