A060681 -id:A060681 - cp's OEIS frontend

A318517 a(n) = A032742(n) XOR n-A032742(n), where XOR is bitwise-xor (A003987) and A032742 = the largest proper divisor of n.

1, 0, 3, 0, 5, 0, 7, 0, 5, 0, 11, 0, 13, 0, 15, 0, 17, 0, 19, 0, 9, 0, 23, 0, 17, 0, 27, 0, 29, 0, 31, 0, 29, 0, 27, 0, 37, 0, 23, 0, 41, 0, 43, 0, 17, 0, 47, 0, 45, 0, 51, 0, 53, 0, 39, 0, 53, 0, 59, 0, 61, 0, 63, 0, 57, 0, 67, 0, 57, 0, 71, 0, 73, 0, 43, 0, 73, 0, 79, 0, 45, 0, 83, 0, 85, 0, 39, 0, 89, 0, 67, 0, 33, 0, 95

Offset: 1

Antti Karttunen, Aug 28 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A003987, A032742, A060681, A106409, A318507, A318516, A318518.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A318517(n) = bitxor(A032742(n),n-A032742(n));

a(n) = A003987(A032742(n), A060681(n)).

a(n) = n - 2*A318518(n).

A322809 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = floor(n/2) for all other numbers.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 26 2018

nonn

This sequence is a restricted growth sequence transform of a function f which is defined as f(n) = A004526(n), unless n is an odd prime, in which case f(n) = -1, which is a constant not in range of A004526. See the Crossrefs section for a list of similar sequences.
For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A039636(i) = A039636(j).
For all i, j: a(i) = a(j) <=> A323161(i+1) = A323161(j+1).
The shifted version of this filter, A323161, has a remarkable ability to find many sequences related to primes and prime chains. - Antti Karttunen, Jan 06 2019

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A004526, A039636, A305801, A323161.
A list of few similarly constructed sequences follows, where each sequence is an rgs-transform of such function f, for which the value of f(n) is the n-th term of the sequence whose A-number follows after a parenthesis, unless n is of the form ..., in which case f(n) is given a constant value outside of the range of that sequence:
  A322809 (A004526, unless an odd prime) [This sequence],
  A322589 (A007913, unless an odd prime),
  A322591 (A007947, unless an odd prime),
  A322805 (A252463, unless an odd prime),
  A323082 (A300840, unless an odd prime),
  A322822 (A300840, unless n > 2 and a Fermi-Dirac prime, A050376),
  A322988 (A322990, unless a prime power > 2),
  A323078 (A097246, unless an odd prime),
  A322808 (A097246, unless a squarefree number > 2),
  A322816 (A048675, unless an odd prime),
  A322807 (A285330, unless an odd prime),
  A322814 (A286621, unless an odd prime),
  A322824 (A242424, unless an odd prime),
  A322973 (A006370, unless an odd prime),
  A322974 (A049820, unless n > 1 and n is in A046642),
  A323079 (A060681, unless an odd prime),
  A322587 (A295887, unless an odd prime),
  A322588 (A291751, unless an odd prime),
  A322592 (A289625, unless an odd prime),
  A323369 (A323368, unless an odd prime),
  A323371 (A295886, unless an odd prime),
  A323374 (A323373, unless an odd prime),
  A323401 (A323372, unless an odd prime),
  A323405 (A323404, unless an odd prime).

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; };
A322809aux(n) = if((n>2)&&isprime(n),-1,(n>>1));
v322809 = rgs_transform(vector(up_to,n,A322809aux(n)));
A322809(n) = v322809[n];

a(n) = A323161(n+1) - 1.

A326064 Odd composite numbers n, not squares of primes, such that (A001065(n) - A032742(n)) divides (n - A032742(n)), where A032742 gives the largest proper divisor, and A001065 is the sum of proper divisors.

Original entry on oeis.org

117, 775, 10309, 56347, 88723, 2896363, 9597529, 12326221, 12654079, 25774633, 29817121, 63455131, 105100903, 203822581, 261019543, 296765173, 422857021, 573332713, 782481673, 900952687, 1129152721, 3350861677, 3703086229, 7395290407, 9347001661, 9350506057

