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 1-7 of 7 results.

A091079 Numbers n which when converted to base 5, reversed and converted back to base 10 yield a number m such that n mod m = 0. Cases which are trivial or result in digit loss are excluded.

Original entry on oeis.org

16, 96, 416, 496, 576, 2016, 2496, 2976, 10016, 10416, 12096, 12496, 14976, 50016, 52416, 60096, 62496, 74976, 250016, 252016, 260416, 262416, 300096, 302096, 310496, 312496, 360576, 374976, 1250016, 1262016, 1300416, 1312416, 1500096, 1512096, 1550496
Offset: 1

Views

Author

Chuck Seggelin, Dec 18 2003

Keywords

Comments

Trivial cases are those numbers which upon conversion result in a number which is palindromic (m = reverse(m)), or a palindrome plus trailing zeros such that m = reverse(m)*10^z where z=number of lost zeros. Nontrivial digit loss occurs when a converted number has trailing zeros that drop off when the number is reversed.
n/m must be either 2 or 4. - Robert Israel, Apr 22 2021

Examples

			a(1) = 16 because: 16 in base 5 is 31; 31 reversed is 13; 13 converted back to base 10 is 8 and 16 mod 8 = 0.

Links

Crossrefs

Cf. A091077 (same in base 3), A091078 (base 4), A091080 (base 6), A091081 (base 7), A091082 (base 8), A091083 (base 9), A031877 (base 10).
See also A222816, A214927.

Programs

  • Maple
    F:= proc(d) local eq,m,R;
  R:= NULL;
  for m in [2,4] do
    eq:= m*add(a[i]*5^i,i=0..d)-add(a[d-i]*5^i,i=0..d);
    R:= R, F1(eq,[],d);
  od;
sort([R]);
end proc:
F1:= proc(eq,A,d) local V,s,e1,i1,i2,vlo,R,v1,v2,Vp,Vm,emax,emin;
      V:= indets(eq);
      if nops(V) = 0 then
         if eq = 0 then subs(A,add(a[d-i]*5^i,i=0..d))
         else NULL
         fi
      elif nops(V) = 1 then
         s:= solve(eq,V[1]);
         if member(s,[$0..4]) then
            subs([op(A),V[1]=s],add(a[d-i]*5^i,i=0..d));
         fi
      else
          Vp,Vm:= selectremove(t -> coeff(eq,t)>0, V);
         emax:= subs(map(`=`,Vp,4),map(`=`,Vm,0),eq);
         if emax < 0 then return NULL fi;
         emin:= subs(map(`=`,Vp,0),map(`=`,Vm,4),eq);
         if emin > 0 then return NULL fi;
           e1:= eq mod 5;
         V:= indets(e1);
         if nops(V) = 0 then procname(e1/5,A,d)
         elif nops(V) = 1 then
           s:= msolve(e1, 5);
           procname(subs(s,eq)/5, [op(A),op(s)], d)
         else
           i1:= op(1,V[1]); i2:= op(1,V[2]);
           if i1 = 0 or i2 = 0 then vlo:= 1 else vlo:= 0 fi;
           R:= NULL;
           for v1 from vlo to 4 do
             s:= msolve(eval(e1, a[i1]=v1),5);
             R:= R, procname(subs(a[i1]=v1, op(s), eq)/5, [op(A),a[i1]=v1,op(s)],d)
           od;
           R
      fi fi
end proc:
seq(op(F(d)),d=1..8); # Robert Israel, Apr 22 2021
  • PARI
    /* See A091077 and use PARI script with b=5 */

Extensions

More terms from Michel Marcus, Oct 10 2014

A091078 Numbers n which when converted to base 4, reversed and converted back to base 10 yield a number m such that n mod m = 0. Cases which are trivial or result in digit loss are excluded.

