A300891 Left-truncatable nonzero triangular numbers.

1, 3, 6, 21, 36, 66, 91, 136, 406, 666, 703, 903, 3003, 6903, 7021, 8001, 5000703

Offset

1,2

Comments

Leading zeros in postfix strings are ignored (e.g., 003 and 03 are both equivalent to 3). Zero was excluded from the set because otherwise any integer starting with 1, 3, or 6 followed by any number of zeros would be a member.

Compare this sequence with the right-truncatable triangular numbers listed in A202269.

This sequence is finite and full. - Conjectured by Stanislav Sykora, Mar 14 2018; proved by Max Alekseyev, Jun 03 2026

Solving equation t + d*10^k = y*(y+1)/2 for any term t, a nonzero digit d, and a positive integer k reduces to computing integral points on a finite number of elliptic curve. Computation shows that no new terms can be obtained this way. Hence the sequence is finite and complete. - Max Alekseyev, Jun 03 2026

Both the source number and the result must be nonzero, otherwise 10 would be a term of the sequence. - Georg Fischer, Dec 02 2023

Links

Table of n, a(n) for n=1..17.

Example

6903 is a term because it is a nonzero triangular number, and 903 is a term of the sequence.

Maple

isA300891 := proc(n)
    option remember ;
    if n in {1, 3, 6} then
        return true;
    elif n < 10 then
        return false;
    end if;
    if isA000217(n) then
        dgs := max(1, ilog10(n)+1) ;
        return procname( modp(n, 10^(dgs-1))) ;
    else
        return false;
    end if;
end proc:
for i from 1 do
    t := A000217(i) ;
    if isA300891(t) then
        print(t) ;
    end if;
end do: # R. J. Mathar, May 02 2018

Crossrefs

Cf. A000217, A069693, A078254, A087598, A202269.

Sequence in context: A174461 A050611 A270510 * A056499 A056489 A015649

Adjacent sequences: A300888 A300889 A300890 * A300892 A300893 A300894

Keyword

nonn,base,fini,full,changed

Author

Stanislav Sykora, Mar 14 2018

Status

approved