A379221 -id:A379221 - 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)).

A379220 Square array A(n, k) = sigma((2n-1)^2) * sigma((2k-1)^2), read by antidiagonals.

Original entry on oeis.org

1, 13, 13, 31, 169, 31, 57, 403, 403, 57, 121, 741, 961, 741, 121, 133, 1573, 1767, 1767, 1573, 133, 183, 1729, 3751, 3249, 3751, 1729, 183, 403, 2379, 4123, 6897, 6897, 4123, 2379, 403, 307, 5239, 5673, 7581, 14641, 7581, 5673, 5239, 307, 381, 3991, 12493, 10431, 16093, 16093, 10431, 12493, 3991, 381, 741, 4953, 9517, 22971, 22143, 17689, 22143, 22971, 9517, 4953, 741

Offset: 1

Antti Karttunen, Dec 22 2024

Array is symmetric.

			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 |   13,   169,   403,   741,   1573,   1729,   2379,   5239,   3991,
   3  25 |   31,   403,   961,  1767,   3751,   4123,   5673,  12493,   9517,
   4  49 |   57,   741,  1767,  3249,   6897,   7581,  10431,  22971,  17499,
   5  81 |  121,  1573,  3751,  6897,  14641,  16093,  22143,  48763,  37147,
   6 121 |  133,  1729,  4123,  7581,  16093,  17689,  24339,  53599,  40831,
   7 169 |  183,  2379,  5673, 10431,  22143,  24339,  33489,  73749,  56181,
   8 225 |  403,  5239, 12493, 22971,  48763,  53599,  73749, 162409, 123721,
   9 289 |  307,  3991,  9517, 17499,  37147,  40831,  56181, 123721,  94249,
  10 361 |  381,  4953, 11811, 21717,  46101,  50673,  69723, 153543, 116967,
  11 441 |  741,  9633, 22971, 42237,  89661,  98553, 135603, 298623, 227487,
  12 529 |  553,  7189, 17143, 31521,  66913,  73549, 101199, 222859, 169771,
  13 625 |  781, 10153, 24211, 44517,  94501, 103873, 142923, 314743, 239767,
  14 729 | 1093, 14209, 33883, 62301, 132253, 145369, 200019, 440479, 335551,
  15 841 |  871, 11323, 27001, 49647, 105391, 115843, 159393, 351013, 267397,
  16 961 |  993, 12909, 30783, 56601, 120153, 132069, 181719, 400179, 304851,

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals
Index entries for sequences related to sigma(n).

Cf. A000203, A016754.
Cf. A379223 (the first row and the first column).
Cf. also A379221.

up_to = 66;
A379220sq(x,y) = (sigma((x+x-1)^2) * sigma((y+y-1)^2));
A379220list(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] = A379220sq(col,(a-(col-1))))); (v); };
v379220 = A379220list(up_to);
A379220(n) = v379220[n];

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

A(n, k) = A000203(A016754(n-1)) * A000203(A016754(k-1)). [NB: A016754 uses 0-based indexing]

A379224 The reversing binary representation of the sum of the divisors of the n-th odd square: a(n) = A065621(A379223(n)).

Original entry on oeis.org

1, 21, 35, 73, 137, 397, 475, 695, 855, 901, 1837, 1657, 1301, 3277, 1451, 1057, 2881, 2859, 3971, 7135, 3023, 2477, 5099, 6513, 7953, 4283, 7539, 12335, 13801, 5757, 4939, 12049, 14969, 12885, 9277, 13321, 16175, 26873, 9893, 10705, 27281, 11589, 28533, 29775, 8671, 31171, 22197, 29287, 28519, 17253, 30787, 31337

Offset: 1

Antti Karttunen, Dec 21 2024

nonn

The first column of square array A379221.

Cf. A065621, A379223.

A379224[n_] := BitXor[# - 1, 2*# - 1] & [DivisorSigma[1, (2*n - 1)^2]];
Array[A379224, 50] (* Paolo Xausa, Jan 23 2025 *)

A065621(n) = bitxor(n-1, n+n-1);
A379223(n) = sigma((2*n-1)^2);
A379224(n) = A065621(A379223(n));

a(n) = A065621(A379223(n)).

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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A379220 Square array A(n, k) = sigma((2n-1)^2) * sigma((2k-1)^2), read by antidiagonals.

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A379224 The reversing binary representation of the sum of the divisors of the n-th odd square: a(n) = A065621(A379223(n)).

Views

Author

Keywords

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Crossrefs

Programs

Mathematica

PARI

Formula