A051266 -id:A051266 - cp's OEIS frontend

A051265 Maximal value of prime divisors of numbers in reduced residue system for n.

0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2

Offset: 1

Labos Elemer

The smallest number for which a(n)=k is the n-th Euclid number (A006862=A002110 + 1).

Largest value of A001221(k) for 1 <= k <= n such that gcd(k, n) = 1. - Michael De Vlieger, Aug 10 2017

			For n=60 a(n)=1 since in RRS[ 60 ] only 1 and prime powers occur (see A051250).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Z. Abel, E. Demaine, M. Demaine, H. Matsui and G. Rote, Common Developments of Several Different Orthogonal Boxes, CCCG 2011, Toronto ON, August 10-12, 2011.

Records are A006862.

Cf. A001221, A002110, A051250, A051266, A051267, A051268.

Table[Max@ Map[PrimeNu, Cases[Range[n - 1], k_ /; CoprimeQ[n, k]]] /. k_ /; ! IntegerQ@ k -> 0, {n, 105}] (* Michael De Vlieger, Aug 10 2017 *)

a(n)=my(k=1,s); forprime(p=2,, if(n%p==0, next); k*=p; if(k>n, return(s)); s++) \\ Charles R Greathouse IV, Aug 10 2017

a(n) << log n/log log n. - Charles R Greathouse IV, Aug 10 2017

A051267 Numbers n such that maximal value of prime divisors of reduced residue system for n is 3.

Original entry on oeis.org

31, 37, 41, 43, 47, 49, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 106, 107, 109, 111, 113, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 127, 128, 129, 131, 133, 134, 136, 137, 139, 141, 142, 143, 145, 146, 147

Offset: 1

Labos Elemer

nonn

Largest value of A001221(k) = 3 for 1 <= k <= n such that gcd(k, n) = 1, i.e., k in row n of A038566. - Michael De Vlieger, Aug 10 2017

Michael De Vlieger, Table of n, a(n) for n = 1..2446

Cf. A001221, A002110, A038566, A048597, A051250, A051265, A051266, A051268.

Block[{n = 3, P}, P = Product[Prime@ i, {i, n}]; P + Position[#, n][[All, 1]] &@ Array[Max@ Map[PrimeNu, Cases[Range@ #, k_ /; CoprimeQ[#, k]]] &, 117, P + 1]] (* Michael De Vlieger, Aug 10 2017 *)

A051268 Numbers n such that maximal value of prime divisors of reduced residue system for n is 4.

Original entry on oeis.org

211, 221, 223, 227, 229, 233, 239, 241, 247, 251, 253, 257, 263, 269, 271, 277, 281, 283, 289, 293, 299, 307, 311, 313, 317, 319, 323, 331, 337, 341, 343, 347, 349, 353, 359, 361, 367, 371, 373, 377, 379, 383, 389, 391, 397, 401, 403, 407, 409, 413

Offset: 1

Labos Elemer

nonn

Largest value of A001221(k) = 4 for 1 <= k <= n such that gcd(k, n) = 1, i.e., k in row n of A038566. - Michael De Vlieger, Aug 10 2017

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

Cf. A001221, A002110, A006862, A038566, A048597, A051250, A051265, A051266, A051267.

Block[{n = 4, P}, P = Product[Prime@ i, {i, n}]; P + Position[#, n][[All, 1]] &@ Array[Max@ Map[PrimeNu, Cases[Range@ #, k_ /; CoprimeQ[#, k]]] &, 175, P + 1]] (* Michael De Vlieger, Aug 10 2017 *)

More terms from Michael De Vlieger, Aug 10 2017

A289172 Irregular triangle read by rows: row n lists terms m of A038566(n) such that A001221(m) = A051265(n), with a(1) = 1.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 4, 5, 6, 3, 5, 7, 2, 4, 5, 7, 8, 3, 7, 9, 6, 10, 5, 7, 11, 6, 10, 12, 3, 5, 9, 11, 13, 14, 15, 6, 10, 12, 14, 15, 5, 7, 11, 13, 17, 6, 10, 12, 14, 15, 18, 3, 7, 9, 11, 13, 17, 19, 10, 20, 15, 21, 6, 10, 12, 14, 15, 18, 20, 21, 22, 5, 7, 11

Offset: 1

Michael De Vlieger, Aug 11 2017

Consider A051265(n), the largest value of A001221(m) for 1 <= m <= n such that gcd(m, n) = 1 (i.e., m is in the reduced residue system or RRS of n, or m is a totative of n). Row n of this sequence consists of m in RRS(n) such that omega(m) = A051265(n).

			Triangle begins:
   n    T(n,m)                        A051265(n)
   1:   1                                     0
   2:   1                                     0
   3:   2                                     1
   4:   3                                     1
   5:   2    3    4                           1
   6:   5                                     1
   7:   6                                     2
   8:   3    5    7                           1
   9:   2    4    5    7    8                 1
  10:   3    7    9                           1
  11:   6   10                                2
  12:   5    7   11                           1
  13:   6   10   12                           2
  14:   3    5    9   11   13                 1
  15:  14                                     2
  16:  15                                     2
  17:   6   10   12   14   15                 2
  18:   5    7   11   13   17                 1
  19:   6   10   12   14   15   18            2
  20:   3    7    9   11   13   17   19       1

Michael De Vlieger, Table of n, a(n) for the first 1000 rows, flattened

Cf. A001221, A002110, A038566, A048597, A051250, A051265, A051266, A051267, A051268.

Table[MaximalBy[#, Last][[All, 1]] &@ Map[{#, PrimeNu@ #} &, Cases[Range[n - 1], k_ /; CoprimeQ[n, k]]] /. {} -> {1}, {n, 30}] // Flatten (* Michael De Vlieger, Aug 11 2017 *)

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A051265 Maximal value of prime divisors of numbers in reduced residue system for n.

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A051267 Numbers n such that maximal value of prime divisors of reduced residue system for n is 3.

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A051268 Numbers n such that maximal value of prime divisors of reduced residue system for n is 4.

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A289172 Irregular triangle read by rows: row n lists terms m of A038566(n) such that A001221(m) = A051265(n), with a(1) = 1.

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