A161351 -id:A161351 - cp's OEIS frontend

A358350 Numbers that can be written as (m + sum of digits of m + product of digits of m) for some m.

3, 6, 9, 11, 12, 14, 15, 17, 18, 20, 21, 22, 23, 24, 26, 27, 29, 30, 32, 33, 34, 35, 38, 42, 43, 44, 46, 48, 50, 53, 54, 55, 56, 58, 62, 63, 66, 68, 69, 73, 74, 76, 77, 78, 80, 82, 83, 86, 88, 90, 92, 95, 97, 98, 99, 101, 103, 104, 105, 106, 107, 108, 109, 110

Offset: 1

Bernard Schott, Nov 11 2022

Integers that are in A161351.
(i) Can arbitrarily long sets of consecutive integers be found in this sequence?
(ii) Is the gap between two consecutive terms bounded?
A000533 \ {1} is a subsequence.
This has the same asymptotic density, approximately 0.9022222, as A176995, since the asymptotic density of non-pandigital numbers is 0. - Charles R Greathouse IV, Nov 16 2022

			A161351(23) = 23 + (2+3) + (2*3) = 34 so 34 is a term.
There is no integer du_10 such that du + (d+u) + (d*u) = 31, so 31 is not a term.

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

Range of A161351.
Similar: A176995 (m+digitsum), A336826 (m*digitprod), A337718 (m+digitprod).
Cf. A000533.

f[n_] := n + Total[(d = IntegerDigits[n])] + Times @@ d; With[{m = 110}, Select[Union[Table[f[n], {n, 1, m}]], # <= m &]] (* Amiram Eldar, Nov 11 2022 *)

f(n) = my(d=digits(n)); n + vecsum(d) + vecprod(d); \\ A161351
lista(nn) = select(x->(x<=nn), Set(vector(nn, k, f(k)))); \\ Michel Marcus, Nov 12 2022

from math import prod
def sp(n): d = list(map(int, str(n))); return sum(d) + prod(d)
def ok(n): return any(m + sp(m) == n for m in range(n))
print([k for k in range(111) if ok(k)]) # Michael S. Branicky, Dec 19 2022

a(n) ~ kn with k approximately 1.108374, see comments. - Charles R Greathouse IV, Nov 16 2022

A358351 Number of values of m such that m + (sum of digits of m) + (product of digits of m) is n.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 3, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 1, 1, 1, 0, 2, 0, 0, 0, 2, 1, 0, 0, 1, 0, 2, 1, 0, 0, 0, 1, 2, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 2, 0

Offset: 1

Bernard Schott, Nov 16 2022

a(n) is the number of times n occurs in A161351.

a(n) > 0 iff n is in A358350.

			A161351(15) = 15 + (1+5) + (1*5) = 26 =  21 + (2+1) + (2*1) = A161351(21), so, a(26) = 2.

Cf. A011540, A161351, A358350.

Similar: A230093 (m+digitsum), A230103 (m+digitprod).

f[n_] := n + Total[(d = IntegerDigits[n])] + Times @@ d; With[{m = 100}, BinCounts[Table[f[n], {n, 1, m}], {1, m, 1}]] (* Amiram Eldar, Nov 16 2022 *)

first(n) = {my(res = vector(n)); for(i = 1, n, c = i + sumdigits(i) + vecprod(digits(i)); if(c <= n, res[c]++ ) ); res } \\ David A. Corneth, Nov 16 2022

from itertools import combinations_with_replacement
from math import prod
def A358351(n):
    c = set()
    for l in range(1,len(str(n))+1):
        l1, l2 = 10**l, 10**(l-1)
        for d in combinations_with_replacement(tuple(range(10)),l):
            s, p = sum(d), prod(d)
            if l1>(m:=n-s-p)>=l2 and sorted(int(d) for d in str(m)) == list(d):
                c.add(m)
    return len(c) # Chai Wah Wu, Nov 20 2022

A358352 a(n) is the smallest number k such that A358351(k) = n.

