A307861 Numbers that set records on longest chains of "Chained Divisors" of A306661 and A307858.

1, 11, 121, 1111, 12221, 134431, 1234321, 1478741, 1600951, 17610461, 161696051, 193715071, 1634570971

Offset: 1

Author

Giorgos Kalogeropoulos, May 05 2019

Keywords

• nonn
• base
• more

Comments

Here are the divisor chains:

Length---m----------Divisors

1 1 {1}

2 11 {1,11}

3 121 {1,11,121}

4 1111 {1,11,101,1111}

6 12221 {1,11,101,121,1111,12221}

8 134431 {1,11,101,121,1111,1331,12221,134431}

9 1234321 {1,11,101,121,1111,10201,12221,112211,1234321}

10 1478741 {1,11,101,121,1111,1331,12221,14641,134431,1478741}

The longest chain known is of number 1634570971 of length 24.

Under the assumption that all divisors begin and end with 1, the next terms are 19565222171, 153203879191, 165091668071, 197783087491, and 1685242671101, with 30, 32, 36, 40, and 48 divisors, respectively. - Charlie Neder, May 08 2019

Crossrefs

Cf. A307858, A306661.

Programs

PARI

{ mx=0; for (n=1, oo, if (#(d=divisors(n))>mx, ok=1; d=apply(digits,d); for (i=1, #d-1, if (d[i][#d[i]]!=d[i+1][1], ok=0; break)); if (ok,
print1 (n ", "); mx=#d))) } \\ Rémy Sigrist, May 06 2019

Extensions

a(11)-a(13) from Rémy Sigrist, May 06 2019