A247888 -id:A247888 - cp's OEIS frontend

A327750 Numbers without zero digits such that after adding the product of its digits to it, a number with the same product of digits is obtained.

28, 128, 214, 239, 266, 318, 326, 364, 494, 497, 563, 598, 613, 637, 695, 819, 1128, 1214, 1239, 1266, 1318, 1326, 1364, 1494, 1497, 1563, 1598, 1613, 1637, 1695, 1819, 2114, 2139, 2168, 2285, 2313, 2356, 2369, 2419, 2594, 2639, 2791, 3118, 3126, 3148, 3213, 3235

Offset: 1

Bernard Schott, Sep 24 2019

The idea of this sequence comes from a problem in the annual Moscow Mathematical Olympiad (MMO) in 2003: Level A, problem 2. The problem only asks to find a ten-digit number that has the property of the name.
When an integer k belongs to this sequence, the integer 111..11//k obtained by concatenation // of 111..11 and k is also a term; hence, there are primitive terms as 28, 214, 239, 266, 318, 326, ... (A340908).
A subset of it is formed by the numbers 239, 326, 364, 497, 563, 598, 613, 637, 695, 819, 1239, 1326, 1364, 1497, 1563, 1598, 1613, 1637, 1695, 1819, 2139, 2313, 2356, 2369, ... for which the number obtained after adding the product of the digits has exactly the same digits (they are obtained by permuting the digits of the initial number). So, 239 + 2*3*9 = 239 + 54 = 293, 326 + 3*2*6 = 326 + 36 = 362, 3235 + 3*2*3*5 = 3235 + 90 = 3325, 23286 + 2*3*2*8*6 = 23286 + 576 = 23862. - Marius A. Burtea, Sep 24 2019
This subset is A247888. - Bernard Schott, Jul 22 2020

			28 + 2*8 = 44 and 2*8 = 4*4 hence 28 is a term.
326 + 3*2*6 = 362 and 3*2*6 = 3*6*2 hence 326 is another term.

Michel Marcus, Table of n, a(n) for n = 1..10000
Roman Fedorov, Alexei Belov, Alexander Kovaldzhi, Ivan Yashchenko, Moscow Mathematical Olympiads, 2000-2005, Level A, Problem 2, 2003; MSRI, 2011, p. 15 and 97.
Index to sequences related to Olympiads.

Cf. A007954, A009994, A052382.

Subsequences: A247888, A340907, A340908 (primitives).

[k:k in [1..3500]| not 0 in Intseq(k) and &*Intseq(k) eq &*(Intseq(k+&*Intseq(k)))]; // Marius A. Burtea, Sep 24 2019

pd[n_] := Times @@ IntegerDigits[n]; aQ[n_] := (p = pd[n]) > 0 && pd[n + p] == p; Select[Range[5000], aQ] (* Amiram Eldar, Sep 24 2019 *)

isok(n) = my(d = digits(n), p); vecmin(d) && (p=vecprod(d)) && (vecprod(digits(n+p)) == p); \\ Michel Marcus, Sep 24 2019

def test(n):
    m, p = n, 1
    while m > 0:
        m, p = m//10, p*(m%10)
    if p == 0:
        return 0
    m, q = n+p, 1
    while m > 0:
        m, q = m//10, q*(m%10)
    return p == q
n, a = 0, 0
while n < 100:
    a = a+1
    if test(a):
        n = n+1
    print(n,a) # A.H.M. Smeets, Sep 25 2019

More terms from Amiram Eldar, Sep 24 2019

A340907 Numbers m without zero digits such that pod(q) = pod(k) = pod(m) where q = k + pod(k) and k = m + pod(m) where pod is the product of digits, A007954.

Original entry on oeis.org

262713, 267338, 283628, 342713, 351678, 432713, 451676, 516469, 516657, 516675, 622713, 634838, 651674, 716655, 728364, 851673, 857297, 916465, 1262713, 1267338, 1283628, 1342713, 1351678, 1432713, 1451676, 1516469, 1516657, 1516675, 1622713, 1634838, 1651674

Offset: 1

Bernard Schott, Jan 26 2021

The idea of this sequence comes from a remark of Amiram Eldar in Discussion section of A327750 (m + pod(m) = k with pod(k) = pod(m)) in September 2019.
Question: is it possible to get a longer string of integers with this rule?
From David A. Corneth, Feb 01 2021: (Start)
The product of digits of a(n) is a multiple of 6. In terms below 10^10 however all products of digits of a(n) are a multiple of 36. Is that product a multiple of 36 for all a(n)?
The least term k such that k + 6 is here is k = 56516718. Are there consecutive terms that differ by less than 6? (End)

			262713 + pod(262713) = 262713 + 504 = 263217, whose product of digits is also 504, and 263217 + 504 = 263721 whose product of digits is again 504; hence, m=262713, k=263217, q=263721 and pod(m)=pod(k)=pod(q)=504, so 262713 is a term.

