A379223 -id:A379223 - cp's OEIS frontend

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

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)).

A065768 Numbers that are sums of divisors of the odd squares; Intersection of A065764 and A065766, written in ascending order and duplicates removed.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Nov 19 2001

nonn

Terms are the sum of the odd divisors (A000593) of the odd squares (A016754), written in ascending order. Subsequence of the odd terms of A274790. - Timothy L. Tiffin, Feb 12 2022

Equally, the sum of divisors (A000203) as only odd divisors are present in odd squares. - Antti Karttunen, Dec 22 2024

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

Cf. A000203, A000593, A016754, A065764, A065766, A274790.

Sequence A379223 sorted into ascending order, with duplicates removed.

f1[p_, e_] := (p^(2*e + 1) - 1)/(p - 1); s1[1] = 1; s1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := (p^(2*e + 1) - 1)/(p - 1); f2[2, e_] := (4^(e + 1) - 1)/3; s2[1] = 1; s2[n_] := Times @@ f2 @@@ FactorInteger[n]; seq[max_] := Intersection[Select[Array[s1, max], # < max^2 &], Select[Array[s2, max], # < max^2 &]]; seq[101] (* Amiram Eldar, Aug 24 2024 *)

Old definition clarified and Timothy L. Tiffin's comment adopted as a new primary definition - Antti Karttunen, Dec 22 2024

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)).

A379482 a(n) = sigma(A003961(n^2)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 13, 31, 121, 57, 403, 133, 1093, 781, 741, 183, 3751, 307, 1729, 1767, 9841, 381, 10153, 553, 6897, 4123, 2379, 871, 33883, 2801, 3991, 19531, 16093, 993, 22971, 1407, 88573, 5673, 4953, 7581, 94501, 1723, 7189, 9517, 62301, 1893, 53599, 2257, 22143, 44517, 11323, 2863, 305071, 16105, 36413, 11811, 37147, 3541

Offset: 1

Antti Karttunen, Dec 27 2024

Cf. A000290, A003973, A048673, A151800, A378231, A379223, A379481, A379483, A379484 [= A038500(a(n))].

Cf. also A337336, A337337.

{1}~Join~Array[DivisorSigma[1, #] &[Apply[Times, Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]] ]^2] &, 52, 2] (* Michael De Vlieger, Dec 27 2024 *)

A379482(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1); f[i, 2] *= 2); sigma(factorback(f)); };

Multiplicative with a(p^e) = (q^(2e+1) - 1)/(q-1), where q = nextprime(p) = A151800(p).
a(n) = A000203(A379481(n)) = A003973(A000290(n)).
a(n) = A379223(A048673(n)).
a(n) = 2*A379481(n) - A378231(n).

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]

cp's OEIS Frontend

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

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A065768 Numbers that are sums of divisors of the odd squares; Intersection of A065764 and A065766, written in ascending order and duplicates removed.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A379482 a(n) = sigma(A003961(n^2)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula