A307858 -id:A307858 - cp's OEIS frontend

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

Giorgos Kalogeropoulos, May 05 2019

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

Cf. A307858, A306661.

{ 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

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

A308099 Numbers k with 2 or more chained distinct prime factors: the last digit of every prime factor is the same as the first digit of the next prime factor. Prime factors must be in ascending order.

Original entry on oeis.org

46, 58, 92, 93, 111, 116, 143, 184, 187, 209, 232, 265, 279, 295, 333, 368, 403, 422, 446, 454, 458, 464, 466, 478, 481, 482, 497, 502, 511, 514, 526, 538, 542, 553, 554, 562, 566, 586, 713, 736, 837, 844, 851, 892, 908, 916, 921, 928, 932, 933, 939, 951, 956, 964, 993, 999

Offset: 1

Giorgos Kalogeropoulos, May 12 2019

			144026 is such a number because its distinct prime factors in ascending order are 2, 23, 31, 101 and the last digit of each prime factor is the same as the first digit of the next one.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A307858, A308252, A308101.

filter:= proc(n) local F,i;
  F:= sort(convert(numtheory:-factorset(n),list));
  nops(F) >= 2 and andmap(i -> F[i] mod 10 = floor(F[i+1]/10^ilog10(F[i+1])),[$1..nops(F)-1])
end proc:
select(filter, [$1..1000]); # Robert Israel, Jun 21 2019

Select[Range@1000,PrimeNu@#>1&&And@@(Last@#[[1]]==First@#[[2]]&/@Partition[IntegerDigits@*First/@FactorInteger@#,2,1])&]

isok(n) = {my(f=factor(n)[, 1]); if (#f <= 1, return(0)); my(vd=digits(f[1]), d=vd[#vd], vd2, d2); for (k=2, #f, vd2 = digits(f[k]); d2 = vd2[1]; if (d2 != d, return (0)); vd = vd2; d = vd[#vd];); return (1);} \\ Michel Marcus, May 18 2019

A306661 Numbers with chained divisors: Numbers k with divisors such that the last digit of every divisor is the same as the first digit of the next divisor.

Original entry on oeis.org

1, 11, 13, 17, 19, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1111

Offset: 1

Giorgos Kalogeropoulos, May 05 2019

All prime numbers whose first digit is 1 (A045707) have this property.

The first composite numbers having this property are A307858: 121, 1111, 1207, ...

			14641 is such a number because its divisors are 1, 11, 121, 1331, 14641.
Also, 90043 is in the sequence because its divisors are 1, 127, 709, 90043 and the last digit of every divisor is the first digit of the next one.

A307858 and A045707 are subsequences.

Select[Range@1500,And@@(Last@#[[1]]==First@#[[2]]&/@Partition[IntegerDigits/@Divisors@#,2,1])&]

cp's OEIS Frontend

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

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A308099 Numbers k with 2 or more chained distinct prime factors: the last digit of every prime factor is the same as the first digit of the next prime factor. Prime factors must be in ascending order.

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PARI

A306661 Numbers with chained divisors: Numbers k with divisors such that the last digit of every divisor is the same as the first digit of the next divisor.

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Mathematica