Original entry on oeis.org

1, 3, 26, 38, 380, 1116, 12912, 95131, 342038, 3320210, 494204209, 773089018

Offset: 0

Bernard Schott, Nov 19 2022

			19+sod(19)+pod(19) = 24+sod(24)+pod(24) = 31+sod(31)+pod(31) = 38, and there is no integer < 38 for which function A161351 has 3 preimages, so a(3) = 38.

Rémy Sigrist, C program

Cf. A006064, A007953, A007954, A161351, A358350, A358351.

C
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See Links section.
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f[n_] := n + Total[(d = IntegerDigits[n])] + Times @@ d; s = With[{m = 10^7}, BinCounts[Table[f[n], {n, 1, m}], {1, m, 1}]]; FirstPosition[s, #] & /@ Range[0, Max[s]] // Flatten (* Amiram Eldar, Nov 19 2022 *)

first(n) = my(res = vector(n)); for(i = 1, n, c = i + sumdigits(i) + vecprod(digits(i)); if(c <= n, res[c]++ ) ); res; \\ A358351
lista(nn) = my(v=first(nn)); for (n=0, 20, my(vs = select(x->(x==n), v, 1)); if (#vs, print1(vs[1], ", "), break);); \\ Michel Marcus, Nov 20 2022

a(4)-a(5) from Michel Marcus, Nov 19 2022
a(6)-a(9) from Amiram Eldar, Nov 19 2022
a(10)-a(11) from Rémy Sigrist, Nov 20 2022

A358353 Numbers that are not of the form m + (sum of digits of m) + (product of digits of m) for any m.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 13, 16, 19, 25, 28, 31, 36, 37, 39, 40, 41, 45, 47, 49, 51, 52, 57, 59, 60, 61, 64, 65, 67, 70, 71, 72, 75, 79, 81, 84, 85, 87, 89, 91, 93, 94, 96, 100, 102, 116, 120, 125, 126, 129, 137, 141, 142, 146, 150, 152, 153, 160, 161, 162, 166, 171, 172, 173, 180

Offset: 1

Bernard Schott, Dec 19 2022

Numbers missing from A358350.

The first differences show some periodicity, for example those for values 2184-3811 repeat at terms 5513-7140. - Bill McEachen, Jan 08 2023

			There is no term du_10 < 36 such that du + (d+u) + (d*u) = 36, so 36 is a term.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for Colombian or self numbers and related sequences.

Cf. A161351, A358350, A358351, A358352.

Similar: A003052 (m+digitsum), A230104 (m+digitprod).

f:= proc(n) local L; L:= convert(n,base,10); n + convert(L,`+`)+convert(L,`*`) end proc:
sort(convert({$1..200} minus map(f, {$1..200}),list)); # Robert Israel, Dec 22 2022

f[n_] := n + Total[(d = IntegerDigits[n])] + Times @@ d; With[{m = 180}, Complement[Range[m], Table[f[n], {n, 1, m}]]] (* Amiram Eldar, Dec 19 2022 *)

f(n) = my(d=digits(n)); vecsum(d)+vecprod(d)+n; \\ A161351
isok(m) = for(i=1, m, if (f(i) == m, return(0))); return(1); \\ Michel Marcus, Jan 09 2023

from math import prod
def sp(n): d = list(map(int, str(n))); return sum(d) + prod(d)
def ok(n): return all(m + sp(m) != n for m in range(n+1))
print([k for k in range(181) if ok(k)]) # Michael S. Branicky, Dec 19 2022

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A358350 Numbers that can be written as (m + sum of digits of m + product of digits of m) for some m.

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A358351 Number of values of m such that m + (sum of digits of m) + (product of digits of m) is n.

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A358352 a(n) is the smallest number k such that A358351(k) = n.

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A358353 Numbers that are not of the form m + (sum of digits of m) + (product of digits of m) for any m.

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