A337184 - OEIS

%I #54 Dec 03 2024 11:47:41

%S 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,

%T 104,105,111,112,115,121,122,123,124,125,126,128,131,132,135,141,142,

%U 144,145,147,151,152,153,155,156,161,162,164,165,168,171,172,175,181,182

%N Numbers divisible by their first digit and their last digit.

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

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

%C All positive repdigits (A010785) are terms.

%C 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

%H David A. Corneth, <a href="/A337184/b337184.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Ar#10-automatic">Index entries for 10-automatic sequences</a>.

%F (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

%F Conjecture: 3n < a(n) < 7n for n > 75. - _Charles R Greathouse IV_, Dec 02 2024

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

%o (Python)

%o def ok(n): s = str(n); return s[-1] != '0' and n%int(s[0])+n%int(s[-1]) == 0

%o print([m for m in range(180) if ok(m)]) # _Michael S. Branicky_, Jan 29 2021

%o (PARI) is(n) = n%10>0 && n%(n%10)==0 && n % (n\10^logint(n,10)) == 0 \\ _David A. Corneth_, Jan 29 2021

%Y Intersection of A034709 and A034837.

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

%Y Cf. A139138.

%K nonn,base,easy

%O 1,2

%A _Bernard Schott_, Jan 29 2021