A213919 Triangle read by rows: T(n,m) = (n/k)^(k-1), where k is the m-th divisor of n, 1 <= m <= tau(n).

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 4, 8, 1, 1, 9, 1, 1, 5, 16, 1, 1, 1, 1, 6, 16, 27, 32, 1, 1, 1, 1, 7, 64, 1, 1, 25, 81, 1, 1, 8, 64, 128, 1, 1, 1, 1, 9, 36, 243, 256, 1, 1, 1, 1, 10, 125, 256, 512, 1, 1, 49, 729, 1, 1, 11, 1024, 1, 1, 1, 1, 12, 64, 216, 1024, 2187, 2096, 1, 1, 625, 1, 1, 13, 4096, 1, 1, 81, 6561, 1, 1, 14, 343, 4096

Offset

1,7

Comments

Row lengths are tau = A000005 and n-th row sum is equal to A087909(n) = sum_[k divides n}(n/k)^(k-1). If n is the prime then n-th row is 2.

Divisor k of composite number n with maximal value (n/k)^(k-1): 2, 3, 4, 3, 5, 6, 7, 5, 8, 9, 10, 7, 11, 8, 5, 13, 9, 14,...

(Sum_{k|n} (n/k)^(n-1)) mod n: 0, 0, 2, 0, 2, 3, 2, 6, 2, 3, 2, 11, 2, 3, 3, 10, 2, 6, 2, 5, 3, 3, 2, 7, 2, 3, 2, 19, 2, 2, 18, 3, 3, 3, 32, 2, 3, 3, 35, 2, 17, 2, 27, 27, 3, 2, 5, 2, 23, 6, 5, 2,...

Numbers n such that (sum_{k divides n} (n/k)^(k-1)) mod n is no prime: 1, 2, 4, 8, 16, 18, 32, 36, 40, 44, 45,...

Numbers n such that tau(n) = (sum_{k divides n}(n/k)^(k-1)) mod n: 3, 5, 7, 11, 13, 17, 18, 19, 23, 29, 31, 37, 41, 43, 47, 53,...

Sum_{k divides n}((n/k)^(k-1) mod k): 0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 7, 1, 3, 3, 1, 1, 9, 1, 5, 3, 3, 1, 17, 1, 3, 1, 3, 1,...

(Sum_(k divides n}(n/k)^k)) mod n: 0, 1, 1, 1, 1, 0, 1, 1, 1, 8, 1, 6, 1, 10, 9, 1, 1, 9, 1, 14, 11, 14, 1, 2, 1, 16, 1, 2, 1, 14, 1, 1, 15,...

Links

Table of n, a(n) for n=1..99.

Formula

T(n,k) = A027750(n, A000005(n) + 1 - k)/(A027750(n,k) - 1), 1 <= k <= A000005(n).

Example

Triangle begins:
1;
1, 1;
1, 1;
1, 2, 1;
1, 1;
1, 3, 4, 1;
1, 1;
1, 4, 8, 1;
1, 9, 1;
1, 5, 16, 1;
1, 1;
1, 6, 16, 27, 32, 1.

Crossrefs

Cf. A000005, A027750, Ao55225, A087909, A167401, A208239.

Sequence in context: A120621 A201080 A039754 * A362043 A337220 A062277

Adjacent sequences: A213916 A213917 A213918 * A213920 A213921 A213922

Keyword

nonn,tabf

Author

Gerasimov Sergey, Mar 05 2013

Status

approved