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A198067 Square array read by antidiagonals, n>=1, k>=1; T(n,k) is the number of nonprime numbers which are prime to n and are not strong divisors of k. 2
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 2, 1, 1, 1, 1, 2, 3, 1, 2, 1, 2, 1, 1, 1, 1, 6, 2, 3, 1, 3, 1, 2, 1, 1, 1, 1, 1, 6, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 7, 1, 6, 2, 3, 1, 3, 1, 2 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,11

COMMENTS

We say d is a strong divisor of n iff d is a divisor of n and d > 1. Let alpha(n) be number of nonprime numbers in the reduced residue system of n. Then alpha(n) = T(n,1) = T(n,n).

LINKS

Table of n, a(n) for n=1..87.

Peter Luschny, Euler's totient function

EXAMPLE

T(15, 22) = card({1,4,8,14}) = 4 because the coprimes of 15 are {1,2,4,7,8,11,13,14} and the strong divisors of 22 are {2,11,22}.

-

[x][1][2][3][4][5][6][7][8]

[1] 1, 1, 1, 1, 1, 1, 1, 1

[2] 1, 1, 1, 1, 1, 1, 1, 1

[3] 1, 1, 1, 1, 1, 1, 1, 1

[4] 1, 1, 1, 1, 1, 1, 1, 1

[5] 2, 2, 2, 1, 2, 2, 2, 1

[6] 1, 1, 1, 1, 1, 1, 1, 1

[7] 3, 3, 3, 2, 3, 2, 3, 2

[8] 1, 1, 1, 1, 1, 1, 1, 1

-

Triangle k=1..n, n>=1:

[1]           1

[2]          1, 1

[3]        1, 1, 1

[4]       1, 1, 1, 1

[5]     2, 2, 2, 1, 2

[6]    1, 1, 1, 1, 1, 1

[7]  3, 3, 3, 2, 3, 2, 3

[8] 1, 1, 1, 1, 1, 1, 1, 1

-

Triangle n=1..k, k>=1:

[1]           1

[2]          1, 1

[3]        1, 1, 1

[4]       1, 1, 1, 1

[5]     1, 1, 1, 1, 2

[6]    1, 1, 1, 1, 2, 1

[7]  1, 1, 1, 1, 2, 1, 3

[8] 1, 1, 1, 1, 1, 1, 2, 1

MAPLE

strongdivisors := n -> numtheory[divisors](n) minus {1}:

coprimes  := n -> select(k->igcd(k, n)=1, {$1..n}):

nonprimes := n -> remove(isprime, {$1..n});

T := (n, k) -> nops(nonprimes(n) intersect (coprimes(n) minus strongdivisors(k))):

seq(seq(T(n-k+1, k), k=1..n), n=1..13);  # Square array by antidiagonals.

seq(print(seq(T(n, k), k=1..n)), n=1..8); # Lower triangle.

seq(print(seq(T(n, k), n=1..k)), k=1..8); # Upper triangle.

CROSSREFS

Cf. A000010, A048864, A193804, A193805, A198066.

Sequence in context: A211111 A074971 A344008 * A282749 A132587 A318930

Adjacent sequences:  A198064 A198065 A198066 * A198068 A198069 A198070

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Nov 07 2011

STATUS

approved

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Last modified September 21 07:28 EDT 2021. Contains 347596 sequences. (Running on oeis4.)