A353007 -id:A353007 - cp's OEIS frontend

A353736 Number of n-digit terms in A353007.

5, 21, 122, 765, 5000, 34581, 249152, 1843485, 14184320, 111117141, 890892032, 7338139005, 60554071040, 518102719701, 4409318285312, 38356828343325, 341939662684160, 2933245707834261, 28085287524564992, 229163829314312445, 2425706018857287680, 18151248585662332821

Offset: 1

Michael S. Branicky, May 06 2022

From Bernard Schott, Jul 14 2022: (Start)
Conjecture 1: lim_{n->oo} a(2n+1)/a(2n-1) = 100.
Conjecture 2: lim_{n->oo} a(2n+2)/a(2n) = 81.
These conjectures are the same as for A353735. (End)

			There are five 1-digit terms in A353007: 0, 2, 4, 6, 8. Thus, a(1) = 5.

Michael S. Branicky, Table of n, a(n) for n = 1..1002

Cf. A333369, A353007, A353735.

def isA353007(n):
    digits = list(map(int, str(n)))
    return all(digits.count(d)%2 != d%2 for d in set(digits))
def a(n):
    start = 0 if n == 1 else 10**(n-1)
    return sum(1 for i in range(start, 10**n) if isA353007(i))
print([a(n) for n in range(1, 7)]) # Michael S. Branicky, May 06 2022

A333369 Positive integers in which any odd digit, if present, occurs an odd number of times, and any even digit, if present, occurs an even number of times.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 17, 19, 22, 31, 35, 37, 39, 44, 51, 53, 57, 59, 66, 71, 73, 75, 79, 88, 91, 93, 95, 97, 100, 111, 122, 135, 137, 139, 144, 153, 157, 159, 166, 173, 175, 179, 188, 193, 195, 197, 212, 221, 223, 225, 227, 229, 232, 252, 272, 292, 300, 315, 317, 319, 322

Offset: 1

Bernard Schott, Mar 17 2020

Inspired by the 520th problem of Project Euler (see link) where such a number is called a "simber".
This sequence has little mathematical interest. The name "simber", which might be interpreted as "silly number", is deprecated. - N. J. A. Sloane, Aug 04 2022
The number of terms with respectively 1, 2, 3, ... digits is 5, 24, 130, ...

			656 is a 3-digit term because it has one 5 and two 6's.
447977 is a 6-digit term because it has one 9, two 4's and three 7's.

Michel Marcus, Table of n, a(n) for n = 1..10000
Project Euler, Problem 520: Simbers.

Cf. A108571 (finite subsequence), A353007.

seqQ[n_] := AllTrue[Tally @ IntegerDigits[n], EvenQ[Plus @@ #] &]; Select[Range[300], seqQ] (* Amiram Eldar, Mar 17 2020 *)

isok(m) = my(d=digits(m), s=Set(d)); for (i=1, #s, if (#select(x->(x==s[i]), d) % 2 != (s[i] % 2), return (0))); return (1); \\ Michel Marcus, Mar 17 2020

def ok(n): s = str(n); return all(s.count(d)%2 == int(d)%2 for d in set(s))
print([k for k in range(323) if ok(k)]) # Michael S. Branicky, Apr 15 2022

A353735 Number of n-digit terms in A333369.

Original entry on oeis.org

5, 24, 130, 792, 5080, 34584, 247360, 1817112, 13918720, 108848664, 869866240, 7169995032, 59085276160, 505735077144, 4311229112320, 37428004374552, 335520388710400, 2861870689152024, 27669446179225600, 223578655251963672, 2398913308953149440, 17708639883984065304

Offset: 1

Michael S. Branicky, May 06 2022

Also the number of n-digit simbers, where a simber is a positive integer in which any odd digit, if present, occurs an odd number of times, and any even digit, if present, occurs an even number of times.

			There are five 1-digit terms in A333369: 1, 3, 5, 7, 9. Thus, a(1) = 5.

Michael S. Branicky, Table of n, a(n) for n = 1..1002
Project Euler, Problem 520: Simbers.

Cf. A333369, A353007, A353736.

def isA333369(n):
    digits = list(map(int, str(n)))
    return all(digits.count(d)%2 == d%2 for d in set(digits))
def a(n): return sum(1 for i in range(10**(n-1), 10**n) if isA333369(i))
print([a(n) for n in range(1, 7)]) # Michael S. Branicky, May 06 2022

From Bernard Schott, Jul 11 2022: (Start)
Conjecture 1: lim_{n->oo} a(2n+1)/a(2n-1) = 100.
Conjecture 2: lim_{n->oo} a(2n+2)/a(2n) = 81. (End)

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A353736 Number of n-digit terms in A353007.

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A333369 Positive integers in which any odd digit, if present, occurs an odd number of times, and any even digit, if present, occurs an even number of times.

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A353735 Number of n-digit terms in A333369.

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