A293215 -id:A293215 - cp's OEIS frontend

A218403 Bitwise OR of all proper divisors of n; a(1) = 0 by convention.

0, 1, 1, 3, 1, 3, 1, 7, 3, 7, 1, 7, 1, 7, 7, 15, 1, 15, 1, 15, 7, 11, 1, 15, 5, 15, 11, 15, 1, 15, 1, 31, 11, 19, 7, 31, 1, 19, 15, 31, 1, 31, 1, 31, 15, 23, 1, 31, 7, 31, 19, 31, 1, 31, 15, 31, 19, 31, 1, 31, 1, 31, 31, 63, 13, 63, 1, 55, 23, 47, 1, 63, 1

Offset: 1

Reinhard Zumkeller, Oct 28 2012

			n=20: properDivisors(20) = {1, 2, 4, 5, 10}, 0001 OR 0010 OR 0100 OR 0101 OR 1010 = 1111 -> a(20) = 15;
n=21: properDivisors(21) = {1, 3, 7}, 001 OR 011 OR 111 = 111 -> a(21) = 7;
n=22: properDivisors(22) = {1, 2, 11}, 0001 OR 0010 OR 1011 = 1111 -> a(22) = 11;
n=23: properDivisors(23) = {1} -> a(23) = 23;
n=24: properDivisors(24) = {1, 2, 3, 4, 6, 8, 12}, 0001 OR 0010 OR 0011 OR 0100 OR 0110 OR 1000 OR 1100 = 1111 -> a(24) = 15;
n=25: properDivisors(25) = {1, 5}, 001 OR 101 = 101 -> a(25) = 5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..8191
Eric Weisstein's World of Mathematics, OR
Wikipedia, Bitwise operation OR
Index entries for sequences related to binary expansion of n

Cf. A001065, A027751, A000225 (subsequence), A123345, A227320, A293214, A293215.

import Data.Bits ((.|.))
a218403 = foldl (.|.)  0 . a027751_row :: Integer -> Integer

Table[BitOr@@Most[Divisors[n]],{n,80}] (* Harvey P. Dale, Nov 09 2012 *)

A218403(n) = { my(s=0); fordiv(n,d,if(dAntti Karttunen, Oct 08 2017

a(n) <= A218388(n).
a(A000040(n)) = 1.
From Antti Karttunen, Oct 08 2017: (Start)
a(n) = A087207(A293214(n)).
A227320(n) <= a(n) <= A001065(n).
(End)

A318835 Restricted growth sequence transform of A318834, product_{d|n, dA019565(A000010(d)).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 5, 6, 2, 7, 2, 8, 9, 8, 2, 10, 2, 11, 12, 13, 2, 14, 15, 16, 12, 14, 2, 17, 2, 18, 19, 20, 21, 22, 2, 23, 24, 25, 2, 26, 2, 27, 28, 29, 2, 30, 9, 31, 32, 33, 2, 34, 24, 35, 36, 37, 2, 38, 2, 39, 40, 39, 41, 42, 2, 43, 44, 45, 2, 46, 2, 47, 48, 49, 50, 51, 2, 52, 53, 54, 2, 55, 56, 57, 58, 59, 2, 60, 61, 62, 63, 64, 65, 66, 2

Offset: 1

Antti Karttunen, Sep 04 2018

nonn

For all i, j: a(i) = a(j) => A051953(i) = A051953(j).

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

Cf. A000010, A019565, A318831, A318834,

Cf. also A293215, A293217, A293223, A293224, A293232, A300833, A300835.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A318834(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(eulerphi(d)))); m; };
v318835 = rgs_transform(vector(up_to,n,A318834(n)));
A318835(n) = v318835[n];

A320014 Filter sequence combining the binary expansions of proper divisors of n, grouped by their residue classes mod 3.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 39, 40, 41, 42, 43, 2, 44, 2, 45, 46, 47, 48, 49, 2, 50, 51, 52, 2, 53, 2, 54, 55, 56, 57, 58, 2, 59, 60, 61, 2, 62, 63, 64, 65, 66, 2, 67, 68, 69, 70, 71, 72, 73, 2, 74, 75

Offset: 1

Antti Karttunen, Oct 03 2018

nonn

Restricted growth sequence transform of triple [A319990(n), A319991(n), A319992(n)], or equally, of ordered pair [A320010(n), A320013(n)].
Apart from trivial cases of primes, all other duplicates in range 1 .. 65537 seem to be squarefree semiprimes of the form 3k+1, i.e., both prime factors are either of the form 3k+1 or of the form 3k+2. Question: Is there any reason that more complicated cases would not occur later?
For all i, j: a(i) = a(j) => A293215(i) = A293215(j).
Differs from A319693 first for n = 108. - Georg Fischer, Oct 16 2018

			The first set of numbers that forms a nontrivial equivalence class is [295, 583, 799, 943] = [5*59, 11*53, 17*47, 23*41]. The prime factors in these are all of the form 3k+2, and when the binary expansions of the factors (like e.g., "101" for 5 and "111011" for 59 or "10111" for 23 and "101001" for 41) are overlaid, the resulting bit vector is always [1, 1, 1, 1, 1, 1^2], with the least significant bit-position containing 2 copies of 1's. Thus we have a(295) = a(583) = a(799) = a(943).

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

Cf. A019565, A319990, A319991, A319992, A320010, A320011, A320012, A320013.
Cf. also A293214, A293215, A293226, A300833.
Differs from A305800 for the first time at n=583, where a(583) = 234, while A305800(478).

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A319990(n) = { my(m=1); fordiv(n,d,if((dA019565(d))); m; };
A319991(n) = { my(m=1); fordiv(n,d,if((dA019565(d))); m; };
A319992(n) = { my(m=1); fordiv(n,d,if((dA019565(d))); m; };
v320014 = rgs_transform(vector(up_to,n,[A319990(n),A319991(n),A319992(n)]));
A320014(n) = v320014[n];

A319709 Filter sequence combining primorial base representations of the proper divisors of n; Restricted growth sequence transform of A319708.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 4, 17, 18, 19, 2, 20, 2, 21, 22, 23, 18, 24, 2, 25, 26, 27, 2, 28, 2, 29, 30, 31, 2, 32, 33, 34, 12, 35, 2, 36, 37, 38, 39, 40, 2, 41, 2, 42, 43, 44, 45, 46, 2, 47, 48, 49, 2, 50, 2, 51, 52, 53, 45, 54, 2, 55, 56, 57, 2, 58, 59, 60, 61, 62, 2, 63, 64, 65, 66, 67, 68, 69, 2, 70, 71

Offset: 1

Antti Karttunen, Oct 03 2018

nonn

For all i, j:
  a(i) = a(j) => A001065(i) = A001065(j),
  a(i) = a(j) => A319713(i) = A319713(j).

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

Cf. A276086, A319708, A319713.

Cf. A293215, A293226, A300835 for similar constructions for other bases.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A319708(n) = { my(m=1); fordiv(n, d, if(dA276086(d))); (m); };
v319709 = rgs_transform(vector(up_to,n,A319708(n)));
A319709(n) = v319709[n];

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A218403 Bitwise OR of all proper divisors of n; a(1) = 0 by convention.

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A318835 Restricted growth sequence transform of A318834, product_{d|n, dA019565(A000010(d)).

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A320014 Filter sequence combining the binary expansions of proper divisors of n, grouped by their residue classes mod 3.

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A319709 Filter sequence combining primorial base representations of the proper divisors of n; Restricted growth sequence transform of A319708.

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