Original entry on oeis.org

225, 945, 3825, 15345, 57825, 61425, 230625, 245745, 921825, 968625, 983025, 368, 6625, 3871665, 3932145, 14745825, 14803425, 15483825, 15671025, 15728625, 589826, 25, 59224545, 61932465, 62672625, 62914545
Offset: 1

Views

Author

Chuck Seggelin, Dec 18 2003

Keywords

Comments

Trivial cases are those numbers which upon conversion result in a number which is palindromic (m = reverse(m)), or a palindrome plus trailing zeros such that m = reverse(m)*10^z where z=number of lost zeros. Nontrivial digit loss occurs when a converted number has trailing zeros that drop off when the number is reversed.

Examples

			a(1) = 225 because: 225 in base 4 is 3201; 3201 reversed is 1023; 1023 converted back to base 10 is 75 and 225 mod 75 = 0.

Links

Crossrefs

Cf. A091077 (same in base 3), A091079 (base 5), A091080 (base 6), A091081 (base 7), A091082 (base 8), A091083 (base 9), A031877 (base 10).

Programs

  • PARI
    /* See A091077 and use PARI script with b=4 */

Extensions

More terms from Michel Marcus, Oct 10 2014

A091080 Numbers n which when converted to base 6, reversed and converted back to base 10 yield a number m such that n mod m = 0. Cases which are trivial or result in digit loss are excluded.

Original entry on oeis.org

980, 1225, 6020, 7525, 36260, 45325, 217700, 272125, 1271060, 1306340, 1588825, 1632925, 7621460, 7838180, 9526825, 9797725, 45723860, 46817540, 47029220, 57154825, 58521925, 58786525
Offset: 1

Views

Author

Chuck Seggelin, Dec 18 2003

Keywords

Comments

Trivial cases are those numbers which upon conversion result in a number which is palindromic (m = reverse(m)), or a palindrome plus trailing zeros such that m = reverse(m)*10^z where z=number of lost zeros. Nontrivial digit loss occurs when a converted number has trailing zeros that drop off when the number is reversed.

Examples

			a(1) = 980 because: 980 in base 6 is 4312; 4312 reversed is 2134; 2134 converted back to base 10 is 490 and 980 mod 490 = 0.

Links

Crossrefs

Cf. A091077 (same in base 3), A091078 (base 4), A091079 (base 5), A091081 (base 7), A091082 (base 8), A091083 (base 9), A031877 (base 10).

Programs

  • PARI
    /* See A091077 and use PARI script with b=6 */

Extensions

More terms from Michel Marcus, Oct 10 2014

A091081 Numbers n which when converted to base 7, reversed and converted back to base 10 yield a number m such that n mod m = 0. Cases which are trivial or result in digit loss are excluded.

Original entry on oeis.org

36, 288, 1800, 2052, 2304, 12384, 14400, 16416, 86472, 88236, 99072, 100836, 115200, 605088, 619200, 691776, 705888, 806688, 4235400, 4247748, 4323600, 4335948, 4840704, 4853052, 4928904, 4941252, 5534208, 5647104, 29647584, 29746368, 30254400, 30353184
Offset: 1

Views

Author

Chuck Seggelin, Dec 18 2003

Keywords

Comments

Trivial cases are those numbers which upon conversion result in a number which is palindromic (m = reverse(m)), or a palindrome plus trailing zeros such that m = reverse(m)*10^z where z=number of lost zeros. Nontrivial digit loss occurs when a converted number has trailing zeros that drop off when the number is reversed.

Examples

			a(1) = 36 because: 36 in base 7 is 51; 51 reversed is 15; 15 converted back to base 10 is 12 and 36 mod 12 = 0.

Links

Crossrefs

Cf. A091077 (same in base 3), A091078 (base 4), A091079 (base 5), A091080 (base 6), A091082 (base 8), A091083 (base 9), A031877 (base 10).

Programs

  • PARI
    /* See A091077 and use PARI script with b=7 */

Extensions

More terms from Michel Marcus, Oct 10 2014

A091082 Numbers n which when converted to base 8, reversed and converted back to base 10 yield a number m such that n mod m = 0. Cases which are trivial or result in digit loss are excluded.

