cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A379121 Odd squares k for which A379113(k) > 1, i.e., k that have a proper unitary divisor d > 1 such that A048720(A065621(sigma(d)),sigma(k/d)) is equal to sigma(k).

Original entry on oeis.org

225, 3025, 3249, 12321, 29241, 38025, 91809, 216225, 247009, 354025, 408321, 751689, 772641, 855625, 919681, 1366561, 1595169, 3814209, 9828225, 11189025, 12173121, 12709225, 29430625, 47927929, 52403121, 66471409, 67486225, 77457601, 80263681, 94148209, 100661089, 110397049, 126540001, 204232681, 264875625, 328878225
Offset: 1

Views

Author

Antti Karttunen, Dec 18 2024

Keywords

Comments

Of the first 2025 terms, only two, a(520) and a(1087) have multiple solutions. See the examples.
See also comments in A379123.

Examples

			k = 225 = 15^2 is included, because x = A379113(k) = 9, y = A379119(k) = 225/9 = 25, and A048720(A065621(sigma(9)), sigma(25)) = A048720(A065621(13), 31) = A048720(21, 31) = 403 = sigma(225).
a(8) = k = 216225 = 465^2 = (3*5*31)^2 is included, because x = A379113(k) = 9, y = A379119(k) = k/9 = 24025, sigma(9) = 13, A065621(13) = 21, sigma(24025) = 30783 and A048720(21, 30783) = 400179 = sigma(k). Note that pair x = 31^2 = 961, y = k / 961 = 225 is not among the solutions (we have A379129(k) = 1, not 2), because A048720(A065621(sigma(961)), sigma(k/961)) = 425971 > 400179.
a(520) = k = 383942431613601 = 19594449^2 is included, because x = A379113(k) = 16129,  y = A379119(k) = 23804478369, and A048720(A065621(sigma(x)),sigma(y)) = 703777973774337 = sigma(k). This is the first term that has more than one such solution (A379129(k) = 2), the other solution pair being x=961 and y=399523862241.
a(1087) = k = 19012955210325729 = 137887473^2 is included, because x = A379113(k) = 8649, y = k/8649 = 2198283640921, and A048720(A065621(sigma(x)),sigma(y)) = A048720(22197, 2198285123583) = sigma(x)*sigma(y) = 28377662660332947 = A379125(1087). Note that 8649 = 9*961 and here also x=961 and x=9 satisfy the condition, so there are three solutions in total.

Links

Crossrefs

Intersection of A016754 and A379114.
Cf. A000203, A048720, A065621, A277320, A379113, A379122 (square roots).
Cf. A379123 [= A379113(a(n))], A379124 [= A379119(a(n))], A379125 [= sigma(a(n))], A379129.

Programs

Formula

{k such that k is an odd square and A379113(k) > 1 (or equally, A379129(k) > 0)}.
a(n) = A379122(n)^2.
a(n) = A379123(n)*A379124(n).
For all n, A379125(n) = sigma(a(n)) = A277320(sigma(A379123(n)), sigma(A379124(n))).

A379123 a(n) = A379113(A379121(n)), where A379121 gives those odd squares k for which A379113(k) > 1.

Original entry on oeis.org

9, 121, 9, 9, 81, 1521, 9, 9, 49, 49, 81, 9, 9, 625, 49, 49, 9, 961, 9, 9, 9, 961, 961, 49, 9, 961, 961, 169, 961, 961, 16129, 49, 49, 961, 961, 961, 961, 961, 49, 9, 9, 9, 9, 625, 961, 16129, 16129, 961, 961, 961, 49, 9, 49, 16129, 961, 49, 961, 9, 49, 49, 49, 49, 9, 9, 9, 9, 49, 9, 16129, 9, 9, 49, 49, 9, 49, 9
Offset: 1

Views

Author

Antti Karttunen, Dec 18 2024

Keywords

Comments

All terms are odd squares (A016754) by definition.
Among the initial 2025 terms, only the following 12 terms occur:
Term Occurs Where
n times
---------------------------------------------------------------
9 699
49 665
81 2 a(5) and a(11)
121 1 a(2)
169 2 a(28) and a(926)
625 9 at n=14, 44, 85, 110, 155, 447, 654, 896, 1217.
961 390
1521 1 a(6) NB: 1521 = 9*169.
8649 1 a(1087). NB: 8649 = 9*961.
16129 246
67092481 8 First occurrence at a(1120)
3287531569 1 a(1636). NB: 3287531569 = 49*67092481.
Questions: Is this sequence infinite? Do all terms of A133049 eventually appear here? Or any 4th or higher powers of Mersenne and other primes, apart from 81 and 625?

Examples

			See examples in A379121.

Links

Crossrefs

Cf. A016754, A114390, A133049, A379113, A379121, A379124, A379125.

Programs

  • PARI
    forstep(n=1,2^18,2,d=A379113(n^2); if(d>1, print1(d,", ")));

Formula

a(n) = A379121(n) / A379124(n).

A379124 a(n) = A379119(A379121(n)), where A379121 gives those odd squares k for which A379113(k) > 1 and A379119(n) = n/A379113(n).

Original entry on oeis.org

25, 25, 361, 1369, 361, 25, 10201, 24025, 5041, 7225, 5041, 83521, 85849, 1369, 18769, 27889, 177241, 3969, 1092025, 1243225, 1352569, 13225, 30625, 978121, 5822569, 69169, 70225, 458329, 83521, 97969, 6241, 2253001, 2582449, 212521, 275625, 342225, 358801, 363609, 7502121, 45657049, 63696361, 65626201, 78659161
Offset: 1

Views

Author

Antti Karttunen, Dec 18 2024

Keywords

Comments

All terms are odd squares (A016754) by definition.

Links

Crossrefs

Cf. A016754, A114390, A133049, A379113, A379121, A379123, A379125.

Programs

  • PARI
    forstep(n=1,2^18,2,d=A379113(n^2); if(d>1, print1((n^2)/d,", ")));

Formula

a(n) = A379121(n) / A379123(n).

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

Views

Author

Antti Karttunen, Dec 22 2024

Keywords

Examples

			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,

Links

Crossrefs

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

Programs

  • PARI
    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];

Formula

A(n, k) = A277320(A379223(n), A379223(k)).
Showing 1-4 of 4 results.

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