A347475 -id:A347475 - cp's OEIS frontend

A355276 Number of n-digit terms in A347475.

2, 2, 1, 4, 4, 6, 3, 8, 9, 12, 11, 18, 33, 37, 40, 43, 64, 77, 71, 118, 135, 167, 241

Offset: 1

M. F. Hasler, Sep 08 2022

Cf. A000217 (triangular numbers), A014261 (numbers with only odd digits), A117960 (triangular numbers with only odd digits), A349243 (indices of the former), A347475 (such indices with only odd digits), A349247 (least such k-digit term), A355277 (largest such k-digit term).

a(20)-a(23) from Michael S. Branicky, Sep 09 2022

A349247 Least n-digit number k with only odd digits such that the k-th triangular number also has only odd digits.

Original entry on oeis.org

1, 13, 177, 1777, 15173, 135173, 3397973, 13535137, 135157537, 1193111377, 11979759377, 119595919137, 1195991117973, 11979931335173, 119777591993777, 1199999593111377, 11977793913551137, 119593573333335733, 1195935733333335733, 11977593393931151137, 119759371717733717537

Offset: 1

M. F. Hasler, Nov 23 2021

It appears that all a(n), n > 9, have initial digits "119".

It also appears that the sequence of digits of the terms converges to a limit, (1, 1, 9, 3, 1, ...). Can this be proved or disproved?

S. S. Gupta, Can You Find (CYF) no. 55, Nov 11 2021.
A. Zimmermann, Al Zimmermann's Programming Contests: Oddly Triangular, Sep. 7-8, 2022

Cf. A000217 (triangular numbers), A014261 (numbers with only odd digits), A117960 (triangular numbers with only odd digits), A349243 (indices of the former), A347475 (such indices with only odd digits), A355277 (largest such k-digit term).

apply( A349247(n)=A347475_next(10^n\9), [1..15]) \\ Edited (moved function body to A347475) by M. F. Hasler, Sep 13 2022

from itertools import product
def A349247(n):
    for a in product('13579',repeat=n):
        if set(str((m:=int(''.join(a)))*(m+1)>>1)) <= {'1', '3', '5', '7', '9'}:
            return m # Chai Wah Wu, Sep 08 2022

A349247 = lambda n: next_A347475(10**n//9) # M. F. Hasler, Sep 10 2022

a(n) = min { k in A347475 | k >= 10^(n-1) }.

A355277 Largest n-digit number k with only odd digits such that the k-th triangular number also has only odd digits.

Original entry on oeis.org

5, 17, 177, 5573, 79137, 791377, 7913777, 79971937, 557335733, 5995957537, 59995599137, 599591791137, 7991739957973, 79971739957537, 799739357539937, 7991713197753777, 79991971791119137, 799999173991317537, 7997391313911797973

Offset: 1

M. F. Hasler, Sep 07 2022

It appears that all a(n), n > 12, have initial digits "799".
The first digit of a(n) is never 9. - Chai Wah Wu, Sep 08 2022
As in A347475, all terms with more than 2 digits end in 33, 37, 73 or 77. - M. F. Hasler, Sep 12 2022

			T(5) = A000217(5) = 5*6/2 = 5*3 = 15 has only odd digits, and neither T(7) nor T(9) have this property, therefore a(1) = 5.

M. F. Hasler, Table of n, a(n) for n = 1..24, Sep 08 2022
S. S. Gupta, Can You Find (CYF) no. 55, Nov 11 2021.

Cf. A000217 (triangular numbers), A014261 (numbers with only odd digits), A117960 (triangular numbers with only odd digits), A349243 (indices of the former), A347475 (such indices with only odd digits), A349247 (least k-digit term).

apply( A355277(n)=A347475_prec(10^n), [1..15]) \\ M. F. Hasler, Sep 08 2022

from itertools import product
def A355277(n):
    for a in '7531':
        for b in product('97531',repeat=n-1):
            m = int(a+''.join(b))
            if set(str(m*(m+1)>>1)) <= {'1', '3', '5', '7', '9'}:
                return m # Chai Wah Wu, Sep 08 2022

a(n) = max { k in A347475 | k < 10^n }.

A383939 Numbers k such that k and the k-th triangular number T(k) = k*(k+1)/2 have only even digits.

Original entry on oeis.org

0, 28, 40, 64, 400, 2828, 4000, 4064, 6428, 22840, 24028, 40000, 202428, 240028, 400000, 2040040, 2400028, 4000000, 6422840, 6428064, 6646624, 20044064, 20202080, 20400040, 20406080, 24000028, 40000000, 66400064, 200042428, 204000040, 228406080, 240000028

Offset: 1

Shyam Sunder Gupta, Aug 18 2025

The sequence also contains the infinite subsequence 4*10^n for n >= 1.

			64 is a term since it and T(64) = 2080 both have only even digits.

Shyam Sunder Gupta, Triangular Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 3, 82-125.

Cf. A000217, A014263, A117978, A347475.

q[k_] := And @@ (AllTrue[IntegerDigits[#], EvenQ] & /@ {k, k*(k+1)/2}); Select[Range[0, 4*10^6], q] (* Amiram Eldar, Aug 18 2025 *)

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A355276 Number of n-digit terms in A347475.

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A349247 Least n-digit number k with only odd digits such that the k-th triangular number also has only odd digits.

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A355277 Largest n-digit number k with only odd digits such that the k-th triangular number also has only odd digits.

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PARI

Python

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A383939 Numbers k such that k and the k-th triangular number T(k) = k*(k+1)/2 have only even digits.

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Mathematica