A249515 -id:A249515 - cp's OEIS frontend

A249516 Numbers k for which the digital product A007954(k) contains the same distinct digits as the number k.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 111, 1111, 1288, 1557, 1575, 1755, 1828, 1882, 2188, 2818, 2881, 3448, 3484, 3577, 3757, 3775, 3844, 4348, 4384, 4438, 4483, 4834, 4843, 5157, 5175, 5377, 5517, 5571, 5715, 5737, 5751, 5773, 7155, 7357, 7375, 7515, 7537, 7551

Offset: 1

Jaroslav Krizek, Oct 31 2014

			1288 is a member since 1288 and its digital product 1*2*8*8 = 128 have the same digit set {1,2,8}.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A007954, A249515, A249517.

[0] cat [n: n in [0..100000] | Set(Intseq(n)) eq Set(Intseq(&*Intseq(n)))];

Select[Range[0,8000],Union[IntegerDigits[Times@@IntegerDigits[#]]] == Union[ IntegerDigits[#]]&] (* Harvey P. Dale, Aug 05 2018 *)

print1(0,", ");for(n=1,10^4,d=digits(n);p=prod(i=1,#d,d[i]);if(vecsort(digits(p),,8)==vecsort(d,,8),print1(n,", "))) \\ Derek Orr, Nov 05 2014

A249517 Numbers k for which the digital sum A007953(k) and the digital product A007954(k) both contain the same distinct digits as the number k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11111111111

Offset: 1

Jaroslav Krizek, Oct 31 2014

a(12) = (10^106-1)/9 + 122222222. - Max Alekseyev, Nov 15 2014

Other entries include (10^111-1)/9, (10^113-1)/9 + 177, (10^115-1)/9 + 122222222, (10^117-1)/9 + 11117, (10^125-1)/9 + 2224, (10^126-1)/9 + 333335, (10^135-1)/9 + 4666, (10^143-1)/9 + 446, (10^143-1)/9 + 2224, (10^144-1)/9 + 33335. All other entries with 150 or fewer digits are formed by permutations of the decimal digits of these entries (including a(12)). (10^((10^m-1)/9)-1)/9 are entries of the sequence for m >= 0. - Chai Wah Wu, Nov 15 2014

			11111111111 is a term since A007953(11111111111) = 11 and A007954(11111111111) = 1.

Intersection of A249515 and A249516. Subsequence of A249334.

Cf. A007954, A249334, A249515, A249516.

[0] cat [n: n in [0..10^7] | Set(Intseq(n)) eq Set(Intseq(&*Intseq(n))) and Set(Intseq(n)) eq Set(Intseq(&+Intseq(n)))];

is(n)=if(n<=9,return(1)); my(d=digits(n),s=Set(d)); s==Set(digits(sum(i=1,#d,d[i]))) && s==Set(digits(prod(i=1,#d,d[i]))) \\ Charles R Greathouse IV, Nov 13 2014

from itertools import product
from operator import mul
from functools import reduce
A249517_list = [0]
for g in range(1,15):
    xp, ylist = [], []
    for i in range(9*g,-1,-1):
        x = set(str(i))
        if not (('0' in x) or (x in xp)):
            xv = [int(d) for d in x]
            imin = int(''.join(sorted(str(i))))
            if max(xv)*(g-len(x)) >= imin-sum(xv) and i-sum(xv) >=  min(xv)*(g-len(x)):
                xp.append(x)
                for y in product(x,repeat=g):
                    if set(y) == x:
                        yd = [int(d) for d in y]
                        if set(str(sum(yd))) == x == set(str(reduce(mul, yd, 1))):
                            ylist.append(int(''.join(y)))
    A249517_list.extend(sorted(ylist)) # Chai Wah Wu, Nov 15 2014

a(11) = 11111111111 confirmed by Sean A. Irvine, Nov 13 2014, by direct search.

A360986 Primes whose sum of decimal digits has the same set of decimal digits as the prime.

Original entry on oeis.org

2, 3, 5, 7, 199, 919, 991, 2999, 9929, 11177, 11717, 17117, 31333, 33331, 71171, 71711, 161611, 616111, 999499, 1111333, 1131133, 1131331, 1133131, 1313311, 3111313, 3111331, 3131113, 3131311, 3133111, 3311131, 3337777, 3377377, 3773377, 3773773, 7377373, 7733377, 7737337, 7737733, 32333333

Offset: 1

Robert Israel, Feb 27 2023

			a(5) = 199 is a term because 199 is prime and 1+9+9 = 19 has the same set {1,9} of decimal digits as 199.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A158473.

Primes in A249515.

dmax:= 7: # for terms with up to dmax digits
dsets:= proc(s, S) option remember;
# nondecreasing lists [x_1, ..., x_n] with sum s and set of elements S
   local i, x1;
   if S = {} then if s = 0 then return {[]} else return {} fi fi;
   x1:= min(S);
   `union`(seq(map(t -> [x1$i, op(t)], procname(s-i*x1, S minus {x1})), i=1..`if`(x1=0,dmax,floor(s/x1))))
end proc:
R:= {2,3,5,7}: count:= 4:
for s from 2 to 9*dmax-1 do
  if s mod 3 = 0 then next fi;
  ds:= convert(convert(s,base,10),set);
  DS:= select (t -> nops(t) > 1 and nops(t) <= dmax, dsets(s,ds));
  for r in DS do
     for v in remove(t -> member(t[1],[0,2,4,5,6,8]) or t[-1]=0,combinat:-permute(r)) do
       p:= add(v[i]*10^(i-1),i=1..nops(v));
       if isprime(p) then R:= R union {p}; count:= count+1;
      fi
od od od:
sort(convert(R,list));

isok(p) = if (isprime(p), my(d=digits(p)); Set(d) == Set(digits(vecsum(d)))); \\ Michel Marcus, Feb 28 2023

A383328 Numbers that have the same set of digits as the sum of the squares of their digits.

