A048864 -id:A048864 - cp's OEIS frontend

A048984 As n runs through composite numbers, a(n) = number of nonprime d < n such that gcd(d,n) = 1.

1, 1, 1, 3, 2, 1, 2, 4, 3, 1, 2, 6, 4, 1, 12, 5, 10, 5, 1, 6, 11, 7, 15, 3, 8, 14, 6, 2, 8, 12, 10, 3, 28, 7, 19, 11, 4, 26, 10, 22, 14, 2, 14, 20, 15, 32, 5, 15, 27, 8, 6, 17, 21, 17, 41, 6, 12, 33, 20, 4, 43, 21, 35, 19, 3, 50, 22, 38, 24, 50, 10, 19, 37

Offset: 1

Naohiro Nomoto

It appears that a(n) = A048864(A002808(n)). - Michel Marcus, Jul 14 2013

			9 is 4th composite number, gcd(9,1) = gcd(9,4) = gcd(9,8) = 1, so a(4) = 3.

Cf. A000010, A048983, A002808.

ai(nn) = {forcomposite (n=1, nn, nb = 0; for(d=1, n-1, if (! isprime(d) && gcd(n, d) == 1, nb++)); print1(nb, ", "););} \\ Michel Marcus, Jul 14 2013

a(n) = A048983(n) + 1. - Michel Marcus, Jul 14 2013

A285788 Irregular triangle T(n,m): nonprime 1 <= k <= n such that n and k are coprime.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 4, 6, 1, 1, 4, 8, 1, 9, 1, 4, 6, 8, 9, 10, 1, 1, 4, 6, 8, 9, 10, 12, 1, 9, 1, 4, 8, 14, 1, 9, 15, 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 1, 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 1, 9, 1, 4, 8, 10, 16, 20, 1, 9, 15, 21, 1, 4, 6, 8, 9, 10

Offset: 1

Michael De Vlieger, Apr 26 2017

Row n is a subset of A038566(n) such that the union of a(n) and A112484(n) = A038566(n).
Row lengths are A048864(n) = A000010(n)-(A000720(n)-A001221(n)), i.e., phi(n)-(pi(n)-omega(n)).
1 appears in every row since 1 is not prime and coprime to all n.
4 is the smallest composite and appears first in row 5 since 4 divides 4.
Rows that contain the single term 1 are in A048597; the largest n = 30 such that the only term is 1.
For prime p, row p contains 1 and all composites k < p, since 1 < m < p are coprime to p.

			Triangle begins:
  n\m  1  2   3   4  5   6   7
   1:  1
   2:  1
   3:  1
   4:  1
   5:  1  4
   6:  1
   7:  1  4   6
   8:  1
   9:  1  4   8
  10:  1  9
  11:  1  4   6   8  9  10
  12:  1
  13:  1  4   6   8  9  10  12
  14:  1  9
  15:  1  4   8  14
  16:  1  9  15
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11055 (rows 1 <= n <= 240)

Cf. A038566, A048597, A048864, A112484.

Table[Select[Range@ n, And[! PrimeQ@ #, CoprimeQ[#, n]] &], {n, 23}] // Flatten

from sympy import gcd, isprime
def a(n): return list(filter(lambda k: isprime(k)==0 and gcd(k, n)==1, range(1, n + 1)))
for n in range(1, 21): print(a(n)) # Indranil Ghosh, Apr 26 2017

A048982 Number of numbers which have a "prime-rich" reduced residue system (RRS) and binary order n.

Original entry on oeis.org

0, 0, 0, 1, 3, 8, 15, 22, 32, 50, 85, 80, 98, 84, 59, 37, 10, 2, 0, 0, 0

Offset: 0

Labos Elemer

It is remarkable that in exponentially increasing ranges these occurrences increase to n=13 and thereafter decline to zero. So A048868 is believed to be finite.

			In binary order (A029837) zone of 7, i.e., in [65,128], 22 numbers belong to A048868: 66, 68, 70, 72, 74, 76, 78, 80, 84, 88, 90, 96, 98, 100, 102, 104, 108, 110, 112, 114, 120, and 126. The largest term is 90090. The largest 4 are divisible by 2310, the largest 28 by 210, and the largest 103 by 30.

Cf. A048862, A048863, A048864, A048865, A048866, A048867, A048868, A048869.

Cf. A002110, A029837, A048597.

cp's OEIS Frontend

A048984 As n runs through composite numbers, a(n) = number of nonprime d < n such that gcd(d,n) = 1.

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A285788 Irregular triangle T(n,m): nonprime 1 <= k <= n such that n and k are coprime.

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A048982 Number of numbers which have a "prime-rich" reduced residue system (RRS) and binary order n.

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