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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.
8
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
1,1
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
LINKS
EXAMPLE
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.
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
PROG
(PARI) /* See A091077 and use PARI script with b=5 */
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.
Sequence in context: A014344 A239613 A100313 * A321851 A185789 A322617
KEYWORD
base,nonn
AUTHOR
Chuck Seggelin (barkeep(AT)plastereddragon.com), Dec 18 2003
EXTENSIONS
More terms from Michel Marcus, Oct 10 2014
STATUS
approved