A117960 -id:A117960 - cp's OEIS frontend

A349243 Indices of triangular numbers A000217 with only odd digits.

1, 2, 5, 10, 13, 17, 18, 26, 34, 58, 62, 101, 109, 138, 149, 154, 177, 178, 186, 189, 250, 257, 266, 382, 554, 586, 589, 621, 622, 862, 893, 1013, 1050, 1057, 1069, 1258, 1354, 1370, 1634, 1658, 1738, 1754, 1777, 1786, 1853, 1885, 1965, 2657, 2666, 2741, 2818, 3218, 3346, 3445, 3457, 3794, 3845

Offset: 1

M. F. Hasler, Nov 20 2021

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).

a:= proc(n) option remember; local k; for k from
      1+`if`(n=1, 0, b(n-1)) while 0=mul(irem(i, 2),
      i=convert(k*(k+1)/2, base, 10)) do od; k
    end:
seq(a(n), n=1..57);  # Alois P. Heinz, Nov 22 2021

Select[Range[4000], AllTrue[IntegerDigits[#*(# + 1)/2], OddQ] &] (* Amiram Eldar, Nov 20 2021 *)
Position[Accumulate[Range[4000]],?(AllTrue[IntegerDigits[#],OddQ]&)]//Flatten (* _Harvey P. Dale, Sep 06 2023 *)

select( {is_A349243(n)=Set(digits(n*(n+1)\2)%2)==[1]}, [1..9999])

from itertools import islice, count
def A349243(): return filter(lambda n: set(str(n*(n+1)//2)) <= {'1','3','5','7','9'}, count(0))
A349243_list = list(islice(A349243(),20)) # Chai Wah Wu, Nov 22 2021

a(n) = floor(sqrt(2*A117960(n))).

A347475 Numbers k such that k and the k-th triangular number T(k) = k*(k+1)/2 have only odd digits.

Original entry on oeis.org

1, 5, 13, 17, 177, 1777, 3937, 5537, 5573, 15173, 55377, 55733, 79137, 135173, 195937, 339173, 377777, 399377, 791377, 3397973, 5199137, 7913777, 13535137, 17397537, 33993973, 37735377, 39993777, 59591173, 59919137, 79971937, 135157537, 139713973, 153177777

Offset: 1

M. F. Hasler, Nov 20 2021

There is only 1 term with 3 digits and there are only 3 terms with 7 digits. It appears that this (7 digits) is the only length where no term starts with digit 1, and for any length L > 9, the smallest L-digit term (cf. A349247) starts with digits "119...".
Can it be proved that the number of L-digit terms (cf. A355276) tends to infinity as L -> oo?
Can it be proved (or disproved) that the sequence of initial digits of the smallest L-digit term A349247(L) converge, maybe to (1, 1, 9, 3, 1, 1, ...)?
The sequence contains all numbers of the form 33(9{n}7){k}3{n}, where {x} means to repeat the preceding digit or parenthesized sequence of digits x times, for n >= 1 and k = 2, 3 or 4, and for k = 5 with only one initial '3'. - M. F. Hasler, Sep 10 2022
The sequence also contains the infinite subsequence s(k) = 4*10^(1+2*k) - 10^(1+k) - 10^(2+2*k) + 34*10^(3+3*k) + (22*10^k-1)/3. - Kebbaj Mohamed Reda, Sep 11 2022
In the notation of the earlier comment, the above s(k) = 339{k+1}39{k}73{k}. - M. F. Hasler, Sep 13 2022

			The numbers k = 1, 5, 13, 17, 177, 1777, ... have only odd digits, and the associated triangular numbers T(k) = k*(k+1)/2 = 1, 15, 91, 153, 15753, 1579753, 7751953, ... also have only odd digits.
The same is true for k = 119311115937719393371311137, the smallest 27-digit term.
Any number of the form n = 339{k}79{k}73{k} yields T(n) = A000217(n) = 79{k}19{k}13{k-1}453{k+1}5{k}1{k} and therefore is in the sequence, where {k} means k times (the preceding digit), for any k >= 1.

