A048720 -id:A048720 - cp's OEIS frontend

A325560 a(n) is the number of divisors d of n such that A048720(d,k) = n for some k.

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 3, 4, 2, 8, 2, 4, 4, 6, 2, 8, 2, 6, 3, 4, 3, 9, 2, 4, 3, 8, 2, 6, 2, 6, 6, 4, 2, 10, 3, 4, 4, 6, 2, 8, 2, 8, 3, 4, 2, 12, 2, 4, 6, 7, 3, 6, 2, 6, 2, 6, 2, 12, 2, 4, 5, 6, 2, 6, 2, 10, 2, 4, 2, 9, 4, 4, 2, 8, 2, 12, 2, 6, 3, 4, 4, 12, 2, 6, 4, 6, 2, 8, 2, 8, 5

Offset: 1

Antti Karttunen, May 11 2019

nonn

a(n) is the number of divisors d of n such that when the binary expansion of d is converted to a (0,1)-polynomial (e.g., 13=1101[2] encodes X^3 + X^2 + 1), that polynomial is a divisor of the (0,1)-polynomial similarly converted from n, when the polynomial division is done over field GF(2).

			39 = 3*13 has four divisors 1, 3, 13, 39, of which all other divisors except 13 are counted because we have A048720(1,39) = A048720(39,1) = A048720(3,29) = 39, but A048720(13,u) is not equal to 39 for any u, thus a(39) = 3. See also the example in A325563.

Cf. A000005, A048720, A091220, A325559 (positions of 2's), A325563, A325565.

A325560(n) = { my(p = Pol(binary(n))*Mod(1, 2)); sumdiv(n,d,my(q = Pol(binary(d))*Mod(1, 2)); !(p%q)); };

For all n, A325565(n) <= a(n) <= min(A000005(n), A091220(n)).

A325571 Composite numbers n that have no divisor d > 1 such that A048720(A065621(d),n/d) = n.

Original entry on oeis.org

15, 25, 27, 39, 51, 55, 57, 63, 69, 77, 81, 85, 87, 91, 95, 99, 111, 115, 117, 119, 121, 123, 125, 141, 143, 145, 147, 159, 169, 171, 175, 177, 183, 185, 187, 201, 203, 205, 207, 209, 213, 215, 219, 221, 231, 235, 237, 243, 245, 247, 249, 253, 255, 261, 265, 267, 275, 285, 287, 289, 291, 295, 299, 301, 303, 305, 319, 321

Offset: 1

Antti Karttunen, May 10 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A048720, A065621, A325573.

Intersection of A002808 and A325570.

A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1,n+n-1);
isA325571(n) = ((n>1)&&!isprime(n)&&fordiv(n,d,if(A048720(A065621(n/d),d)==n,return(d==n))));

A379119 a(1) = 1; for n > 1, a(n) is the smallest unitary divisor d > 1 of n such that A048720(A065621(sigma(n/d)),sigma(d)) is equal to sigma(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 7, 8, 9, 2, 11, 4, 13, 2, 5, 16, 17, 18, 19, 4, 3, 2, 23, 8, 25, 13, 27, 4, 29, 2, 31, 32, 11, 34, 5, 36, 37, 38, 13, 8, 41, 2, 43, 4, 45, 2, 47, 16, 49, 25, 17, 4, 53, 54, 5, 8, 19, 29, 59, 4, 61, 2, 9, 64, 13, 2, 67, 68, 23, 2, 71, 8, 73, 74, 25, 4, 11, 13, 79, 16, 81, 82, 83, 4, 85, 2, 29, 8, 89

Offset: 1

Antti Karttunen, Dec 17 2024

nonn

Cf. A000203, A048720, A065621, A379113, A379114 (positions of terms such that a(n) < n).

Cf. also A379120.

A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1,n+n-1);
A379119(n) = if(1==n,n,my(s=sigma(n)); fordiv(n,d,if((d>1) && 1==gcd(d,n/d) && A048720(A065621(sigma(n/d)),sigma(d))==s,return(d))));

a(n) = n / A379113(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)).

