cp's OEIS Frontend

A237424 Numbers of the form (10^a + 10^b + 1)/3.

%I A237424 #57 Apr 08 2025 13:20:53
%S A237424 1,4,7,34,37,67,334,337,367,667,3334,3337,3367,3667,6667,33334,33337,
%T A237424 33367,33667,36667,66667,333334,333337,333367,333667,336667,366667,
%U A237424 666667,3333334,3333337,3333367,3333667,3336667
%N A237424 Numbers of the form (10^a + 10^b + 1)/3.
%C A237424 Has the property that the product of any two (not necessarily distinct) terms has digits in nondecreasing order.
%C A237424 Conjecture: This sequence is in a sense the maximally dense sequence with this nondecreasing products property. That is, it appears that every maximal sequence is either (i) A237424, (ii) a finite set of "extra" terms plus A237424 restricted to b=0 (which is A093137), or (iii) a finite set of "extra" terms plus A237424 restricted to a=b (which is A067275). (There might be one more case, not yet identified.) - _David Applegate_, Feb 12 2014
%C A237424 See A254143 and link for products a(i)*a(j) in natural order. - _Reinhard Zumkeller_, Jan 28 2015
%H A237424 Reinhard Zumkeller, <a href="/A237424/b237424.txt">Table of n, a(n) for n = 1..10000</a> (first 1035 terms from Robert G. Wilson v)
%H A237424 Reinhard Zumkeller, <a href="/A254143/a254143.txt">First 10000 products of any two terms of A237424</a>
%F A237424 a(n) = (A052216(n) + 1)/3. - _Reinhard Zumkeller_, Jan 28 2015
%t A237424 Union@ Flatten@ Table[(10^a + 10^b + 1)/3, {a, 0, 8}, {b, a, 8}] (* _Robert G. Wilson v_, Jan 26 2015 *)
%t A237424 (10^#[[1]]+10^#[[2]]+1)/3&/@Tuples[Range[0,8],2]//Union (* _Harvey P. Dale_, May 28 2019 *)
%o A237424 (Haskell)
%o A237424 a237424 = flip div 3 . (+ 1) . a052216
%o A237424 -- _Reinhard Zumkeller_, Jan 28 2015
%o A237424 (PARI) list(lim)=my(v=List(),a,t); while(1, for(b=0,a, t=(10^a+10^b+1)/3; if(t>lim, return(Set(v))); listput(v, t)); a++) \\ _Charles R Greathouse IV_, May 13 2015
%o A237424 (Magma)
%o A237424 A052216:=[10^(n-1) + 10^(k-1): k in [1..n], n in [1..100]];
%o A237424 A237424:= func< n | (A052216[n]+1)/3 >;
%o A237424 [A237424(n): n in [1..100]]; // _G. C. Greubel_, Feb 22 2024
%o A237424 (SageMath)
%o A237424 A052216=flatten([[10^(n-1) + 10^(k-1) for k in range(1,n+1)] for n in range(1,101)])
%o A237424 def A237424(n): return (A052216[n-1]+1)//3
%o A237424 [A237424(n) for n in range(1,101)] # _G. C. Greubel_, Feb 22 2024
%o A237424 (Python)
%o A237424 from math import isqrt
%o A237424 def A237424(n): return (10**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+10**(n-1-(a*(a+1)>>1))+1)//3 # _Chai Wah Wu_, Apr 08 2025
%Y A237424 Cf. A028819, A028820, A028821, A052216, A067275, A093137, A234841, A254143.
%K A237424 nonn
%O A237424 1,2
%A A237424 _Ahmad J. Masad_, Feb 07 2014
%E A237424 Edited by _David Applegate_, Feb 07 2014