A379121 -id:A379121 - cp's OEIS frontend

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

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

Antti Karttunen, Dec 18 2024

nonn

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?

			See examples in A379121.

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

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

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

Antti Karttunen, Dec 18 2024

nonn

All terms are odd squares (A016754) by definition.

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

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

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

A379125 Sum of divisors of those 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

403, 4123, 4953, 18291, 46101, 73749, 133939, 400179, 291441, 542469, 618673, 1153633, 1119859, 1098867, 1077699, 1599249, 2309619, 6848721, 20421219, 20131059, 17598529, 17022999, 44205381, 59669253, 80520921, 68946969, 88131729, 83998281, 88119813, 97595019, 102760497, 137273157, 147291249, 211492119, 574669953

Offset: 1

Antti Karttunen, Dec 18 2024

nonn

Cf. A000203, A048720, A065621, A277320, A379113, A379121, A379123, A379124.

forstep(n=1,oo,2,if(A379113(n^2)>1, k++; print1(sigma(n^2), ", ")));

a(n) = A000203(A379121(n)).
a(n) = A277320(sigma(A379123(n)), sigma(A379124(n))).
a(n) = sigma(A379123(n)) * sigma(A379124(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

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)).

A379122 Odd numbers m for which A379113(m^2) > 1, i.e., k = m^2 has 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

15, 55, 57, 111, 171, 195, 303, 465, 497, 595, 639, 867, 879, 925, 959, 1169, 1263, 1953, 3135, 3345, 3489, 3565, 5425, 6923, 7239, 8153, 8215, 8801, 8959, 9703, 10033, 10507, 11249, 14291, 16275, 18135, 18569, 18693, 19173, 20271, 23943, 24303, 26607, 28325, 32581, 33655, 34163, 40393, 43927, 46221, 47649, 55281

Offset: 1

Antti Karttunen, Dec 18 2024

nonn

Square roots of A379121.

Cf. A379113.

is_A379122(n) = (n%2 && A379113(n^2)>1);

{Odd k such that A379113(k^2) > 1}.

a(n) = A000196(A379121(n)).

A379129 a(n) is the number of unitary proper divisors d > 1 of n for which A048720(A065621(sigma(d)),sigma(n/d)) is equal to sigma(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 2, 2, 0, 1, 0, 1, 0, 1, 0, 5, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 1, 1, 2, 0, 0, 2, 1, 1, 1, 0, 3, 0, 1, 1, 0, 1, 5, 0, 0, 1, 5, 0, 1, 0, 0, 1, 1, 1, 3, 0, 1, 0, 0, 0, 3, 0, 1, 1, 1, 0, 0, 1, 1, 2, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 3

Offset: 1

Antti Karttunen, Dec 18 2024

nonn

			For n = 21 = 3*7, both A048720(A065621(sigma(3)),sigma(7)) [= A048720(4,8)] and A048720(A065621(sigma(7)),sigma(3)) [= A048720(8,4)] yield the decided result, which is 32 = sigma(21), therefore a(21) = 2.
For n = 34 = 2*17, neither A048720(A065621(sigma(2)),sigma(17)) = A048720(7,18) = 126 nor A048720(A065621(sigma(17)),sigma(2)) = A048720(50,3) = 86 is the decided result, 54 = sigma(34), therefore a(34) = 0.
See example in A379121 why a(383942431613601) = 2.

Cf. A000203, A048720, A065621, A277320, A379113, A379114 (positions of terms > 0), A379118, A379130.

A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1,n+n-1);
A379129(n) = { my(s=sigma(n)); sumdiv(n,d,if(1==d || d==n || 1!=gcd(d,n/d), 0, A048720(A065621(sigma(n/d)),sigma(d))==s)); };

a(n) = Sum_{d|n, gcd(d, n/d)=1, 1A048720(A065621(sigma(d)),sigma(n/d)) == sigma(n)], where [ ] is the Iverson bracket.

a(n) <= A379130(n).

cp's OEIS Frontend

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

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

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A379125 Sum of divisors of those 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).

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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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A379122 Odd numbers m for which A379113(m^2) > 1, i.e., k = m^2 has a proper unitary divisor d > 1 such that A048720(A065621(sigma(d)),sigma(k/d)) is equal to sigma(k).

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PARI

Formula

A379129 a(n) is the number of unitary proper divisors d > 1 of n for which A048720(A065621(sigma(d)),sigma(n/d)) is equal to sigma(n).

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