A060681 -id:A060681 - cp's OEIS frontend

A073932 Define f(n) = n - largest nontrivial divisor of n or f(n) = n-1 if n is a prime [that is, f(n) = A060681(n)]. Form a triangle in which the n-th row contains terms n, f(n), f(f(n)), ... until a 1 is reached; sequence gives triangle read by rows.

1, 2, 1, 3, 2, 1, 4, 2, 1, 5, 4, 2, 1, 6, 3, 2, 1, 7, 6, 3, 2, 1, 8, 4, 2, 1, 9, 6, 3, 2, 1, 10, 5, 4, 2, 1, 11, 10, 5, 4, 2, 1, 12, 6, 3, 2, 1, 13, 12, 6, 3, 2, 1, 14, 7, 6, 3, 2, 1, 15, 10, 5, 4, 2, 1, 16, 8, 4, 2, 1, 17, 16, 8, 4, 2, 1, 18, 9, 6, 3, 2, 1, 19, 18, 9, 6, 3, 2, 1, 20, 10, 5, 4, 2, 1

Offset: 1

Amarnath Murthy, Aug 19 2002

			Triangle begins:
   1;
   2, 1;
   3, 2, 1;
   4, 2, 1;
   5, 4, 2, 1;
   6, 3, 2, 1;
   7, 6, 3, 2, 1;
   8, 4, 2, 1;
   9, 6, 3, 2, 1;
  10, 5, 4, 2, 1;

Michael De Vlieger, Table of n, a(n) for n = 1..12386 (rows 1 <= n <= 1000, flattened)
John Machacek, Egyptian Fractions and Prime Power Divisors, Journal of Integer Sequences, Vol. 21 (2018), Article 18.3.7.

Cf. A032742, A060681, A073933, A073934, A073935.

j := 1:a[1] := 1:for i from 2 to 50 do n := i:j := j+1:a[j] := n:while(n>1) do if isprime(n) then r := n-1: else r := n-n/ifactors(n)[2][1][1]; fi; n := r:j := j+1:a[j] := n: od:od:seq(a[k],k=1..j);

Array[If[# == 1, {1}, NestWhileList[If[PrimeQ@ #, # - 1, # - #/FactorInteger[#][[1, 1]] ] &, #, # > 1 &]] &, 20] // Flatten  (* Michael De Vlieger, Apr 15 2020 *)

More terms from Sascha Kurz, Aug 23 2002

Offset corrected from 0 to 1 by Antti Karttunen, Aug 23 2017

A323071 a(n) = gcd(n, 1+A060681(n)).

Original entry on oeis.org

1, 2, 3, 1, 5, 2, 7, 1, 1, 2, 11, 1, 13, 2, 1, 1, 17, 2, 19, 1, 3, 2, 23, 1, 1, 2, 1, 1, 29, 2, 31, 1, 1, 2, 1, 1, 37, 2, 3, 1, 41, 2, 43, 1, 1, 2, 47, 1, 1, 2, 1, 1, 53, 2, 5, 1, 3, 2, 59, 1, 61, 2, 1, 1, 1, 2, 67, 1, 1, 2, 71, 1, 73, 2, 3, 1, 1, 2, 79, 1, 1, 2, 83, 1, 1, 2, 1, 1, 89, 2, 1, 1, 3, 2, 1, 1, 97, 2, 1, 1, 101, 2, 103, 1, 1

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

Differs from A055023 at n = 55, 105, 155, ..., (A323070).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A055023, A060681, A323070, A323072.

A060681(n) = (n-if(1==n,n,n/vecmin(factor(n)[,1])));
A323071(n) = gcd(n,1+A060681(n));

a(n) = gcd(n, 1+A060681(n)).

a(n) = n/A323072(n).

A323072 a(n) = n/A323071(n) = n/gcd(n, 1+A060681(n)).

Original entry on oeis.org

1, 1, 1, 4, 1, 3, 1, 8, 9, 5, 1, 12, 1, 7, 15, 16, 1, 9, 1, 20, 7, 11, 1, 24, 25, 13, 27, 28, 1, 15, 1, 32, 33, 17, 35, 36, 1, 19, 13, 40, 1, 21, 1, 44, 45, 23, 1, 48, 49, 25, 51, 52, 1, 27, 11, 56, 19, 29, 1, 60, 1, 31, 63, 64, 65, 33, 1, 68, 69, 35, 1, 72, 1, 37, 25, 76, 77, 39, 1, 80, 81, 41, 1, 84, 85, 43, 87, 88, 1, 45, 91, 92, 31, 47

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

Differs from A055032 at n = 55, 105, 155, ..., (A323070).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A055032, A060681, A323070, A323071.

