A072455 -id:A072455 - cp's OEIS frontend

A072106 The number of nontotients (even and odd) in the reduced residue system of n.

0, 0, 0, 1, 1, 1, 2, 3, 2, 3, 4, 3, 5, 5, 4, 7, 8, 5, 9, 7, 5, 9, 11, 7, 10, 11, 10, 11, 15, 7, 16, 15, 11, 15, 13, 11, 20, 17, 14, 15, 23, 11, 24, 19, 15, 21, 26, 15, 23, 19, 19, 23, 30, 17, 23, 23, 21, 27, 33, 15, 34, 29, 22, 31, 27, 19, 38, 31, 28, 23, 41, 23, 42, 35, 26, 35, 37

Offset: 1

Labos Elemer, Jun 19 2002

nonn

			For n=113: the reduced residue system consists of 112 numbers: Card[OddNonTotients(113)] = 56 - 1, EvenNonTotients = {14,26,34,38,50,62,68,74,76,86,90,94,98}, i.e., 13 terms, therefore a(113) = 56 - 1 + 13 = 68.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A002202, A005277, A037225, A072451, A072455.

a(n) = sum(k = 1, n, gcd(n, k) == 1 && !istotient(k)); \\ Amiram Eldar, Nov 07 2024

A072454 Number of nontotients in the reduced residue system of 2n-1.

Original entry on oeis.org

0, 0, 1, 2, 2, 4, 5, 4, 8, 9, 5, 11, 10, 10, 15, 16, 11, 13, 20, 14, 23, 24, 15, 26, 23, 19, 30, 23, 21, 33, 34, 22, 27, 38, 28, 41, 42, 26, 37, 47, 35, 49, 37, 37, 53, 44, 38, 43, 59, 41, 62, 63, 32, 65, 66, 46, 68, 55, 46, 58, 69, 53, 64, 79, 55, 81, 65, 50, 85, 86, 60, 77, 72

Offset: 1

Labos Elemer, Jun 19 2002

nonn

			For n=105: phi(105) = 48 with 24 odd, 24 even terms in the reduced residue system, of which 9 even terms and (all but 1) odd term is nontotient: a((105+1)/2) = a(53) = 24-1+9 = 32.
For n=21: reduced residue system(21) = Union({1,5,11,13,17,19}, {2,4,8,16,20}) includes 6 odd and 5 even numbers. No even nontotients terms in the reduced residue system(21), so 6-1 = 5 odd terms give all nontotients, so a((21+1)/2) = a(11) = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A002202, A005277, A037225, A072451, A072455.

Bisection of A072106.

a(n) = {my(m = 2*n-1); sum(k = 1, m, gcd(m, k) == 1 && !istotient(k));} \\ Amiram Eldar, Nov 07 2024

a(n) = A072106(2*n-1). - Amiram Eldar, Nov 07 2024

cp's OEIS Frontend

A072106 The number of nontotients (even and odd) in the reduced residue system of n.

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