A117960 Triangular numbers with only odd digits.

1, 3, 15, 55, 91, 153, 171, 351, 595, 1711, 1953, 5151, 5995, 9591, 11175, 11935, 15753, 15931, 17391, 17955, 31375, 33153, 35511, 73153, 153735, 171991, 173755, 193131, 193753, 371953, 399171, 513591, 551775, 559153, 571915, 791911, 917335, 939135, 1335795

Offset: 1

Author

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

Keywords

• base
• nonn
• changed

Comments

There are infinitely many terms. See Al Zimmermann link. - Shyam Sunder Gupta, Aug 16 2025

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from Alois P. Heinz)
• Shyam Sunder Gupta, Triangular Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 3, 82-125.
• Al Zimmermann's Programming Contests, Oddly Triangular, September 2022.

Crossrefs

Intersection of A000217 (triangular) and A014261 (only odd digits).

Cf. A117978.

Programs

Maple

b:= 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:
a:= n-> (t-> t*(t+1)/2)(b(n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Jul 12 2015

Mathematica

Select[Table[n(n+1)/2,{n,0,1650}],ContainsOnly[IntegerDigits[#],{1,3,5,7,9}]&] (* James C. McMahon, Sep 24 2024 *)

PARI

select( {is_A117960(n)=is_A000217(n)&&is_A014261(n)}, [2*n+1|n<-[0..99999]]) \\ M. F. Hasler, Nov 20 2021

PARI

apply( {A117960_row(n,t=10^n\9,L=List())=forvec(v=vector(n,i,[0,4]), is_A000217(n=t+fromdigits(v)*2)&&listput(L,n));L}, [1..6]) \\ row(n) = terms with n digits. Use concat(%) to flatten. - M. F. Hasler, Nov 23 2021

Python

from itertools import islice, count
def A117960(): return filter(lambda n: set(str(n)) <= {'1','3','5','7','9'}, (m*(m+1)//2 for m in count(0)))
A117960_list = list(islice(A117960(),20)) # Chai Wah Wu, Nov 22 2021

Formula

Intersection of A000217 and A014261. - M. F. Hasler, Nov 20 2021

Extensions

Some terms corrected by Alois P. Heinz, Jul 12 2015