A325565 -id:A325565 - cp's OEIS frontend

A325569 Numbers n for which A325565(n) < A325568(n).

135, 279, 525, 567, 651, 1143, 1323, 2079, 2205, 2295, 2667, 2835, 3087, 4123, 4165, 4185, 4191, 4445, 4599, 5355, 5715, 6223, 8211, 8255, 8370, 8415, 8505, 8925, 9207, 9359, 9765, 10731, 11475, 11811, 12397, 12495, 13335, 16575, 16863, 17010, 17325, 17577, 17885, 18423, 18767, 19467, 21483, 22995, 23715, 24583, 24941, 25039

Offset: 1

Antti Karttunen, May 09 2019

nonn

			135 has eight divisors: 1, 3, 5, 9, 15, 27, 45, 135. Of these only two, d = 1 and d = 9 are such that A048720(A065621(d),135/d) = 135, thus A325565(135) = 2. On the other hand, when iterating A325567 starting from 135 we get: A325567(135) = 9, A325567(9) = 3 and A325567(3) = 1, so A325568(135) = 3 > A325565(135), thus 135 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..1001

Cf. A325565, A325567, A325568.

PARI
```
isA325569(n) = (A325565(n)<A325568(n));
```

A115872 Square array where row n gives all solutions k > 0 to the cross-domain congruence n*k = A048720(A065621(n),k), zero sequence (A000004) if no such solutions exist.

Original entry on oeis.org

1, 2, 1, 3, 2, 3, 4, 3, 6, 1, 5, 4, 7, 2, 7, 6, 5, 12, 3, 14, 3, 7, 6, 14, 4, 15, 6, 7, 8, 7, 15, 5, 28, 7, 14, 1, 9, 8, 24, 6, 30, 12, 15, 2, 15, 10, 9, 28, 7, 31, 14, 28, 3, 30, 7, 11, 10, 30, 8, 56, 15, 30, 4, 31, 14, 3, 12, 11, 31, 9, 60, 24, 31, 5, 60, 15, 6, 3, 13, 12, 48, 10, 62, 28, 56, 6, 62, 28, 12, 6, 5, 14, 13, 51, 11, 63, 30, 60, 7, 63, 30, 15, 7, 10, 7

Offset: 1

Antti Karttunen, Feb 07 2006

Here * stands for ordinary multiplication and X means carryless (GF(2)[X]) multiplication (A048720).
Square array is read by descending antidiagonals, as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
Rows at positions 2^k are 1, 2, 3, ..., (A000027). Row 2n is equal to row n.
Numbers on each row give a subset of positions of zeros at the corresponding row of A284270. - Antti Karttunen, May 08 2019

			Fifteen initial terms of rows 1 - 19 are listed below:
   1:  1,  2,  3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,  15, ...
   2:  1,  2,  3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,  15, ...
   3:  3,  6,  7,  12,  14,  15,  24,  28,  30,  31,  48,  51,  56,  60,  62, ...
   4:  1,  2,  3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,  15, ...
   5:  7, 14, 15,  28,  30,  31,  56,  60,  62,  63, 112, 120, 124, 126, 127, ...
   6:  3,  6,  7,  12,  14,  15,  24,  28,  30,  31,  48,  51,  56,  60,  62, ...
   7:  7, 14, 15,  28,  30,  31,  56,  60,  62,  63, 112, 120, 124, 126, 127, ...
   8:  1,  2,  3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,  15, ...
   9: 15, 30, 31,  60,  62,  63, 120, 124, 126, 127, 240, 248, 252, 254, 255, ...
  10:  7, 14, 15,  28,  30,  31,  56,  60,  62,  63, 112, 120, 124, 126, 127, ...
  11:  3,  6, 12,  15,  24,  27,  30,  31,  48,  51,  54,  60,  62,  63,  96, ...
  12:  3,  6,  7,  12,  14,  15,  24,  28,  30,  31,  48,  51,  56,  60,  62, ...
  13:  5, 10, 15,  20,  21,  30,  31,  40,  42,  45,  47,  60,  61,  62,  63, ...
  14:  7, 14, 15,  28,  30,  31,  56,  60,  62,  63, 112, 120, 124, 126, 127, ...
  15: 15, 30, 31,  60,  62,  63, 120, 124, 126, 127, 240, 248, 252, 254, 255, ...
  16:  1,  2,  3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,  15, ...
  17: 31, 62, 63, 124, 126, 127, 248, 252, 254, 255, 496, 504, 508, 510, 511, ...
  18: 15, 30, 31,  60,  62,  63, 120, 124, 126, 127, 240, 248, 252, 254, 255, ...
  19:  7, 14, 28,  31,  56,  62,  63, 112, 119, 124, 126, 127, 224, 238, 248, ...