A325641 a(1) = 1; for n > 1, a(n) = n/d for the least divisor d > 1 of n such that A048720(d,k) = n for some k.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6, 1, 7, 5, 8, 1, 9, 1, 10, 3, 11, 1, 12, 1, 13, 9, 14, 1, 15, 1, 16, 11, 17, 5, 18, 1, 19, 13, 20, 1, 21, 1, 22, 15, 23, 1, 24, 7, 25, 17, 26, 1, 27, 1, 28, 19, 29, 1, 30, 1, 31, 21, 32, 13, 33, 1, 34, 1, 35, 1, 36, 1, 37, 25, 38, 1, 39, 1, 40, 1, 41, 1, 42, 17, 43, 1, 44, 1, 45, 1, 46, 3, 47, 19

Offset: 1

Antti Karttunen, May 11 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to polynomials in ring GF(2)[X]

Cf. A048720, A325642, A325643.

Cf. A325559 (positions of ones after the initial 1).

A325641(n) = if(1==n,n, my(p = Pol(binary(n))*Mod(1, 2)); fordiv(n,d,if((d>1),my(q = Pol(binary(d))*Mod(1, 2)); if(0==(p%q), return(n/d)))));

a(n) = n / A325643(n).

A325642 a(1) = 1; for n > 1, a(n) = k for the least divisor d > 1 of n such that A048720(d,k) = n for some k.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 7, 5, 1, 6, 1, 7, 5, 8, 1, 9, 1, 10, 7, 11, 1, 12, 1, 13, 9, 14, 1, 15, 1, 16, 31, 17, 13, 18, 1, 19, 29, 20, 1, 21, 1, 22, 27, 23, 1, 24, 11, 25, 17, 26, 1, 27, 1, 28, 23, 29, 1, 30, 1, 31, 21, 32, 21, 33, 1, 34, 1, 35, 1, 36, 1, 37, 57, 38, 1, 39, 1, 40, 1, 41, 1, 42, 17, 43, 1, 44, 1, 45, 1, 46, 7, 47, 19

Offset: 1

Antti Karttunen, May 11 2019

nonn

For n > 1, we first find the least divisor d of n that is larger than 1 and for which it holds that when the binary expansion of d is converted to a (0,1)-polynomial (e.g., 13=1101[2] encodes X^3 + X^2 + 1), then that polynomial is a divisor of (0,1)-polynomial similarly converted from n, when the division is done over GF(2). a(n) is then the quotient polynomial converted back to decimal via its binary encoding. See the example.

			For n = 9, its least nontrivial divisor is 3, and we find that 3 (in binary "11") corresponds to polynomial X + 1, which in this case is a factor of polynomial X^3 + 1 (corresponding to 9 as 9 is "1001" in binary) as the latter factorizes as (X + 1)(X^2 + X + 1) over GF(2), that is, 9 = A048720(3,7). Thus a(9) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to polynomials in ring GF(2)[X]

Cf. A048720, A325641, A325643.

A325642(n) = if(1==n,n, my(p = Pol(binary(n))*Mod(1, 2)); fordiv(n,d,if((d>1),my(q = Pol(binary(d))*Mod(1, 2)); if(0==(p%q), return(fromdigits(Vec(lift(p/q)),2))))));

For all n >= 1, A048720(a(n), A325643(n)) = 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)).

A277807 Numbers n such that A048720(n, A065621(n)) is a perfect square, but n is not in A023758.

Original entry on oeis.org

83, 166, 332, 365, 664, 730, 1328, 1460, 2656, 2920, 5312, 5840, 10624, 11680, 21248, 23360, 33051, 42496, 46720, 66102, 84992, 93440, 115785, 132204, 169984, 186880, 231570, 264408, 279099, 339968, 373760, 388731, 463140, 528816, 558198, 679936, 747520, 777462, 926280, 1057632, 1116396, 1359872, 1495040, 1554924, 1677591

Offset: 1

Antti Karttunen, Nov 01 2016

Not yet proved: Equally, numbers n such that A048720(n, A065621(n)) = k^2 for some k different from n.