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

Nineteen initial terms factored:
   n       a(n)   factorization   A060681(a(n))/A318505(a(n))
      117 =   3^2 *    13,           (3)
      775 =   5^2 *    31,          (10)
    10309 =  13^2 *    61,          (39)
    56347 =  29^2 *    67,          (58)
    88723 =  17^2 *   307,         (136)
  2896363 =  41^2 *  1723,         (820)
  9597529 =  73^2 *  1801,        (1314)
 12326221 =  59^2 *  3541,        (1711)
 12654079 = 113^2 *   991,         (904)
 25774633 =  71^2 *  5113,        (2485)
 29817121 =  97^2 *  3169,        (2328)
 63455131 =  89^2 *  8011,        (3916)
105100903 = 101^2 * 10303,        (5050)
203822581 = 157^2 *  8269,        (6123)
261019543 = 349^2 *  2143,        (2094)
296765173 = 131^2 * 17293,        (8515)
422857021 = 233^2 *  7789,        (6757)
573332713 = 331^2 *  5233,        (4965)
782481673 = 167^2 * 28057,       (13861).
Note how the quotient (in the rightmost column) seems always to be a multiple of non-unitary prime factor and less than the unitary prime factor.
For p, q prime, if p^2+p+1 = kq and k+1|p-1, then p^2*q is in this sequence. - Charlie Neder, Jun 09 2019

Subsequence of A326063.
Cf. A032742, A060681, A246282, A318505.
Cf. also A228058, A325981, A326131, A326141.

Select[Range[15, 10^6 + 1, 2], And[! PrimePowerQ@ #1, Mod[#1 - #2, #2 - #3] == 0] & @@ {#, DivisorSigma[1, #] - #, Divisors[#][[-2]]} &] (* Michael De Vlieger, Jun 22 2019 *)

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A060681(n) = (n-A032742(n));
A318505(n) = if(1==n,0,(sigma(n)-A032742(n))-n);
isA326064(n) = if((n%2)&&(2!=isprimepower(n)), my(s=A032742(n), t=sigma(n)-s); (gcd(t-n, n-A032742(n)) == t-n), 0);

More terms from Amiram Eldar, Dec 24 2020

A328449 Smallest number in whose divisors the longest run is of length n, and 0 if none exists.

Original entry on oeis.org

0, 1, 2, 6, 12, 0, 60, 420, 840, 0, 2520, 0, 27720, 0, 0, 360360, 720720, 0, 12252240, 0, 0, 0, 232792560, 0, 5354228880, 0, 26771144400, 0, 80313433200, 0, 2329089562800, 72201776446800, 0, 0, 0, 0, 144403552893600, 0, 0, 0, 5342931457063200, 0

Offset: 0

Gus Wiseman, Oct 16 2019

nonn

Wikipedia, Run (cards)

Positions of 0's are 0 followed by A024619 - 1.
The version that looks only at all divisors > 1 is A328448.
The longest run of divisors of n has length A055874.
The longest run of divisors of n greater than one has length A328457.
Numbers whose divisors have no non-singleton runs are A005408.
The number of successive pairs of divisors of n is A129308(n).
The Heinz number of the multiset of run-lengths of divisors of n is A328166(n).
The smallest number whose divisors have a (not necessarily longest) maximal run of length n is A181063.
Cf. A000005, A003601, A027750, A033676, A060680, A060681, A199970, A328195.

tav=Table[Max@@Length/@Split[Divisors[n],#2==#1+1&],{n,10000}];
Table[If[FreeQ[tav,i],0,Position[tav,i][[1,1]]],{i,0,Max@@tav}]

a(n) = LCM(1,2,...,n) = A003418(n) if n + 1 is a prime power, otherwise a(n) = 0.

A333790 Smallest path sum when iterating from n to 1 with nondeterministic map k -> k - k/p, where p is any prime factor of k.

Original entry on oeis.org

1, 3, 6, 7, 12, 12, 19, 15, 21, 22, 33, 24, 37, 33, 37, 31, 48, 39, 58, 42, 54, 55, 78, 48, 67, 63, 66, 61, 90, 67, 98, 63, 88, 82, 96, 75, 112, 96, 102, 82, 123, 96, 139, 99, 112, 124, 171, 96, 145, 117, 133, 115, 168, 120, 154, 117, 153, 148, 207, 127, 188, 160, 159, 127, 180, 154, 221, 150, 193, 166, 237, 147, 220, 186, 192, 172, 231

Offset: 1

Antti Karttunen, Apr 06 2020

nonn

Note that although in many cases a simple heuristics of always subtracting the largest proper divisor (i.e., iterating with A060681) gives the path with the minimal sum, this does not hold for the following numbers 119, 143, 187, 209, 221, ..., A333789, on which this sequence differs from A073934.

