A362070 Let m_min(n, k) be the smallest m such that n divides Product_{t=1..m} RisingFactorial(t, k). a(n) = Sum_{r=1..K(n)} m_min(n, r), where K(n) is the Kempner number A002034(n).

1, 3, 6, 9, 15, 6, 28, 10, 16, 15, 66, 9, 91, 28, 15, 16, 153, 16, 190, 15, 28, 66, 276, 10, 33, 91, 29, 28, 435, 15, 496, 24, 66, 153, 28, 16, 703, 190, 91, 15, 861, 28, 946, 66, 18, 276, 1128, 16, 54, 33, 153, 91, 1431, 29, 66, 28, 190

Offset

1,2

Comments

The first two solutions of the equation a(n) = n which are not consecutive triangular numbers with odd prime indices are 1, 16. Is there a larger n? If such a number n exists, it is larger than 10^4.

Conjecture: the equation a(n) = a(n+1) has no solutions. This holds up to at least n = 10^4.

Conjecture: the constant Sum_{n >= 2} 1/a(n)! = 0.16945... is irrational.

Links

Table of n, a(n) for n=1..57.

J. Sondow and E. W. Weisstein, MathWorld: Smarandache Function

Formula

a(1) = 1.

a(p) = p*(p + 1)/2 for p prime.

a(p_1*p_2*...*p_u) = p_u*(p_u + 1)/2, where p_i's are distinct primes and p_1 < p_2 < ... < p_u.

a(P) = P, where P is a perfect number.

a(p*(p + 1)/2) = p*(p + 1)/2 for p prime.

a(n!) = 3*n + ((gpf(n!)^2 - 5*gpf(n!))/2 for n <> 4.

Example

a(18) = 16 because:
- for r = 1: 18 does not divide (1), (1)*(2), (1)*(2)*(3), (1)*(2)*(3)*(4), (1)*(2)*(3)*(4)*(5) and divides (1)*(2)*(3)*(4)*(5)*(6), then m_min(18, 1) = 6 = A002034(18) = K(18);
- for r = 2: 18 does not divide (1*2), (1*2)*(2*3) and divides (1*2)*(2*3)*(3*4), then m_min(18, 2) = 3;
- for r = 3: 18 does not divide (1*2*3) and divides (1*2*3)*(2*3*4), then m_min(18, 3) = 2;
- for r = 4: 18 does not divide (1*2*3*4) and divides (1*2*3*4)*(2*3*4*5), then m_min(18, 4) = 2;
- for r = 5: 18 does not divide (1*2*3*4*5) and divides (1*2*3*4*5)*(2*3*4*5*6), then m_min(18, 5) = 2;
- for r = 6 = K(18): 18 divides (1*2*3*4*5*6), then m_min(18, 6) = 1, hence a(18) = 6 + 3 + 2 + 2 + 2 + 1 = 16.

Prog

(Maxima)
K(u):=(b:1, for i:1 while mod(b, u)#0 do (c:i, b:b*(i+1)), c+1);
a(n):=(s:0, for r:2 thru K(n)-1 do (z:product(j, j, 1, r), for q:1 while mod(z, n)#0 do (z:z*product(y, y, q+1, q+r), m:q+1), s:s+m), s+K(n)+1);
makelist(a(n), n, 2, 100);

Crossrefs

Cf. A002034, A006530, A034953, A048799.

Sequence in context: A310174 A058597 A373447 * A310175 A146562 A342594

Adjacent sequences: A362067 A362068 A362069 * A362071 A362072 A362073

Keyword

nonn

Author

Lechoslaw Ratajczak, May 17 2023

Status

approved