A087599
Smallest nonzero n-digit term of A087597, or 0 if no such number exists.
Original entry on oeis.org
1, 10, 105, 2211, 16836, 105111, 2220778, 14319276, 221098906, 1087061878, 11402689605, 223577504556, 1264725045100, 50869724563503, 111335989114503, 2399795843858155, 11141229266441550, 127955437456464996, 1070124037258522456
Offset: 1
a(4) = 2211, A040115(2211) = 10.
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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) ; } ndigs(n)={ local(nshft,res) ; res=0 ; nshft=n; while(nshft>0, res++ ; nshft=floor(nshft/10) ; ) ; return(res) ; } isA087597(n)={ if( isA000217(n) && isA000217(dd(n)), return(1), return(0) ) ; } A087599(n)={ local(k,T) ; k=floor(-0.5+sqrt(0.25+2*10^(n-1))) ; T=A000217(k) ; if(ndigs(T)A000217(k) ; if(ndigs(T)>n, return(0) ) ; if( isA087597(T), return(T) ) ; k++ ; ) ; } { for(n=2,21, print1(A087599(n),",") ; ) ; } \\ R. J. Mathar, Nov 19 2006
A087600
Largest n-digit term of A087597, or 0 if no such number exists.
Original entry on oeis.org
6, 78, 666, 7503, 82621, 828828, 7552441, 87311505, 557362578, 9901692450, 88893307128, 934624072410, 9836548472766, 99245275962778, 994337011743076, 5535761776004778, 89253915287999385, 865474782199906830, 9888742361454004621
Offset: 1
a(4) = 7503, A040115(7503) = 253 is triangular.
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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) ; } ndigs(n)={ local(nshft,res) ; res=0 ; nshft=n; while(nshft>0, res++ ; nshft=floor(nshft/10) ; ) ; return(res) ; } isA087597(n)={ if( isA000217(n) && isA000217(dd(n)), return(1), return(0) ) ; } A087600(n)={ local(k,T) ; k=ceil(-0.5+sqrt(0.25+2*10^n)) ; T=A000217(k) ; if(ndigs(T)>n, k-- ) ; while(1, T=A000217(k) ; if(ndigs(T)A087597(T), return(T) ) ; k-- ; ) ; } { for(n=2,21, print1(A087600(n),",") ; ) ; } \\ R. J. Mathar, Nov 19 2006
A087598
Numbers m such that all terms in the sequence m, A040115(m), A040115(A040115(m)), ..., 0 are triangular numbers (A000217).
Original entry on oeis.org
0, 1, 3, 6, 10, 21, 28, 36, 45, 55, 66, 78, 171, 465, 528, 666, 2211, 4465, 22791, 333336
Offset: 1
528 is a term since A040115(528) = 36, A040115(36) = 3, A040115(3) = 0, where 528, 36, 3, and 0 are triangular numbers.
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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 *)
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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
Name clarified and terms 0,1,3,6 prepended by
Max Alekseyev, Jul 26 2024
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