			For n=119, the graph obtained is this:
              119
             _/\_
            /    \
          102    112
         _/|\_    | \_
       _/  |  \_  |   \_
      /    |    \ |     \
    51     68    96     56
    /|   _/ |   _/|   _/ |
   / | _/   | _/  | _/   |
  /  |/     |/    |/     |
(48) 34    64     48    28
     |\_    |    _/|   _/|
     |  \_  |  _/  | _/  |
     |    \_|_/    |/    |
    17     32     24    14
      \_    |    _/|   _/|
        \_  |  _/  | _/  |
          \_|_/    |/    |
           16      12    7
            |    _/|    _/
            |  _/  |  _/
            |_/    |_/
            8     _6
            |  __/ |
            |_/    |
            4      3
             \     /
              \_ _/
                2
                |
                1.
By choosing the path that follows the right edge of the above diagram, we obtain the smallest sum for any such path that goes from 119 to 1, thus a(119) = 119+112+56+28+14+7+6+3+2+1 = 348.
Note that if we always subtracted the largest proper divisor (A032742), i.e., iterated with A060681 (starting from 119), we would obtain 119-(119/7) = 102 -> 102-(102/2) -> 51-(51/3) -> 34-(34/2) -> 17-(17/17) -> 16-(16/2) -> 8-(8/2) -> 4-(4/2) -> 2-(2/2) -> 1, with sum 119+102+51+34+17+16+8+4+2+1 = 354 = A073934(119), which is NOT minimal sum in this case.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Michael De Vlieger, Graph montage of k -> k - k/p, with prime p|k for 2 <= k <= 121, red line showing path of least sum.

Cf. A032742, A060681, A332904, A333000, A333001, A333123, A333794.

Differs from A073934 for the first time at n=119, where a(119) = 348, while A073934(119) = 354. (See A333789).

Min@ Map[Total, #] & /@ Nest[Function[{a, n}, Append[a, Join @@ Table[Flatten@ Prepend[#, n] & /@ a[[n - n/p]], {p, FactorInteger[n][[All, 1]]}]]] @@ {#, Length@ # + 1} &, {{{1}}}, 76]   (* Michael De Vlieger, Apr 14 2020 *)

up_to = 65537; \\ 2^20;
A333790list(up_to) = { my(v=vector(up_to)); v[1] = 1; for(n=2, up_to, v[n] = n+vecmin(apply(p -> v[n-n/p], factor(n)[, 1]~))); (v); };
v333790 = A333790list(up_to);
A333790(n) = v333790[n];

a(n) = n + Min a(n - n/p), for p prime and dividing n.

For n >= 1, a(n) <= A333794(n) <= A332904(n), a(n) <= A333001(n).

A060685 Largest difference between consecutive divisors (ordered by size) of 2n+1.

Original entry on oeis.org

2, 4, 6, 6, 10, 12, 10, 16, 18, 14, 22, 20, 18, 28, 30, 22, 28, 36, 26, 40, 42, 30, 46, 42, 34, 52, 44, 38, 58, 60, 42, 52, 66, 46, 70, 72, 50, 66, 78, 54, 82, 68, 58, 88, 78, 62, 76, 96, 66, 100, 102, 70, 106, 108, 74, 112, 92, 78, 102, 110, 82, 100, 126, 86, 130, 114

Offset: 1

Labos Elemer, Apr 19 2001

nonn

Equivalently, a(2n+1) = 2n+1 - (2n+1)/p, where p = A020639(2n+1) is the smallest prime divisor of 2n+1.
The even case is trivial: for 2k the largest difference is k.
Successively greater values of a(n) occur when 2n+1 is prime.

			For n=17, 2n+1=35; divisors={1,5,7,35}; differences={4,2,28}; a(17) = largest difference = 28 = 35 - 35/5.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A060681.

a[n_] := 2n+1-(2n+1)/FactorInteger[2n+1][[1, 1]]
(* Second program: *)
Table[Max@ Differences@ Divisors@ # &[2 n + 1], {n, 66}] (* Michael De Vlieger, Jul 15 2017 *)

a(n) = A060681(2n+1)

Edited by Dean Hickerson, Jan 22 2002

A060766 Least common multiple of differences between consecutive divisors of n (ordered by size).

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 4, 6, 15, 10, 6, 12, 35, 10, 8, 16, 9, 18, 10, 28, 99, 22, 12, 20, 143, 18, 42, 28, 60, 30, 16, 88, 255, 28, 18, 36, 323, 130, 60, 40, 21, 42, 154, 60, 483, 46, 24, 42, 75, 238, 234, 52, 27, 132, 84, 304, 783, 58, 60, 60, 899, 84, 32, 104, 165, 66, 442

Offset: 2

Labos Elemer, Apr 24 2001

nonn

			For n=98, divisors={1,2,7,14,49,98}; differences={1,5,7,35,49}; a(98) = LCM of differences = 245.