A060681(n) = (n-if(1==n,n,n/vecmin(factor(n)[,1])));
A323071(n) = gcd(n,1+A060681(n));
A323072(n) = (n/A323071(n));

a(n) = n/A323071(n) = n/gcd(n, 1+A060681(n)).

A322871 Ordinal transform of A060681, where A060681(n) = n - A032742(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 3, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 3, 2

Offset: 1

Antti Karttunen, Dec 29 2018

nonn

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

Cf. A032742, A060681, A081373, A303756, A322870, A322872, A322873, A322874.

A060681[n_] := n - n/FactorInteger[n][[1, 1]];
b[_] = 1;
a[n_] := a[n] = With[{t = A060681[n]}, b[t]++];
a /@ Range[1, 105] (* Jean-François Alcover, Dec 19 2021 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A060681(n) = if(1==n,0,(n-(n/vecmin(factor(n)[, 1]))));
v322871 = ordinal_transform(vector(up_to,n,A060681(n)));
A322871(n) = v322871[n];

A322873 Ordinal transform of A300721, which is Möbius transform of A060681.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 1, 3, 1, 4, 3, 4, 1, 4, 1, 5, 2, 5, 1, 6, 2, 6, 2, 3, 1, 7, 1, 1, 2, 7, 2, 4, 1, 8, 3, 2, 1, 5, 1, 3, 3, 9, 1, 3, 2, 8, 4, 4, 1, 6, 2, 5, 3, 10, 1, 4, 1, 11, 1, 5, 3, 4, 1, 6, 2, 7, 1, 6, 1, 12, 2, 4, 1, 7, 1, 7, 4, 13, 1, 8, 1, 14, 2, 1, 1, 5, 2, 3, 3, 15, 1, 8, 1, 8, 2, 2, 1, 9, 1, 3, 4

Offset: 1

Antti Karttunen, Dec 29 2018

nonn

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

Cf. A060681, A300721, A081373, A303756, A322871, A322874.

A060681[n_] := n - n/FactorInteger[n][[1, 1]];
A300721[n_] := Sum[MoebiusMu[n/d] A060681[d], {d, Divisors[n]}];
b[_] = 1;
a[n_] := a[n] = With[{t = A300721[n]}, b[t]++];
a /@ Range[1, 105] (* Jean-François Alcover, Dec 19 2021 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A060681(n) = if(1==n,0,(n-(n/vecmin(factor(n)[, 1]))));
A300721(n) = sumdiv(n, d, moebius(n/d)*A060681(d));
v322873 = ordinal_transform(vector(up_to,n,A300721(n)));
A322873(n) = v322873[n];

A323079 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) = A060681(n) for all other numbers.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

For all i, j:
  a(i) = a(j) => A323076(i) = A323076(j),
  a(i) = a(j) => A323077(i) = A323077(j).

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

Cf. A060681, A323076, A323077.

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; };
A060681(n) = (n-if(1==n,n,n/vecmin(factor(n)[,1])));
A323079aux(n) = if((n>2)&&isprime(n),-1,A060681(n));
v323079 = rgs_transform(vector(up_to,n,A323079aux(n)));
A323079(n) = v323079[n];

A300721 Möbius transform of A060681, the largest difference between consecutive divisors of n (ordered by size).

Original entry on oeis.org

0, 1, 2, 1, 4, 0, 6, 2, 4, 0, 10, 2, 12, 0, 4, 4, 16, 2, 18, 4, 6, 0, 22, 4, 16, 0, 12, 6, 28, 4, 30, 8, 10, 0, 18, 6, 36, 0, 12, 8, 40, 6, 42, 10, 16, 0, 46, 8, 36, 4, 16, 12, 52, 6, 30, 12, 18, 0, 58, 8, 60, 0, 24, 16, 36, 10, 66, 16, 22, 6, 70, 12, 72, 0, 24, 18, 50, 12, 78, 16, 36, 0, 82, 12, 48, 0, 28, 20, 88, 8, 60, 22, 30, 0, 54, 16

Offset: 1

Antti Karttunen, Mar 11 2018

nonn

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

Cf. A000010, A008683, A060681, A300236, A300722, A322873 (ordinal transform).

A060681[n_] := n - n/FactorInteger[n][[1, 1]];
a[n_] := Sum[MoebiusMu[n/d]*A060681[d], {d, Divisors[n]}];
Array[a, 100] (* Jean-François Alcover, Jan 07 2022 *)

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A060681(n) = (n-A032742(n));
A300721(n) = sumdiv(n,d,moebius(n/d)*A060681(d));

a(n) = Sum_{d|n} A008683(n/d)*A060681(d).
a(n) = A060681(n) - A300722(n).
a(n) = A000010(n) - A300236(n).