If n is included in this sequence, then also 2n is included (and vice versa), thus the sequence is infinite and wholly determined by its odd terms.

Antti Karttunen, Table of n, a(n) for n = 1..57
Index entries for sequences related to binary expansion of n

Setwise difference of A277704 \ A023758.
Cf. A000265, A010052, A048720, A065621, A277699.
Cf. A277806 (the square roots of the solutions).

A284580 Carryless base-2 product (A048720) of lengths of runs of 1-bits in binary representation of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 2, 2, 3, 4, 1, 1, 1, 2, 1, 1, 2, 3, 2, 2, 2, 4, 3, 3, 4, 5, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 2, 2, 3, 4, 2, 2, 2, 4, 2, 2, 4, 6, 3, 3, 3, 6, 4, 4, 5, 6, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 2, 2, 3, 4, 1, 1, 1, 2, 1, 1, 2, 3, 2, 2, 2, 4, 3, 3, 4, 5, 2, 2, 2, 4, 2, 2, 4, 6, 2, 2, 2, 4, 4, 4, 6, 8, 3, 3, 3, 6, 3, 3, 6, 5, 4

Offset: 0

Antti Karttunen, Apr 14 2017

			a(119) = 5, as 119 is "1110111" in binary, and A048720(3,3) = 5.

Antti Karttunen, Table of n, a(n) for n = 0..10922
Index entries for sequences related to binary expansion of n

Cf. A003714 (positions of ones).
Cf. A048720, A284579.
Differs from similar A227349 for the first time at n=119, where a(119) = 5, while A227349(119) = 9.

(define (A284580 n) (reduce A048720bi 1 (bisect (reverse (binexp->runcount1list n)) (- 1 (modulo n 2))))) ;; Where A048720bi is a two-argument function implementing carryless binary product, A048720. For bisect and binexp->runcount1list, see under A227349.

A325386 Numbers n such that for any divisor d of n and some k, A048720(d,k) = n only for trivial cases d=1 and d=n, despite that n is neither prime nor in A014580.

Original entry on oeis.org

69, 77, 81, 121, 169, 205, 209, 261, 265, 275, 289, 295, 305, 321, 323, 327, 329, 339, 377, 405, 407, 437, 453, 473, 475, 481, 493, 517, 533, 551, 553, 555, 559, 565, 575, 581, 583, 595, 625, 649, 667, 671, 699, 703, 707, 737, 747, 749, 755, 763, 767, 779, 785, 805, 815, 833, 835, 849, 851, 855, 861, 869, 871, 885, 893, 905, 923, 925

Offset: 1

Antti Karttunen, May 11 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to polynomials in ring GF(2)[X]

Terms of A325559 not in A257688.
Subsequence of A005408 (odd numbers).
Cf. A014580, A325560.
Differs from A260428 for the first time at n=32, where a(32) = 555, a value missing from A260428.

A325560(n) = { my(p = Pol(binary(n))*Mod(1, 2)); sumdiv(n,d,my(q = Pol(binary(d))*Mod(1, 2)); !(p%q)); };
isA325386(n) = (!isprime(n) && !polisirreducible(Pol(binary(n))*Mod(1,2)) && (2 == A325560(n)));

cp's OEIS Frontend

A325560 a(n) is the number of divisors d of n such that A048720(d,k) = n for some k.

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A325571 Composite numbers n that have no divisor d > 1 such that A048720(A065621(d),n/d) = n.

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A379119 a(1) = 1; for n > 1, a(n) is the smallest unitary divisor d > 1 of n such that A048720(A065621(sigma(n/d)),sigma(d)) is equal to sigma(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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A325641 a(1) = 1; for n > 1, a(n) = n/d for the least divisor d > 1 of n such that A048720(d,k) = n for some k.

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A325642 a(1) = 1; for n > 1, a(n) = k for the least divisor d > 1 of n such that A048720(d,k) = n for some 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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A277807 Numbers n such that A048720(n, A065621(n)) is a perfect square, but n is not in A023758.

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A284580 Carryless base-2 product (A048720) of lengths of runs of 1-bits in binary representation of n.

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