A277320 -id:A277320 - cp's OEIS frontend

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).

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)).

A379126 a(1) = 1; for n > 1, a(n) is the least number k such that A325567(k) = n, or 0 if no such number exists.

Original entry on oeis.org

1, 4, 9, 8, 35, 18, 49, 16, 135, 70, 33, 36, 65, 98, 225, 32, 527, 270, 133, 140, 651, 66, 161, 72, 775, 130, 837, 196, 899, 450, 961, 64, 2079, 1054, 525, 540, 259, 266, 273, 280, 2583, 1302, 129, 132, 2835, 322, 705, 144, 3087, 1550, 3213, 260, 3339, 1674, 385, 392, 1539, 1798, 3717, 900, 3843, 1922, 3969, 128

Offset: 1

Antti Karttunen, Dec 21 2024

nonn

By definition, sequence is injective (apart from possible 0's) and each a(n) is a multiple of n.

Cf. A048720, A065621, A277320, A325567, A379128 (odd bisection), A379228 [= a(n)/n].

Cf. also A115872, A266195, A266351.

A048720(b, c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1, n+n-1);
memoA325567 = Map();
A325567(n) = if(1==n,1,my(v); if(mapisdefined(memoA325567,n,&v), v, fordiv(n, d, if((d>1)&&A048720(A065621(n/d), d)==n, v = (n/d); break)); mapput(memoA325567,n,v); (v)));
A379126(n) = for(k=1,oo,if(A325567(k)==n, return(k)));

a(n) = n * A379228(n).

A379128 a(1) = 1; for n > 1, a(n) is the least odd number k such that A325567(k) = 2*n-1, or 0 if no such number exists.

Original entry on oeis.org

1, 9, 35, 49, 135, 33, 65, 225, 527, 133, 651, 161, 775, 837, 899, 961, 2079, 525, 259, 273, 2583, 129, 2835, 705, 3087, 3213, 3339, 385, 1539, 3717, 3843, 3969, 8255, 2077, 1035, 1065, 9271, 2325, 1155, 1185, 4131, 2573, 10795, 2697, 11303, 2821, 11811, 2945, 12319, 12573, 12827, 1545, 13335, 13589, 13843, 1665

Offset: 1

Antti Karttunen, Dec 21 2024

nonn

Odd bisection of A379126.

Cf. A048720, A065621, A277320, A325567, A379127.

A048720(b, c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1, n+n-1);
A325567(n) = if(1==n, n, fordiv(n, d, if((d>1)&&A048720(A065621(n/d), d)==n, return(n/d))));
memoA379127 = Map();
A379127(n) = if(1==n,1,my(v); if(mapisdefined(memoA379127,n,&v), v, v = A325567(2*n-1); mapput(memoA379127,n,v); (v)));
A379128(n) = for(k=1,oo,if(A379127(k)==(n+n-1), return(k+k-1)));

a(n) = A379126(2*n-1).

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).

A379127 a(1) = 1; for n > 1, a(n) is the largest proper divisor d of 2n-1 such that A048720(A065621(d),(2n-1)/d) is equal to 2n-1.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 11, 5, 1, 1, 1, 1, 3, 1, 7, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 43, 1, 19, 9, 1, 1, 1, 1, 1, 1, 1, 1, 3, 5, 1, 1, 23, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Dec 21 2024

nonn

The position of the first occurrence of odd numbers k = 1, 3, 5, 7, 9, ... in this sequence is given by (1/2) * (A379128(2*k-1)+1).

Odd bisection of A325567.

Cf. A048720, A065621, A277320, A379128.

A048720(b, c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1, n+n-1);
A325567(n) = if(1==n, n, fordiv(n, d, if((d>1)&&A048720(A065621(n/d), d)==n, return(n/d))));
A379127(n) = A325567(2*n-1);

a(n) = A325567(2*n-1).

A379130 a(n) is the number of unitary divisors d of n for which A048720(A065621(sigma(d)),sigma(n/d)) is equal to sigma(n).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 4, 3, 1, 2, 1, 2, 1, 2, 1, 6, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 2, 2, 3, 1, 1, 3, 2, 2, 2, 1, 4, 1, 2, 2, 1, 2, 6, 1, 1, 2, 6, 1, 2, 1, 1, 2, 2, 2, 4, 1, 2, 1, 1, 1, 4, 1, 2, 2, 2, 1, 1, 2, 2, 4, 3, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 4

Offset: 1

Antti Karttunen, Dec 18 2024

nonn

It seems that A046528 gives all numbers k for which a(k) = A034444(k).

			For every n, a(n) >= 1, because A048720(A065621(sigma(1)), sigma(n)) = A048720(A065621(1), sigma(n)) = A048720(1, sigma(n)) = sigma(n).
For n = 21 = 3*7, after the divisor pair [1,21], all other divisor pairs also satisfy the condition: A048720(A065621(sigma(3)),sigma(7)) [= A048720(4,8)] and A048720(A065621(sigma(7)),sigma(3)) [= A048720(8,4)] and A048720(A065621(sigma(21)),sigma(1)) [= A048720(32,1)] all yield the decided result, 32 = sigma(21), therefore a(21) = 4.
See also examples in A379129.

Cf. A000203, A034444, A046528, A048720, A065621, A277320, A379113, A379129.

Cf. also A325565.

A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1,n+n-1);
A379130(n) = { my(s=sigma(n)); sumdiv(n,d,if(1!=gcd(d,n/d), 0, A048720(A065621(sigma(n/d)),sigma(d))==s)); };

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

A379129(n) <= a(n) <= A034444(n).

cp's OEIS Frontend

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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A379126 a(1) = 1; for n > 1, a(n) is the least number k such that A325567(k) = n, or 0 if no such number exists.

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A379128 a(1) = 1; for n > 1, a(n) is the least odd number k such that A325567(k) = 2*n-1, or 0 if no such number exists.

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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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A379127 a(1) = 1; for n > 1, a(n) is the largest proper divisor d of 2n-1 such that A048720(A065621(d),(2n-1)/d) is equal to 2n-1.

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A379130 a(n) is the number of unitary divisors d of n for which A048720(A065621(sigma(d)),sigma(n/d)) is equal to sigma(n).

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