A358350 -id:A358350 - cp's OEIS frontend

A359248 a(n) is the first number that is the start of a string of exactly n consecutive numbers in A358350.

3, 11, 42, 32, 20, 154, 130, 1240, 515, 1033, 610, 3101, 103, 4010, 56379, 31809, 35212, 23103, 7413, 12101, 1011, 204, 10391, 92109, 25013, 812819, 75099, 8493016, 437016, 775009, 287017, 8029, 457014, 10503, 26148108, 10997, 27445016, 286092, 231135007, 480014

Offset: 1

Robert Israel, Dec 22 2022

a(n) is the least k such that k, k+1, ..., k+n-1 are in A358350 but k-1 and k+n are not.

			a(3) = 42 because the three consecutive numbers 42, 43 and 44 are in A358350 but 41 and 45 are not, and this is the first string of exactly three consecutive numbers in A358350.

Michael S. Branicky, Table of n, a(n) for n = 1..71
Michael S. Branicky, All terms below 3.4*10^10

Cf. A358350.

f:= proc(n) local L; L:= convert(n,base,10); n + convert(L,`+`)+convert(L,`*`) end proc:
S:= select(`<=`,map(f,{$1..10^6}),10^6):
S:= sort(convert(S,list)):
V:= Vector(27):
a:= 1:
for x from 2 to nops(S) do
  if S[x] - S[x-1] > 1 then
    v:= x-a;
    if v <= 27 and V[v] = 0 then V[v]:= S[a]; count:= count+1; fi;
    a:= x;
  fi
od:
convert(V,list);

from math import prod
from itertools import islice
def sp(n): d = list(map(int, str(n))); return sum(d) + prod(d)
def agen(increment=10**6):
    S, L, U, adict, rl, n = set(), 1, increment, dict(), 0, 1
    while True:
        S |= set(i + sp(i) for i in range(L, U))
        for i in range(L, U):
            if i in S: rl += 1
            else:
                if rl not in adict:
                    adict[rl] = i - rl
                    while n in adict: yield adict[n]; n += 1
                rl = 0
        S -= set(range(L, U))
        L, U = U, U + increment
print(list(islice(agen(), 34))) # Michael S. Branicky, Dec 23 2022

a(28)-a(40) from Michael S. Branicky, Dec 22 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
```
See Links section.
```

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

cp's OEIS Frontend

A359248 a(n) is the first number that is the start of a string of exactly n consecutive numbers in A358350.

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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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Mathematica

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Python

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

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Author

Keywords

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Crossrefs

Programs

C

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python