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A087598
Numbers m such that all terms in the sequence m, A040115(m), A040115(A040115(m)), ..., 0 are triangular numbers (A000217).
3
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
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
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