A354047 -id:A354047 - cp's OEIS frontend

A381624 For any nonnegative integer n with ternary expansion Sum_{k >= 0} t_k * 3^k, a(n) = Sum_{k >= 0} t_k * (2^(k+1) - 1).

0, 1, 2, 3, 4, 5, 6, 7, 8, 7, 8, 9, 10, 11, 12, 13, 14, 15, 14, 15, 16, 17, 18, 19, 20, 21, 22, 15, 16, 17, 18, 19, 20, 21, 22, 23, 22, 23, 24, 25, 26, 27, 28, 29, 30, 29, 30, 31, 32, 33, 34, 35, 36, 37, 30, 31, 32, 33, 34, 35, 36, 37, 38, 37, 38, 39, 40, 41

Offset: 0

Rémy Sigrist, Mar 02 2025

Every nonnegative integer appears in the sequence, a finite number of times.

			a(42) = a(3^3 + 3^2 + 2*3) = (2^(3+1) - 1) + (2^(2+1) - 1) + 2*(2^(1+1) - 1) = 28.

Rémy Sigrist, Table of n, a(n) for n = 0..6560

Cf. A354047, A381625.

a(n) = { my (v = 0, k); while (n, n -= 3^k = valuation(n, 3); v += 2^(k+1) - 1;); return (v); }

a(A354047(n)) = n.

A354059 Square array read by ascending antidiagonals: T(n,k) is the number of elements in the multiplicative group of integers modulo n that have order k.

Original entry on oeis.org

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

Offset: 1

Jianing Song, May 16 2022

Row n and Row n' are the same if and only if (Z/nZ)* = (Z/n'Z)*, where (Z/nZ)* is the multiplicative group of integers modulo n.

For the truncated version see A252911.

			The 7th, 9th, 14th and 18th rows of A354047 are {1,2,3,2,1,6,1,2,3,2,1,6,...}, so applying the Moebius transform gives {1,1,2,0,0,2,0,0,0,0,0,0,...}.

Jianing Song, Table of n, a(n) for n = 1..5050

Moebius transform of A354057 applied to each row.

Cf. A327924.

b(n,k)=my(Z=znstar(n)[2]); prod(i=1, #Z, gcd(k, Z[i]));
T(n,k) = sumdiv(k, d, moebius(k/d)*b(n,d))

A327924(n,k) = Sum_{d|k} T(n,k)/phi(d).

cp's OEIS Frontend

A381624 For any nonnegative integer n with ternary expansion Sum_{k >= 0} t_k * 3^k, a(n) = Sum_{k >= 0} t_k * (2^(k+1) - 1).

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