cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A087597 Triangular numbers m such that A040115(m) is also triangular.

Original entry on oeis.org

0, 1, 3, 6, 10, 21, 28, 36, 45, 55, 66, 78, 105, 171, 465, 528, 561, 666, 2211, 4465, 7503, 16836, 18336, 22791, 44850, 53628, 55278, 82621, 105111, 114003, 333336, 427350, 828828, 2220778, 2256750, 3136260, 3373503, 3454506, 3927003, 5443350, 6175855, 7552441
Offset: 1

Views

Author

Amarnath Murthy, Sep 18 2003

Keywords

Comments

Conjecture: Sequence is infinite.

Examples

			171 is a member as A040115(171) = 66 is also a triangular number.

Links

Crossrefs

Cf. A000217, A040115, A087598, A087599, A087600.

Programs

  • Maple
    q:= n-> issqr(1+8*(l-> parse(cat(0,seq(abs(l[-i]-l[1-i])
         , i=2..nops(l)))))(convert(n, base, 10))):
select(q, [i*(i+1)/2$i=0..10000])[];  # Alois P. Heinz, Jul 27 2024
  • 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) ) ; } isA087597(n)={ if( isA000217(n) && isA000217(dd(n)), return(1), return(0) ) ; } { for(k=10,10000000, if(isA087597(k), print1(k,",") ; ) ; ) ; } \\ R. J. Mathar, Nov 19 2006

Extensions

Corrected and extended by R. J. Mathar, Nov 19 2006
Definition clarified, offset corrected, and terms 0,1,3,6 prepended by Max Alekseyev, Jul 27 2024

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

Views

Author

Amarnath Murthy, Sep 18 2003

Keywords

Comments

Conjecture: No term is zero.

Examples

			a(4) = 2211, A040115(2211) = 10.

Crossrefs

Cf. A000217, A040115, A087597, A087598, A087600.

Programs

  • 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) ; } 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

Extensions

Corrected and extended by R. J. Mathar, Nov 19 2006
a(14)-a(18) from Donovan Johnson, May 08 2010
a(19) from Donovan Johnson, Jun 19 2011
a(1)=1 prepended by Max Alekseyev, Jul 27 2024

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

Views

Author

Amarnath Murthy, Sep 18 2003

Keywords

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
Showing 1-3 of 3 results.

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