Harvey P. Dale, Table of n, a(n) for n = 2..1000

The GCD version appears to be A258409.
The LCM of the prime indices of n is A290103(n).
The differences between consecutive divisors of n are row n of A193829.
Cf. A000005, A027750, A060680, A060681, A060683, A328023, A328026, A328219.

a[n_ ] := LCM@@(Drop[d=Divisors[n], 1]-Drop[d, -1])
Table[LCM@@Differences[Divisors[n]],{n,2,70}] (* Harvey P. Dale, Oct 08 2012 *)

a(n) = A290103(A328023(n)). - Gus Wiseman, Oct 16 2019

Edited by Dean Hickerson, Jan 22 2002

A318516 a(n) = A032742(n) OR n-A032742(n), where OR is bitwise-or (A003986) and A032742 = the largest proper divisor of n.

Original entry on oeis.org

1, 1, 3, 2, 5, 3, 7, 4, 7, 5, 11, 6, 13, 7, 15, 8, 17, 9, 19, 10, 15, 11, 23, 12, 21, 13, 27, 14, 29, 15, 31, 16, 31, 17, 31, 18, 37, 19, 31, 20, 41, 21, 43, 22, 31, 23, 47, 24, 47, 25, 51, 26, 53, 27, 47, 28, 55, 29, 59, 30, 61, 31, 63, 32, 61, 33, 67, 34, 63, 35, 71, 36, 73, 37, 59, 38, 75, 39, 79, 40, 63, 41, 83, 42

Offset: 1

Antti Karttunen, Aug 28 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A003986, A032742, A060681, A318506, A318514, A318517, A318518.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A318516(n) = bitor(A032742(n),n-A032742(n));

a(n) = A003986(A032742(n), A060681(n)).

a(n) = n - A318518(n).

A318518 a(n) = A032742(n) AND n-A032742(n), where AND is bitwise-and (A004198) and A032742 = the largest proper divisor of n.

Original entry on oeis.org

0, 1, 0, 2, 0, 3, 0, 4, 2, 5, 0, 6, 0, 7, 0, 8, 0, 9, 0, 10, 6, 11, 0, 12, 4, 13, 0, 14, 0, 15, 0, 16, 2, 17, 4, 18, 0, 19, 8, 20, 0, 21, 0, 22, 14, 23, 0, 24, 2, 25, 0, 26, 0, 27, 8, 28, 2, 29, 0, 30, 0, 31, 0, 32, 4, 33, 0, 34, 6, 35, 0, 36, 0, 37, 16, 38, 2, 39, 0, 40, 18, 41, 0, 42, 0, 43, 24, 44, 0, 45, 12, 46, 30, 47, 0, 48, 0, 49, 0, 50, 0, 51, 0

Offset: 1

Antti Karttunen, Aug 28 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A004198, A032742, A060681, A318515, A318516, A318517.

Cf. also A318508.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A318518(n) = bitand(A032742(n),n-A032742(n));

a(n) = A004198(A032742(n), A060681(n)).

a(n) = n - A318516(n) = (n - A318517(n))/2.

A323076 Number of iterations of map x -> 1+(x-(largest divisor d < x)), starting from x=n, needed to reach a fixed point, which is always either a prime or 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

Differs from A064918 at n = 25, 48, 51, 69, 75, 81, 85, 94, 95, 99, 100, 111, 115, 121, ...

Antti Karttunen, Table of n, a(n) for n = 1..20669
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A060681, A064918, A323075 (the fixed points reached), A323077, A323079.

Cf. also A039651.

{0}~Join~Array[-2 + Length@ NestWhileList[1 + (# - Divisors[#][[-2]]) &, #, UnsameQ, All] &, 104, 2] (* Michael De Vlieger, Jan 04 2019 *)

A060681(n) = (n-if(1==n,n,n/vecmin(factor(n)[,1])));
A323076(n) = { my(nn = 1+A060681(n)); if(nn==n,0,1+A323076(nn)); };

If n == (1+A060681(n)), then a(n) = 0, otherwise a(n) = 1 + a(1+A060681(n)).

cp's OEIS Frontend

A318517 a(n) = A032742(n) XOR n-A032742(n), where XOR is bitwise-xor (A003987) and A032742 = the largest proper divisor of n.

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A322809 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = floor(n/2) for all other numbers.

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A326064 Odd composite numbers n, not squares of primes, such that (A001065(n) - A032742(n)) divides (n - A032742(n)), where A032742 gives the largest proper divisor, and A001065 is the sum of proper divisors.

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A328449 Smallest number in whose divisors the longest run is of length n, and 0 if none exists.

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A333790 Smallest path sum when iterating from n to 1 with nondeterministic map k -> k - k/p, where p is any prime factor of k.

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A060685 Largest difference between consecutive divisors (ordered by size) of 2n+1.

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A060766 Least common multiple of differences between consecutive divisors of n (ordered by size).

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A318516 a(n) = A032742(n) OR n-A032742(n), where OR is bitwise-or (A003986) and A032742 = the largest proper divisor of n.

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