A302043 a(n) = n - A302042(n); an analog of A060681 based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 6, 4, 6, 5, 10, 6, 12, 7, 10, 8, 16, 9, 18, 10, 12, 11, 22, 12, 20, 13, 20, 14, 28, 15, 30, 16, 18, 17, 28, 18, 36, 19, 28, 20, 40, 21, 42, 22, 24, 23, 46, 24, 42, 25, 26, 26, 52, 27, 30, 28, 30, 29, 58, 30, 60, 31, 50, 32, 54, 33, 66, 34, 36, 35, 70, 36, 72, 37, 58, 38, 66, 39, 78, 40, 42, 41, 82, 42, 50, 43, 52, 44, 88, 45, 42

Offset: 1

Antti Karttunen, Mar 31 2018

nonn

An analog of A060681 based on the sieve of Eratosthenes (A083221).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences generated by sieves

Cf. A060681, A302042.

A302042(n) = if(1==n,n,my(k=0); while((n%2), n = A268674(n); k++); n = n/2; while(k>0, n = A250469(n); k--); (n));
A302043(n) = (n - A302042(n));

a(n) = n - A302042(n).

A300722 Difference between A060681 (largest difference between consecutive divisors of n as ordered by size) and its Möbius transform.

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 0, 2, 2, 5, 0, 4, 0, 7, 6, 4, 0, 7, 0, 6, 8, 11, 0, 8, 4, 13, 6, 8, 0, 11, 0, 8, 12, 17, 10, 12, 0, 19, 14, 12, 0, 15, 0, 12, 14, 23, 0, 16, 6, 21, 18, 14, 0, 21, 14, 16, 20, 29, 0, 22, 0, 31, 18, 16, 16, 23, 0, 18, 24, 29, 0, 24, 0, 37, 26, 20, 16, 27, 0, 24, 18, 41, 0, 30, 20, 43, 30, 24, 0, 37, 18, 24, 32, 47, 22, 32

Offset: 1

Antti Karttunen, Mar 11 2018

nonn

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

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A060681(n) = (n-A032742(n));
A300722(n) = -sumdiv(n,d,(dA060681(d));

a(n) = -Sum_{d|n, dA008683(n/d)*A060681(d).

a(n) = A060681(n) - A300721(n).

A323070 Numbers k such that A055023(k) != A323071(k), where A323071(k) = gcd(k, 1+A060681(k)).

Original entry on oeis.org

55, 105, 155, 203, 253, 355, 405, 455, 497, 595, 655, 689, 705, 737, 755, 791, 955, 979, 1005, 1027, 1055, 1081, 1221, 1255, 1305, 1355, 1379, 1555, 1605, 1655, 1673, 1703, 1711, 1751, 1855, 1905, 1955, 1967, 2065, 2155, 2189, 2205, 2255, 2261, 2329, 2455, 2505, 2555, 2755, 2805, 2849, 2855, 3055

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

Equivalently, numbers k for which A055032(k) != A323072(k).

Neither primes nor prime powers present?

Cf. A055023, A055032, A060681, A323071, A323072.

A055023(n) = (n/denominator((sum(m=1, n - 1, m^(n - 1)) + 1)/n)); \\ From A055023.
A060681(n) = (n-if(1==n,n,n/vecmin(factor(n)[,1])));
A323071(n) = gcd(n,1+A060681(n));
is_A323070(n) = (A055023(n)!=A323071(n));

cp's OEIS Frontend

A073932 Define f(n) = n - largest nontrivial divisor of n or f(n) = n-1 if n is a prime [that is, f(n) = A060681(n)]. Form a triangle in which the n-th row contains terms n, f(n), f(f(n)), ... until a 1 is reached; sequence gives triangle read by rows.

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A323071 a(n) = gcd(n, 1+A060681(n)).

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A323072 a(n) = n/A323071(n) = n/gcd(n, 1+A060681(n)).

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A322871 Ordinal transform of A060681, where A060681(n) = n - A032742(n).

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A322873 Ordinal transform of A300721, which is Möbius transform of A060681.

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A323079 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) = A060681(n) for all other numbers.

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A300721 Möbius transform of A060681, the largest difference between consecutive divisors of n (ordered by size).

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A302043 a(n) = n - A302042(n); an analog of A060681 based on the sieve of Eratosthenes (A083221).

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A300722 Difference between A060681 (largest difference between consecutive divisors of n as ordered by size) and its Möbius transform.