A337143 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.

5, 6, 8, 9, 12, 16, 18, 28, 37, 81, 85, 88, 130, 150, 262, 810, 1030, 1032, 4132, 9828, 9832, 10662, 10666, 562576, 562578

Offset: 1

Author

François Marques, Sep 14 2020

Keywords

• nonn
• base
• more

Comments

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

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

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).

Examples

			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.
		

Crossrefs

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).

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).

Cf. A337496, A337535, A337536.