A087598 Numbers m such that all terms in the sequence m, A040115(m), A040115(A040115(m)), ..., 0 are triangular numbers (A000217).

0, 1, 3, 6, 10, 21, 28, 36, 45, 55, 66, 78, 171, 465, 528, 666, 2211, 4465, 22791, 333336

Offset

1,3

Comments

a(21) would need to have A040115(a(21)) among the listed terms. Equation A040115(x) = t for any term t reduces to computing integral points on a finite number of elliptic curve. Computation shows that no any new number can be obtained this way. Hence the sequence is finite and complete. - Max Alekseyev, Aug 02 2024

Links

Table of n, a(n) for n=1..20.

Example

528 is a term since A040115(528) = 36, A040115(36) = 3, A040115(3) = 0, where 528, 36, 3, and 0 are triangular numbers.

Mathematica

trnoQ[n_]:=IntegerQ[(Sqrt[8n+1]-1)/2]; oknQ[n_]:=Module[{ll= NestWhileList[FromDigits[Abs[Differences[IntegerDigits[#]]]]&, n, #>9&]}, Length[ll]>1&&And@@trnoQ/@ll]; Select[Accumulate[Range[ 2000000]], oknQ] (* Harvey P. Dale, May 15 2011 *)

Prog

(PARI) dd(k)={ local(kshf, res, dig, odig, p) ; kshf=k ; res=0 ; odig=kshf % 10 ; p=0 ; while(kshf>9, kshf=floor(kshf/10) ; dig=kshf % 10 ; res += 10^p*abs(dig-odig) ; odig=dig ; p++ ; ) ; return(res) ; } isA000217(n)={ if( issquare(1+8*n), return(1), return(0) ) ; } A000217(n)={ return(n*(n+1)/2) ; } isA087598(n)={ local(nredu) ; nredu=n ; while( nredu>10, if( isA000217(nredu), nredu=dd(nredu), return(0) ) ; ) ; if( isA000217(nredu), return(1), return(0) ) ; } { for(k=4, 1000000, if(isA087598(A000217(k)), print1(A000217(k), ", ") ; ) ; ) ; } \\ R. J. Mathar, Nov 19 2006

Crossrefs

Cf. A000217, A040115, A087597, A087599, A087600.

Sequence in context: A060179 A056411 A087597 * A068855 A068882 A076713

Adjacent sequences: A087595 A087596 A087597 * A087599 A087600 A087601

Keyword

base,nonn,full,fini

Author

Amarnath Murthy, Sep 18 2003

Extensions

Corrected and extended by R. J. Mathar, Nov 19 2006

Name clarified and terms 0,1,3,6 prepended by Max Alekseyev, Jul 26 2024

Status

approved