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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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

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.

EXAMPLE

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

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

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 *)

PROG

(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

CROSSREFS

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