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

Author

Amarnath Murthy, Sep 18 2003

Keywords

• base
• nonn
• full
• fini

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

Examples

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

Crossrefs

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

Programs

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 *)

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

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