A342826 Numbers k such that d(1)^0 + d(2)^1 + ... + d(p)^(p-1) = d(1)^(p-1) + d(2)^(p-2) + ... + d(p)^0, where d(i), i=1..p, are the digits of k.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 444, 454, 464

Offset: 1

Author

Carole Dubois, Mar 23 2021

Keywords

• nonn
• base

Comments

This sequence starts off like other palindromic sequences such as A178354, A002113, A110751, and A227858 but it differs starting at the 110th term: 109th: 1001, 110th: 1011, 111th: 1101, ..., 119th: 1863, etc.

Differs from A297271 (which e.g. has 1021, 1031, 1041,.. 1091 which are absent here). - R. J. Mathar, Sep 27 2021

Contains the palindromes, and palindromes where pairs of digits have been substituted by d(i)=0, d(p-i)=1 or d(i)=1, d(p-1)=0, and "genuine" numbers like 1863, 2450, 2804, 2814, 3681, 4081, 4182, 103221, 113221, 122301, 122311, 142023,.. - R. J. Mathar, Sep 27 2021

Examples

			1863 is in this sequence because 1^0 + 8^1 + 6^2 + 3^3 = 1^3 + 8^2 + 6^1 + 3^0 = 72.
		

Links

• R. J. Mathar, Table of n, a(n) for n = 1..1137 (replacing older b-file which did not contain a(101))
• Carole Dubois, Scatterplot of terms of A178354, A342826 vs Sum of powers in definition.

Crossrefs

Cf. A002113 (subset), A179309, A110751, A227858.

Programs

Maple

isA342826 := proc(n)
    local dgs ;
    dgs := convert(n,base,10) ;
    if  add(op(i,dgs)^(i-1),i=1..nops(dgs)) = add(op(-i,dgs)^(i-1),i=1..nops(dgs)) then
        true;
    else
        false;
    end if;
end proc:
A342826 := proc(n)
    option remember ;
    if n =1 then
        1;
    else
        for a from procname(n-1)+ 1 do
            if isA342826(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Sep 27 2021

Mathematica

Select[Range[500],Mod[#,10]!=0&&Total[IntegerDigits[#]^Range[0,IntegerLength[ #]-1]]==Total[IntegerDigits[#]^Range[IntegerLength[#]-1,0,-1]]&] (* Harvey P. Dale, Jan 18 2023 *)

Python

def digpow(s): return sum(int(d)**i for i, d in enumerate(s))
def aupto(limit):
  alst = []
  for k in range(1, limit+1):
    s = str(k)
    if digpow(s) == digpow(s[::-1]): alst.append(k)
  return alst
print(aupto(464)) # Michael S. Branicky, Mar 23 2021