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A306465
Lexicographically earliest sequence of distinct positive terms such that the product of any two consecutive terms can be computed without carry by long multiplication in base 10.
3
1, 2, 3, 10, 4, 11, 5, 100, 6, 101, 7, 110, 8, 111, 9, 1000, 12, 13, 20, 14, 21, 22, 30, 23, 102, 24, 112, 31, 32, 103, 33, 120, 40, 121, 41, 200, 34, 201, 42, 202, 43, 1001, 15, 1010, 16, 1011, 17, 1100, 18, 1101, 25, 1110, 26, 1111, 27, 10000, 19, 10001, 28
OFFSET
1,2
COMMENTS
This sequence is the variant of A266195 in base 10.
This sequence is a permutation of the natural numbers, with inverse A306466. Proof:
- we can always extend the sequence with a power of ten not yet in the sequence, hence the sequence is well defined and infinite,
- for any k > 0, 10^(k-1) is the first k-digit number appearing in the sequence,
- all powers of ten appear in the sequence, in increasing order,
- a power of ten is always followed by the least number unused so far,
hence every number eventually appears. QED
FORMULA
A007953(a(n) * a(n+1)) = A007953(a(n)) * A007953(a(n+1)).
A054055(a(n)) * A054055(a(n+1)) <= 9.
EXAMPLE
The first terms, alongside their digital sum and the digital sum of the product with the next term, are:
n a(n) ds(a(n)) ds(a(n)*a(n+1))
-- ---- -------- ---------------
1 1 1 2
2 2 2 6
3 3 3 3
4 10 1 4
5 4 4 8
6 11 2 10
7 5 5 5
8 100 1 6
9 6 6 12
10 101 2 14
11 7 7 14
12 110 2 16
13 8 8 24
14 111 3 27
15 9 9 9
16 1000 1 3
17 12 3 12
PROG
(PARI) See Links section.
CROSSREFS
Cf. A007953, A054055, A266195, A306466 (inverse).
Sequence in context: A342045 A307591 A031275 * A276104 A376370 A329804
KEYWORD
nonn,base
AUTHOR
Rémy Sigrist, Feb 17 2019
STATUS
approved