Roman Fedorov, Alexei Belov, Alexander Kovaldzhi, and Ivan Yashchenko, Moscow-Mathematical Olympiads, 2000-2005, Level A, Problem 2, 2003; MSRI, 2011, p. 15 and 97.

David A. Corneth, Table of n, a(n) for n = 1..10000

Subsequence of A327750.

Cf. A007954, A052382, A247888, A340908.

pod[n_] := Times @@ IntegerDigits[n]; seqQ[n_] := Module[{p = pod[n], k, q}, k = n + p; q = k + pod[k]; p > 0 && Equal @@ {p, pod[k], pod[q]}]; Select[Range[2*10^6], seqQ] (* Amiram Eldar, Jan 26 2021 *)

isok(m) = my(pm=vecprod(digits(m)), k=m+pm, pk=vecprod(digits(k)), q=k+pk, pq=vecprod(digits(q))); pm && (pm==pk) && (pk==pq); \\ Michel Marcus, Jan 26 2021

Terms from Amiram Eldar, Jan 26 2021

A340908 Primitive numbers m without zero digits such that pod(m + pod(m)) = pod(m) where pod is the product of digits, A007954.

Original entry on oeis.org

28, 214, 239, 266, 318, 326, 364, 494, 497, 563, 598, 613, 637, 695, 819, 2114, 2139, 2168, 2285, 2313, 2356, 2369, 2419, 2594, 2639, 2791, 3118, 3126, 3148, 3213, 3235, 3238, 3259, 3354, 3365, 3561, 3698, 3786, 4138, 4145, 4188, 4219, 4338, 4346, 4353, 4368, 4395

Offset: 1

Bernard Schott, Jan 31 2021

When a number k belongs to A327750, the integer 111..11//k obtained by concatenation of 111..11 and k is another term; hence, there exist primitive terms as 28, 214, 239, ... that are listed in this sequence.

Equivalently, terms of A327750 that do not begin with 1.

			pod(28 + pod(28)) = pod(28 + 2*8) = pod(28 + 16) = pod(44) = 4*4 = 16 = pod(28), hence 28 that does not begin with 1 is a term.

Roman Fedorov, Alexei Belov, Alexander Kovaldzhi, and Ivan Yashchenko, Moscow-Mathematical Olympiads, 2000-2005, Level A, Problem 2, 2003; MSRI, 2011, pp. 15 and 97.

Index to sequences related to Olympiads.

Cf. A007954, A247888, A340907.

Subsequence of A327750.

pod[n_] := Times @@ IntegerDigits[n]; q[n_] := First[IntegerDigits[n]] > 1 && (p = pod[n]) > 0 && pod[n + p] == p; Select[Range[5000], q] (* Amiram Eldar, Jan 31 2021 *)

isok(n) = my(d = digits(n), p); vecmin(d) && ((d[1]!=1) && p=vecprod(d)) && (vecprod(digits(n+p)) == p); \\ Michel Marcus, Feb 01 2021

A248040 Numbers n such that n*digsum(n) contains the same distinct digits as n.

Original entry on oeis.org

1, 10, 100, 109, 190, 208, 280, 307, 370, 406, 450, 460, 505, 550, 604, 640, 703, 730, 802, 820, 901, 910, 1000, 1009, 1018, 1027, 1036, 1045, 1054, 1063, 1072, 1081, 1090, 1108, 1168, 1180, 1207, 1270, 1286, 1306, 1360, 1405, 1450, 1504, 1540, 1603, 1630, 1702, 1720, 1726, 1801

Offset: 1

Derek Orr, Sep 30 2014

10^k is a subsequence for k >= 0. Thus this sequence is infinite.

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

Cf. A007953 (sum of digits), A011557 (powers of 10).

Cf. A247888 (similar, with n + product of digits), A247887 (similar, with n + digsum).

filter:= proc(n) local L,s,d;
  L:= convert(n,base,10);
  d:= convert(L,set);
  s:= convert(L,`+`);
  evalb(convert(convert(n*s,base,10),set)=d)
end proc:
select(filter, [$1..2000]); # Robert Israel, Mar 06 2018

for(n=1,10^4,d=digits(n);if(vecsort(d,,8)==vecsort(digits(n*sumdigits(n)),,8),print1(n,", ")))

A248210 Zeroless numbers k (numbers in A052382) such that k - DigitProduct(k) contains the same distinct digits as k.

