A178354 -id:A178354 - cp's OEIS frontend

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

Carole Dubois, Mar 23 2021

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

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

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.

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

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

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 *)

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

A342944 Numbers m such that d(1)^0 + d(2)^1 + ... + d(k)^(k-1) = d(1)! + d(2)! + ... + d(k)!, where d(i), i=1..k, are the digits of m.

Original entry on oeis.org

1, 11, 12, 111, 121, 133, 202, 1020, 1111, 1211, 1331, 1403, 2021, 2030, 2120, 2220, 2305, 2413, 3012, 3102, 3115, 3202, 3215, 3322, 3335, 4033, 4123, 4223, 4434, 10165, 10201, 10210, 10300, 10533, 11065, 11111, 11200, 12065, 12111, 12200, 13050, 13265, 13311

Offset: 1

Carole Dubois, Mar 30 2021

			2413 is in this sequence because 2^0 + 4^1 + 1^2 + 3^3 = 2! + 4! + 1! + 3! = 33.

Cf. A009994, A342945, A342826, A178354.

Select[Range@20000,Total[(a=IntegerDigits@#)^Range[0,Length@a-1]]==Total[a!]&] (* Giorgos Kalogeropoulos, Mar 30 2021 *)

is(n) = my(d = digits(n)); sum(i = 1, #d, d[i]!) == sum(i = 1, #d, d[i]^(i-1)) \\ David A. Corneth, Mar 30 2021

from math import factorial
def digfac(s): return sum(factorial(int(d)) for d in s)
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) == digfac(s): alst.append(k)
  return alst
print(aupto(14000)) # Michael S. Branicky, Mar 30 2021

A342945 Numbers m such that d(1)^1 + d(2)^2 + ... + d(p)^k = d(1)! + d(2)! + ... + d(k)!, where d(i), i=1..k, are the digits of m.

Original entry on oeis.org

1, 2, 11, 21, 33, 111, 211, 331, 403, 1065, 1111, 1200, 2065, 2111, 2200, 3050, 3265, 3311, 4031, 4122, 4130, 4543, 5143, 10651, 11111, 11650, 12001, 12010, 12100, 13000, 15330, 20651, 21111, 21650, 22001, 22010, 22100, 23000, 25330, 30200, 30501, 30510, 31500

Offset: 1

Carole Dubois, Mar 30 2021

			3265 is in the sequence because 3^1 + 2^2 + 6^3 + 5^4 = 3! + 2! + 6! + 5! = 848.

Cf. A002275, A342944, A342826, A178354.

Select[Range@40000,Total[(a=IntegerDigits@#)^Range@Length@a]==Total[a!]&] (* Giorgos Kalogeropoulos, Mar 30 2021 *)

from math import factorial
def digfac(s): return sum(factorial(int(d)) for d in s)
def digpow(s): return sum(int(d)**i for i, d in enumerate(s, start=1))
def aupto(limit):
  alst = []
  for k in range(1, limit+1):
    s = str(k)
    if digpow(s) == digfac(s): alst.append(k)
  return alst
print(aupto(32000)) # Michael S. Branicky, Mar 30 2021

cp's OEIS Frontend

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.

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A342944 Numbers m such that d(1)^0 + d(2)^1 + ... + d(k)^(k-1) = d(1)! + d(2)! + ... + d(k)!, where d(i), i=1..k, are the digits of m.

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A342945 Numbers m such that d(1)^1 + d(2)^2 + ... + d(p)^k = d(1)! + d(2)! + ... + d(k)!, where d(i), i=1..k, are the digits of m.

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Mathematica

Python