M. F. Hasler, Table of n, a(n) for n = 1..500, Sep 08 2022.
S. S. Gupta, Can You Find (CYF) no. 55, Nov 11 2021, updated Sep 12 2022
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), A349247 (least k-digit term), A355277 (largest k-digit term), A355276 (number of k-digit terms).

q[n_] := AllTrue[IntegerDigits[n], OddQ]; Select[Range[10^6], And @@ q /@ {#, #*(# + 1)/2} &] (* Amiram Eldar, Nov 20 2021 *)

apply( {A347475_row(n, t=10^n\9, L=List())=forvec(v=vector(n,i,[0,4]), is_A014261((1+n=t+fromdigits(v)*2)*n\2)&& listput(L,n));L}, [1..8]) \\ row(n) = terms with n digits. Use concat(%) to flatten the list.

A347475_first(n)=vector(n,i, n = next_A347475(n*(i>1)+1))
A347475_next(n)={my(t, p, f(v)=for(i=1, #v, bittest(v[i], 0) || return(10^(#v-i)))); while(((p=f(digits(n))) && !n+=p*10\9+if(p>99,22)-n%p) || p=f(digits(t=n*(n+1)\2)), n=max(sqrtint((t+p*10\9-t%p)*2), n+2));n} \\ used in A349247
A347475_prec(n)={my(t, p, f(v)=for(i=1, #v, bittest(v[i], 0) || return(10^(#v-i)))); while(((p=f(digits(n))) && !n-=n%p+if(p>99 && n\p%10, 23, 3)) || p=f(digits(t=n*(n+1)\2)), n=min(sqrtint((t-t%p-1)*2), n-2); if(n>p=n%100, n+=select(t->t<=p,[77,73,37,33,-23])[1]-p)); n} \\ used in A355277. - M. F. Hasler, Sep 13 2022

from itertools import islice, count, product
def A347345gen(): return filter(lambda k: set(str(k*(k+1)//2)) <= {'1','3','5','7','9'}, (int(''.join(d)) for l in count(1) for d in product('13579',repeat=l)))
A347345_list = list(islice(A347345gen(),30)) # Chai Wah Wu, Dec 05 2021

from math import isqrt
def first_even(n):
    "Return 10^k corresponding to first even digit in n."
    for i,c in enumerate(n := str(n), 1):
        if c in "02468": return 10**(len(n)-i)
def next_A347475(n):
    "Return the least term > n."
    if f := first_even(n := n+1): # next larger having only odd digits
        n += f*10//9 - n % f
    while f := first_even(t := n*(n+1)//2):
        if f := first_even(n := max(isqrt((t + 10*f//9 - t % f)*2), n+2)):
            n += 10*f//9 - n % f
    return n  #  M. F. Hasler, Sep 08 2022
N=1 # Example of use of the above function:
for n in range(30): print(N := next_A347475(N), end=", ")

Intersection of A014261 and A349243.

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.

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 }.

A117978 Triangular numbers with only even digits.

Original entry on oeis.org

0, 6, 28, 66, 406, 666, 820, 2080, 2628, 8646, 28680, 42486, 48828, 64620, 66066, 80200, 84666, 200028, 204480, 228826, 264628, 288420, 426426, 446040, 468028, 484620, 600060, 626640, 644680, 686206, 828828, 886446, 2222886, 2248260, 2862028, 2888406

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), May 03 2006

There are infinitely many terms. For example all 88...88 6 44...44 6 with an equal number of 8's and 4's (their triangular indices being A185127). - Shyam Sunder Gupta, Aug 16 2025

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)
Shyam Sunder Gupta, Triangular Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 3, 82-125.

Intersection of A000217 and A014263.

Cf. A117960, A185127.

Select[Accumulate[Range[0,5000]],Count[IntegerDigits[#],?(OddQ)] ==0&] (* _Harvey P. Dale, Oct 21 2011 *)

Corrected (a(32) was in error) and extended by Harvey P. Dale, Oct 21 2011

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A349243 Indices of triangular numbers A000217 with only odd digits.

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A347475 Numbers k such that k and the k-th triangular number T(k) = k*(k+1)/2 have only odd digits.

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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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A117978 Triangular numbers with only even digits.

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A355276 Number of n-digit terms in A347475.

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