Original entry on oeis.org

293, 362, 436, 545, 554, 631, 653, 749, 763, 891, 958, 965, 1293, 1362, 1436, 1545, 1554, 1631, 1653, 1749, 1763, 1891, 1958, 1965, 2193, 2331, 2491, 2536, 2556, 2565, 2693, 2917, 2954, 2963, 3162, 3231, 3325, 3382, 3529, 3534, 3635, 3651, 4291, 4515, 4533, 4551, 4634, 4935, 4952, 4971

Offset: 1

Derek Orr, Oct 03 2014

Numbers that contain zeros trivially have this property. - Tanya Khovanova, Jul 19 2021

			631 - 6*3*1 = 613 contains the same digits as 631. So 631 is a term of this sequence.

Cf. A052382 (zeroless numbers), A007954 (digit product).

Cf. A247888 (similar, with n + digit product).

Select[Range@5000,(d=IntegerDigits@#;FreeQ[d,0]&&Union@IntegerDigits[#-Times@@d]==Union@d)&] (* Giorgos Kalogeropoulos, Jul 20 2021 *)

for(n=1, 10^4, d=digits(n); p=prod(i=1, #d, d[i]); if(p && vecsort(digits(n), , 8)==vecsort(digits(n-p), , 8), print1(n, ", ")))

from math import prod
def ok(n):
    s = str(n)
    return '0' not in s and set(str(n-prod(int(d) for d in s))) == set(s)
print(list(filter(ok, range(5000)))) # Michael S. Branicky, Jul 18 2021

A248039 Numbers n such that n*A007954(n) contains the same distinct digits as n.

Original entry on oeis.org

0, 1, 11, 111, 792, 1111, 1376, 2174, 2841, 11111, 11628, 12168, 12763, 12841, 14213, 14228, 14663, 19842, 24314, 24679, 24738, 24786, 26439, 26731, 26938, 29126, 39117, 39228, 49326, 64113, 76983, 79328, 83694, 83712, 83764, 86429, 87164, 89174, 92387, 92476, 93711, 94831, 98174

Offset: 1

Derek Orr, Sep 30 2014

A002275 is a subsequence, thus this sequence is infinite.

Cf. A002275 (repunits), A007954 (digit product).

Cf. A247887 (similar, with n + digit sum), A247888 (similar, with n + digit product).

[n: n in [0..10^5] | Set(Intseq(n*&*Intseq(n))) eq Set(Intseq(n))]; // Bruno Berselli, Oct 09 2014

for(n=0, 10^6, d=digits(n); p=prod(i=1, #d, d[i]); if(vecsort(digits(n), , 8)==vecsort(digits(n*p), , 8), print1(n, ", ")))

A248718 Numbers n such that n + (sum of digits of n) and n + (product of digits of n) contain the same distinct digits of n.

Original entry on oeis.org

1512, 4346, 5112, 5769, 11215, 11512, 12115, 12313, 12511, 13213, 14346, 14512, 15112, 15211, 15412, 21115, 21313, 21511, 23113, 25111, 27369, 31213, 32113, 34135, 34535, 41346, 41512, 43135, 43535, 45112, 51112, 51211, 51412, 52111, 52569, 53435, 53534, 53958, 54112, 54533, 56925

Offset: 1

Derek Orr, Oct 12 2014

Intersection of A247887 and A247888.

Cf. A247887, A247888, A007954, A007953, A052382.

for(n=0,10^6,d=digits(n);p=prod(i=1,#d,d[i]);vp=vecsort(digits(p+n), ,8);vs=vecsort(digits(sumdigits(n)+n), ,8);if(vs==vp&&vp==vecsort(d, ,8)&&vs==vecsort(d, ,8)&&p,print1(n,", ")))

cp's OEIS Frontend

A327750 Numbers without zero digits such that after adding the product of its digits to it, a number with the same product of digits is obtained.

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A340907 Numbers m without zero digits such that pod(q) = pod(k) = pod(m) where q = k + pod(k) and k = m + pod(m) where pod is the product of digits, A007954.

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A340908 Primitive numbers m without zero digits such that pod(m + pod(m)) = pod(m) where pod is the product of digits, A007954.

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A248040 Numbers n such that n*digsum(n) contains the same distinct digits as n.

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A248210 Zeroless numbers k (numbers in A052382) such that k - DigitProduct(k) contains the same distinct digits as k.

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A248039 Numbers n such that n*A007954(n) contains the same distinct digits as n.

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A248718 Numbers n such that n + (sum of digits of n) and n + (product of digits of n) contain the same distinct digits of n.

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