Transpose: A114388. First column: A115873.
Cf. A048720, A065621, A115857, A115871, A325565, A325566, A325567, A325568, A325570, A325571.
Cf. also arrays A277320, A277810, A277820, A284270.
A few odd-positioned rows: row 1: A000027, Row 3: A048717, Row 5: A115770 (? Checked for all values less than 2^20), Row 7: A115770, Row 9: A115801, Row 11: A115803, Row 13: A115772, Row 15: A115801 (? Checked for all values less than 2^20), Row 17: A115809, Row 19: A115874, Row 49: A114384, Row 57: A114386.

X[a_, b_] := Module[{A, B, C, x},
     A = Reverse@IntegerDigits[a, 2];
     B = Reverse@IntegerDigits[b, 2];
     C = Expand[
        Sum[A[[i]]*x^(i-1), {i, 1, Length[A]}]*
        Sum[B[[i]]*x^(i-1), {i, 1, Length[B]}]];
     PolynomialMod[C, 2] /. x -> 2];
T[n_, k_] := Module[{x = BitXor[n-1, 2n-1], k0 = k},
     For[i = 1, True, i++, If[n*i == X[x, i],
     If[k0 == 1, Return[i], k0--]]]];
Table[T[n-k+1, k], {n, 1, 14}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 04 2022 *)

up_to = 120;
A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1,n+n-1);
A115872sq(n, k) = { my(x = A065621(n)); for(i=1,oo,if((n*i)==A048720(x,i),if(1==k,return(i),k--))); };
A115872list(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] = A115872sq(col,(a-(col-1))))); (v); };
v115872 = A115872list(up_to);
A115872(n) = v115872[n]; \\ (Slow) - Antti Karttunen, May 08 2019

Example section added and the data section extended up to n=105 by Antti Karttunen, May 08 2019

A325567 a(1) = 1; for n > 1, a(n) is the largest proper divisor d of n such that A048720(A065621(d),n/d) is equal to n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 1, 8, 1, 6, 1, 4, 3, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16, 11, 2, 5, 12, 1, 2, 1, 8, 1, 6, 1, 4, 3, 2, 1, 16, 7, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 32, 13, 22, 1, 4, 1, 10, 1, 24, 1, 2, 5, 4, 1, 2, 1, 16, 1, 2, 1, 12, 1, 2, 1, 8, 1, 6, 1, 4, 3, 2, 1, 32, 1, 14, 1, 4, 1, 2, 1, 8, 7

Offset: 1

Antti Karttunen, May 09 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A032742, A048720, A065621, A115872, A325565, A325566, A325568.

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))));

A325566 a(n) is the largest divisor d of n such that A048720(A065621(d),n/d) = n.

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 1, 8, 3, 2, 1, 4, 1, 2, 1, 16, 1, 6, 1, 4, 3, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 32, 11, 2, 5, 12, 1, 2, 1, 8, 1, 6, 1, 4, 3, 2, 1, 16, 7, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 64, 13, 22, 1, 4, 1, 10, 1, 24, 1, 2, 5, 4, 1, 2, 1, 16, 1, 2, 1, 12, 1, 2, 1, 8, 1, 6, 1, 4, 3, 2, 1, 32, 1, 14, 1, 4, 1, 2, 1, 8, 7

Offset: 1

Antti Karttunen, May 09 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A048720, A065621, A325565, A325567, A325570 (positions of ones).

A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1,n+n-1);
A325566(n) = fordiv(n,d,if(A048720(A065621(n/d),d)==n,return(n/d)));

A325570 Numbers n that have no divisor d > 1 such that A048720(A065621(d),n/d) = n.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 37, 39, 41, 43, 47, 51, 53, 55, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 131, 137, 139, 141, 143, 145, 147, 149, 151, 157, 159, 163, 167, 169, 171, 173, 175, 177, 179, 181

Offset: 1

Antti Karttunen, May 10 2019

nonn

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

Cf. A048720, A065621, A115872, A115873.
Positions of ones in A325565 and A325566.
Cf. A065091 (a subsequence), A325571 (the composite terms), A325572 (complement).
Subsequence of A005408 (odd numbers).