Original entry on oeis.org

0, 1, 155, 224, 242, 334, 343, 422, 433, 505, 515, 550, 551, 1388, 1788, 1838, 1878, 1883, 1887, 3188, 3334, 3336, 3343, 3363, 3433, 3633, 3818, 3881, 4333, 5005, 5050, 5500, 6333, 7188, 7818, 7881, 8138, 8178, 8183, 8187, 8318, 8381, 8718, 8781, 8813, 8817, 8831

Offset: 1

Jean-Marc Rebert, Apr 23 2025

			155 and 1^2 + 5^2 + 5^2 = 51 have the same set of digits {1,5}, so 155 is a term.

Cf. A003132, A029793, A249515.

q[k_] := Module[{d = IntegerDigits[k]}, Union[d] == Union[IntegerDigits[Total[d^2]]]]; Select[Range[0, 10000], q] (* Amiram Eldar, Apr 23 2025 *)

isok(k) = my(d=digits(k)); Set(d) == Set(digits(sum(i=1, #d, d[i]^2))); \\ Michel Marcus, May 13 2025

def ok(n): return set(s:=str(n)) == set(str(sum(int(d)**2 for d in s)))
print([k for k in range(10**4) if ok(k)]) # Michael S. Branicky, Apr 23 2025

A383347 Numbers that have the same set of digits as the sum of the cubes of their digits.

Original entry on oeis.org

0, 1, 135, 137, 153, 173, 307, 315, 317, 351, 370, 371, 407, 470, 513, 531, 703, 704, 713, 730, 731, 740, 3007, 3070, 3700, 4007, 4070, 4700, 7003, 7004, 7030, 7040, 7300, 7400, 11112, 11113, 11121, 11131, 11211, 11311, 12111, 12599, 12959, 12995, 13111, 15299

Offset: 1

Jean-Marc Rebert, Apr 24 2025

Contains 3*10^j + 7*10^k and 4*10^j + 7*10^k if j <> k and max(j,k) >= 2. - Robert Israel, Apr 25 2025

			135 and 1^3 + 3^3 + 5^3 = 153 have the same set of digits {1,3,5}, so 135 is a term.

Cf. A055012, A249515.

Cf. A046197 (a subsequence).

filter:= proc(n) local L,t;
  L:= convert(n,base,10);
  convert(L,set) = convert(convert(add(t^3,t=L),base,10),set)
end proc:
select(filter, [$0 .. 10^5]); # Robert Israel, Apr 25 2025

q[k_] := Module[{d = IntegerDigits[k]}, Union[d] == Union[IntegerDigits[Total[d^3]]]]; Select[Range[0, 16000], q] (* Amiram Eldar, Apr 24 2025 *)

isok(k) = my(d=digits(k)); Set(d) == Set(digits(sum(i=1, #d, d[i]^3))); \\ Michel Marcus, Apr 24 2025

def ok(n): return set(s:=str(n)) == set(str(sum(int(d)**3 for d in s)))
print([k for k in range(2*10**4) if ok(k)]) # Michael S. Branicky, Apr 24 2025

A383349 Numbers that have the same set of digits as the sum of 4th powers of its digits.

Original entry on oeis.org

0, 1, 488, 668, 686, 848, 866, 884, 1346, 1364, 1436, 1463, 1634, 1643, 2088, 2556, 2565, 2655, 2808, 2880, 3146, 3164, 3416, 3461, 3614, 3641, 4136, 4163, 4316, 4361, 4479, 4497, 4613, 4631, 4749, 4794, 4947, 4974, 5256, 5265, 5526, 5562, 5625, 5652, 6134, 6143

Offset: 1

Jean-Marc Rebert, Apr 24 2025

			488 and 4^4 + 8^4 + 8^4 = 8448 have the same set of digits {4,8}, so 488 is a term.

Cf. A055013, A383328, A383347, A249515, A003132.

Cf. A052455 (a subsequence).

q[k_] := Module[{d = IntegerDigits[k]}, Union[d] == Union[IntegerDigits[Total[d^4]]]]; Select[Range[0, 7000], q] (* Amiram Eldar, Apr 24 2025 *)

isok(k) = my(d=digits(k)); Set(d) == Set(digits(sum(i=1, #d, d[i]^4))); \\ Michel Marcus, Apr 24 2025

def ok(n): return set(s:=str(n)) == set(str(sum(int(d)**4 for d in s)))
print([k for k in range(10**4) if ok(k)]) # Michael S. Branicky, Apr 24 2025

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A249516 Numbers k for which the digital product A007954(k) contains the same distinct digits as the number k.

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A249517 Numbers k for which the digital sum A007953(k) and the digital product A007954(k) both contain the same distinct digits as the number k.

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A360986 Primes whose sum of decimal digits has the same set of decimal digits as the prime.

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A383328 Numbers that have the same set of digits as the sum of the squares of their digits.

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A383347 Numbers that have the same set of digits as the sum of the cubes of their digits.

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Maple

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A383349 Numbers that have the same set of digits as the sum of 4th powers of its digits.

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