A366048 For n >= 1, a(n) is the least k >= 1 such that 1/d(k) + … + 1/d(k + n - 1) is an integer, d(i) = A000005(i).

1, 2, 1, 25, 54, 7, 53, 65, 6, 22, 51, 49, 343, 209, 416, 624, 17, 18, 338, 410, 1622, 341, 140, 849, 139, 337, 1939, 338, 849, 4365, 2565, 6368, 496, 4366, 132, 8392, 131, 4453, 128, 4173, 127, 487, 123, 4437, 492, 122, 3011, 491, 3724, 4171, 2637, 1231, 1631, 12765, 119

Offset

1,2

Comments

Conjecture : The sum 1/d(k) + … + 1/d(k + n - 1) = C, C an integer, exists for all k >= 1, n >= 1.

Are there, for some fixed n >= 3, infinitely many k's such that 1/d(k) + … + 1/d(k + n - 1) is an integer ?

Links

Table of n, a(n) for n=1..55.

Example

n = 3: 1/d(k) + 1/d(k + 1) + 1/d(k + 2) = C, C an integer, is valid for the least k = 1, thus a(3) = 1.
n = 4: 1/d(k) + 1/d(k + 1) + 1/d(k + 2) + 1/d(k + 3) = C, C an integer, is valid for the least k = 25, thus a(4) = 25.

Prog

(PARI) a(n) = my(k=1); while (denominator(sum(i=0, n-1, 1/numdiv(k+i))) != 1, k++); k; \\ Michel Marcus, Sep 27 2023

Crossrefs

Cf. A000005, A359056.

Sequence in context: A010256 A087452 A317385 * A098878 A235031 A138955

Adjacent sequences: A366045 A366046 A366047 * A366049 A366050 A366051

Keyword

nonn

Author

Ctibor O. Zizka, Sep 27 2023

Extensions

More terms from Michel Marcus, Sep 27 2023

Status

approved