Original entry on oeis.org

42, 378, 2625, 2730, 3066, 3402, 3969, 21546, 23625, 24570, 27594, 32193, 170625, 172074, 174762, 191625, 193914, 196602, 221130, 257985, 1346625, 1376298, 1400490, 1535625, 1548666, 1572858, 1769418, 2064321, 10754625, 10922625, 11010090, 11031594, 11184810
Offset: 1

Views

Author

Chuck Seggelin, Dec 18 2003

Keywords

Comments

Trivial cases are those numbers which upon conversion result in a number which is palindromic (m = reverse(m)), or a palindrome plus trailing zeros such that m = reverse(m)*10^z where z=number of lost zeros. Nontrivial digit loss occurs when a converted number has trailing zeros that drop off when the number is reversed.

Examples

			a(1) = 42 because: 42 in base 8 is 52; 52 reversed is 25; 25 converted back to base 10 is 21 and 42 mod 21 = 0.

Links

Crossrefs

Cf. A091077 (same in base 3), A091078 (base 4), A091079 (base 5), A091080 (base 6), A091081 (base 7), A091083 (base 9), A031877 (base 10).

Programs

  • PARI
    /* See A091077 and use PARI script with b=8 */

Extensions

More terms from Michel Marcus, Oct 10 2014

A091083 Numbers n which when converted to base 9, reversed and converted back to base 10 yield a number m such that n mod m = 0. Cases which are trivial or result in digit loss are excluded.

Original entry on oeis.org

64, 640, 4800, 5248, 5824, 6400, 43680, 46720, 52480, 58240, 393600, 419968, 425152, 467200, 472384, 524800, 3542880, 3779200, 3831040, 4199680, 4251520, 4723840, 31497600, 31886400, 34012288, 34058944, 34437376, 34484032, 37792000, 37838656, 38217088
Offset: 1

Views

Author

Chuck Seggelin, Dec 18 2003

Keywords

Comments

Trivial cases are those numbers which upon conversion result in a number which is palindromic (m = reverse(m)), or a palindrome plus trailing zeros such that m = reverse(m)*10^z where z=number of lost zeros. Nontrivial digit loss occurs when a converted number has trailing zeros that drop off when the number is reversed.

Examples

			a(1) = 64 because 64 in base 9 is 71; 71 reversed is 17; 17 converted back to base 10 is 16 and 64 mod 16 = 0.

Links

Crossrefs

Cf. A091077 (same in base 3), A091078 (base 4), A091079 (base 5), A091080 (base 6), A091081 (base 7), A091082 (base 8), A031877 (base 10).

Programs

  • PARI
    /* See A091077 and use PARI script with b=9 */

Extensions

More terms from Michel Marcus, Oct 10 2014

A354886 Numbers that are palindromes in both ternary and balanced ternary representations with representations that are different.

Original entry on oeis.org

16, 52, 160, 484, 1312, 1456, 3904, 4372, 11680, 12688, 13120, 35008, 37960, 39364, 104992, 106288, 113776, 116800, 118096, 314944, 319156, 341224, 350080, 354292, 944800, 948688, 957760, 1023568, 1027456, 1049920, 1058992, 1062880, 2834368, 2847004, 2873572
Offset: 1

Views

Author

Amiram Eldar, Jun 10 2022

Keywords

Comments

Is a(n) == 16 (mod 18) for all n?

Examples

			16 is a term since its ternary representation is 121 and its balanced ternary representation (with 2 standing for the -1 digit) is 1221, and both are palindromes.

Links

Crossrefs

Equals A354885 \ A005836.
Cf. A007089, A014190, A091077, A134027, A140267.

Programs

  • Mathematica
    Select[Range[0, 3*10^6], PalindromeQ[d3 = IntegerDigits[#, 3]] && PalindromeQ[db3 = balTernDigits[#]] && d3 != db3 &] (* using balTernDigits by Robert G. Wilson v at A134027 *)

Formula

a(n) = A091077(n) / 4. - Hugo Pfoertner, Jun 10 2022
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.