A337184 Numbers divisible by their first digit and their last digit.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 22, 24, 33, 36, 44, 48, 55, 66, 77, 88, 99, 101, 102, 104, 105, 111, 112, 115, 121, 122, 123, 124, 125, 126, 128, 131, 132, 135, 141, 142, 144, 145, 147, 151, 152, 153, 155, 156, 161, 162, 164, 165, 168, 171, 172, 175, 181, 182

Offset

1,2

Comments

The first 23 terms are the same first 23 terms of A034838 then a(24) = 101 while A034838(24) = 111.

Terms of A034709 beginning with 1 and terms of A034837 ending with 1 are terms.

All positive repdigits (A010785) are terms.

There are infinitely many terms m for any of the 53 pairs (first digit, last digit) of m described below: when m begins with {1, 3, 7, 9} then m ends with any digit from 1 to 9; when m begins with {2, 4, 6, 8}, then m must also end with {2, 4, 6, 8}; to finish, when m begins with 5, m must only end with 5. - Metin Sariyar, Jan 29 2021

Links

David A. Corneth, Table of n, a(n) for n = 1..10000

Index entries for 10-automatic sequences.

Formula

(10n-9)/9 <= a(n) < 45n. (I believe the liminf of a(n)/n is 3.18... and the limsup is 6.18....) - Charles R Greathouse IV, Nov 26 2024

Conjecture: 3n < a(n) < 7n for n > 75. - Charles R Greathouse IV, Dec 02 2024

Mathematica

Select[Range[175], Mod[#, 10] > 0 && And @@ Divisible[#, IntegerDigits[#][[{1, -1}]]] &] (* Amiram Eldar, Jan 29 2021 *)

Prog

(Python)
def ok(n): s = str(n); return s[-1] != '0' and n%int(s[0])+n%int(s[-1]) == 0
print([m for m in range(180) if ok(m)]) # Michael S. Branicky, Jan 29 2021
(PARI) is(n) = n%10>0 && n%(n%10)==0 && n % (n\10^logint(n, 10)) == 0 \\ David A. Corneth, Jan 29 2021

Crossrefs

Intersection of A034709 and A034837.

Subsequences: A010785\{0}, A034838, A043037, A043040, A208259, A066622.

Cf. A139138.

Sequence in context: A084434 A034709 A178158 * A034838 A063527 A209933

Adjacent sequences: A337181 A337182 A337183 * A337185 A337186 A337187

Keyword

nonn,base,easy

Author

Bernard Schott, Jan 29 2021

Status

approved