A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1,n+n-1);
isA325570(n) = fordiv(n,d,if(A048720(A065621(n/d),d)==n,return(d==n)));

A380844 The number of divisors of n that have the same binary weight as n.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 4, 2, 2, 1, 3, 1, 2, 1, 5, 1, 4, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 2, 2, 2, 6, 1, 2, 1, 4, 1, 4, 1, 3, 2, 2, 1, 5, 2, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 7, 2, 4, 1, 3, 1, 4, 1, 8, 1, 2, 2, 3, 1, 2, 1, 5, 1, 2, 1, 6, 1, 2, 1

Offset: 1

Amiram Eldar, Feb 05 2025

First differs from A325565 at n = 133: a(133) = 3 while A325565(133) = 2.

The sum of these divisors is A380845(n).

			a(6) = 2 because 6 = 110_2 has binary weight 2, and 2 of its divisors, 3 = 11_2 and 6, have the same binary weight.

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

Cf. A000005, A000043, A000120, A000265, A000396, A007814, A324392, A325565, A380845.

a[n_] := Module[{h = DigitCount[n, 2, 1]}, DivisorSum[n, 1 &, DigitCount[#, 2, 1] == h &]]; Array[a, 100]

a(n) = {my(h = hammingweight(n)); sumdiv(n, d, hammingweight(d) == h);}

a(n) = Sum_{d|n} [A000120(d) = A000120(n)], where [ ] is the Iverson bracket.
a(2^n) = n+1.
a(n) <= A000005(n) with equality if and only if n is a power of 2.
a(n) = a(A000265(n)) * (A007814(n)+1), or equivalently, a(k*2^n) = a(k)*(n+1) for k odd and n >= 0.
In particular, since a(p) = 1 for an odd prime p, a(p*2^n) = n+1 for an odd prime p and n >= 0.
a(A000396(n)) = A000043(n), assuming that odd perfect numbers do no exist.

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

Original entry on oeis.org

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)).

A325568 a(1) = 0; for n > 1, a(n) = 1 + a(A325567(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 1, 4, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 2, 2, 2, 4, 1, 2, 1, 4, 1, 3, 1, 3, 2, 2, 1, 5, 2, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 2, 3, 1, 3, 1, 3, 1, 5, 1, 2, 2, 3, 1, 2, 1, 5, 1, 2, 1, 4, 1, 2, 1, 4, 1, 3, 1, 3, 2, 2, 1, 6, 1, 3, 1, 3, 1, 2, 1, 4, 2

Offset: 1

Antti Karttunen, May 09 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001222, A048720, A065621, A115872, A325565, A325566, A325567, A325569.

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))));
A325568(n) = if(1==n,0,1+A325568(A325567(n)));

a(1) = 0; for n > 1, a(n) = 1 + a(A325567(n)).

a(n) <= A001222(n) for all n.

A379120 a(1) = 1; and for n > 1, a(n) is the smallest divisor d > 1 of n such that A048720(A065621(n/d),d) is equal to n.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 15, 2, 17, 3, 19, 5, 7, 11, 23, 3, 25, 13, 27, 7, 29, 15, 31, 2, 3, 17, 7, 3, 37, 19, 39, 5, 41, 7, 43, 11, 15, 23, 47, 3, 7, 25, 51, 13, 53, 27, 55, 7, 57, 29, 59, 15, 61, 31, 63, 2, 5, 3, 67, 17, 69, 7, 71, 3, 73, 37, 15, 19, 77, 39, 79, 5, 81, 41, 83, 7, 85, 43, 87, 11, 89

Offset: 1

Antti Karttunen, Dec 17 2024

nonn

Cf. A048720, A065621, A115872, A325565, A325566, A325567.

Cf. also A379119.

A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1,n+n-1);
A379120(n) = if(1==n,n,fordiv(n,d,if((d>1)&&A048720(A065621(n/d),d)==n,return(d))));

a(n) = n / A325567(n).

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

A325569 Numbers n for which A325565(n) < A325568(n).

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A115872 Square array where row n gives all solutions k > 0 to the cross-domain congruence n*k = A048720(A065621(n),k), zero sequence (A000004) if no such solutions exist.

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A325567 a(1) = 1; for n > 1, a(n) is the largest proper divisor d of n such that A048720(A065621(d),n/d) is equal to n.

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A325566 a(n) is the largest divisor d of n such that A048720(A065621(d),n/d) = n.

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A325570 Numbers n that have no divisor d > 1 such that A048720(A065621(d),n/d) = n.

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A380844 The number of divisors of n that have the same binary weight as n.

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Formula

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

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A325568 a(1) = 0; for n > 1, a(n) = 1 + a(A325567(n)).

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A379120 a(1) = 1; and for n > 1, a(n) is the smallest divisor d > 1 of n such that A048720(A065621(n/d),d) is equal to n.

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