A115872 -id:A115872 - cp's OEIS frontend

A115874 Integers i such that 19*i = 55 X i.

0, 7, 14, 28, 31, 56, 62, 63, 112, 119, 124, 126, 127, 224, 238, 248, 252, 254, 255, 448, 455, 476, 496, 504, 508, 510, 511, 896, 910, 952, 992, 1008, 1016, 1020, 1022, 1023, 1792, 1799, 1820, 1823, 1904, 1911, 1984, 1991, 2016, 2032, 2040, 2044

Offset: 1

Antti Karttunen, Feb 07 2006

nonn

Here * stands for ordinary multiplication and X means carryless (GF(2)[X]) multiplication (A048720).
From Robert Israel, Apr 08 2018: (Start)
n is in the sequence if and only if 2*n is.
If n is in the sequence, then so is (2^k+1)*n if 2^k > n.
Contains 2^k-1 for k >= 5. (End)

Row 19 of A115872. Superset of A115876? A115875 shows this sequence in binary.

X:= proc(a,b) local A,B,C;
A:= convert(a,base,2);
B:= convert(b,base,2);
C:= expand(add(A[i]*x^(i-1),i=1..nops(A))*add(B[i]*x^(i-1),i=1..nops(B))) mod 2;
eval(C,x=2)
end proc:
select(t -> 19*t = X(55,t), [$0..10^4]); # Robert Israel, Apr 08 2018

X[a_, b_] := Module[{A, B, C},
     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];
Select[Range[0, 10^4], 19*# == 55~X~#&] (* Jean-François Alcover, Jan 04 2022, after Robert Israel *)

Offset corrected by Robert Israel, Apr 08 2018

A325573 Odd numbers n that have divisor d > 1 such that A048720(A065621(d),n/d) = n.

Original entry on oeis.org

9, 21, 33, 35, 45, 49, 65, 75, 93, 105, 129, 133, 135, 153, 155, 161, 165, 189, 195, 217, 225, 259, 273, 279, 297, 309, 315, 341, 345, 381, 385, 403, 441, 465, 513, 525, 527, 561, 567, 585, 589, 597, 611, 621, 635, 645, 651, 681, 693, 705, 713, 729, 765, 775, 793, 819, 837, 889, 899, 945, 961, 1025, 1029, 1035, 1057, 1065

Offset: 1

Antti Karttunen, May 10 2019

nonn

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

Cf. A048720, A065621, A115872, A325566, A325567, A325571.

Subsequence of A071904 and of A325572.

A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1,n+n-1);
isA325573(n) = ((n%2)&&fordiv(n,d,if(A048720(A065621(n/d),d)==n,return(d

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).

A114384 Integers i such that 49*i = 81 X i.

Original entry on oeis.org

0, 21, 42, 63, 84, 85, 126, 127, 168, 170, 189, 191, 252, 253, 254, 255, 336, 340, 341, 378, 382, 383, 504, 506, 508, 509, 510, 511, 672, 680, 682, 693, 701, 703, 756, 757, 764, 765, 766, 767, 1008, 1012, 1013, 1016, 1018, 1020, 1021, 1022, 1023, 1344

Offset: 0

Antti Karttunen, Feb 07 2006

nonn

Here * stands for ordinary multiplication and X means carryless (GF(2)[X]) multiplication (A048720).

Index entries for sequences defined by congruent products between domains N and GF(2)[X]

Row 49 of A115872. A114385 shows this sequence in binary. Cf. A114392.

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).

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A115874 Integers i such that 19*i = 55 X i.

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A325573 Odd numbers n that have divisor d > 1 such that A048720(A065621(d),n/d) = n.

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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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A114384 Integers i such that 49*i = 81 X i.

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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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