A337143 - OEIS

%I #36 Feb 13 2021 01:10:43

%S 5,6,8,9,12,16,18,28,37,81,85,88,130,150,262,810,1030,1032,4132,9828,

%T 9832,10662,10666,562576,562578

%N Numbers k for which there are only 3 bases b (2, k+1 and another one) in which the digits of k contain the digit b-1.

%C This sequence is the list of indices k such that A337496(k)=3.

%C Conjecture: this sequence is finite and full. a(26) > 3.8*10^12 if it exists.

%C All terms of this sequence increased by 1 are either prime numbers, or prime numbers squared, or 2 times a prime number because if b is a strict divisor of k+1, the digit for the units in the expansion of k in base b is b-1 so it must be 2 or the third base. In fact k+1 could have been equal to 8=2*4 but 7 is not a term of the sequence (7 = 111_2 = 21_3 = 13_4 = 7_8).

%e a(7)=18 because there are only 3 bases (2, 19 and 3) which satisfy the condition of the definition (18=200_3) and 18 is the seventh of these numbers.

%Y Cf. Numbers with at least one digit b-1 in base b : A074940 (b=3), A337250 (b=4), A337572 (b=5), A011539 (b=10), A095778 (b=11).

%Y Cf. Numbers with no digit b-1 in base b: A005836 (b=3), A023717 (b=4), A020654 (b=5), A037465 (b=6), A020657 (b=7), A037474 (b=8), A037477 (b=9), A007095 (b=10), A065039 (b=11).

%Y Cf. A337496, A337535, A337536.

%K nonn,base,more

%O 1,1

%A _François Marques_, Sep 14 2020