A333369 -id:A333369 - cp's OEIS frontend

A353735 Number of n-digit terms in A333369.

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)

A355771 a(n) is the smallest integer that has exactly n divisors from A333369.

Original entry on oeis.org

1, 3, 9, 15, 45, 105, 195, 315, 945, 900, 1575, 2100, 3900, 6825, 11655, 10500, 6300, 18900, 25200, 35100, 27300, 31500, 44100, 94500, 157500, 107100, 81900, 233100, 220500, 598500, 245700, 333900, 409500, 491400, 900900, 573300, 600600, 1228500, 1669500, 1965600

Offset: 1

Bernard Schott, Jul 17 2022

			15 has 4 divisors: {1, 3, 5, 15} all of which are in A333369 integers, and no smaller number has this property, hence a(4) = 15.

Michael S. Branicky, Table of n, a(n) for n = 1..87

Cf. A333369, A353735, A355770, A355772, A355773.
Cf. A005231, A053624.
Similar sequences: A087997, A333456, A355303, A355699.

q[n_] := AllTrue[Tally @ IntegerDigits[n], EvenQ[Plus @@ #] &]; f[n_] := DivisorSum[n, 1 &, q[#] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n]; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[40, 10^7] (* Amiram Eldar, Jul 17 2022 *)

issimber(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); \\ A333369
a(n) = my(k=1); while (sumdiv(k, d, issimber(d)) != n, k++); k; \\ Michel Marcus, Jul 18 2022

from sympy import divisors
from itertools import count, islice
def c(n): s = str(n); return all(s.count(d)%2 == int(d)%2 for d in set(s))
def f(n): return sum(1 for d in divisors(n, generator=True) if c(d))
def agen():
    n, adict = 1, dict()
    for k in count(1):
        fk = f(k)
        if fk not in adict: adict[fk] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 29))) # Michael S. Branicky, Jul 23 2022

More terms from Amiram Eldar, Jul 17 2022

A355773 Numbers all of whose divisors are members of A333369.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 17, 19, 31, 35, 37, 39, 51, 53, 57, 59, 71, 73, 79, 91, 93, 95, 97, 111, 137, 139, 153, 157, 159, 173, 179, 193, 197, 221, 223, 227, 229, 317, 333, 359, 371, 379, 395, 397, 443, 449, 519, 537, 571, 579, 591, 593, 661, 663, 669, 719, 739

Offset: 1

Bernard Schott, Jul 18 2022

All terms are necessarily odd because 2 is not in A333369

			111 is a term since all the divisors of 111, i.e., 1, 3, 37 and 111, are in A333369.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Project Euler, Problem 520: Simbers.

Cf. A333369, A353735, A355770, A355771, A355772.
Similar sequences: A062687, A190217, A329419, A337741
.
Subsequences: A155045, A355853.

simQ[n_] := AllTrue[Tally @ IntegerDigits[n], EvenQ[Plus @@ #] &]; Select[Range[1000], AllTrue[Divisors[#], simQ] &] (* Amiram Eldar, Jul 19 2022 *)

issimber(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); \\ A333369
isok(k) = fordiv(k, d, if (!issimber(d), return(0))); return(1); \\ Michel Marcus, Jul 19 2022

from sympy import divisors, isprime
def c(n): s = str(n); return all(s.count(d)%2 == int(d)%2 for d in set(s))
def ok(n): return n > 0 and all(c(d) for d in divisors(n, generator=True))
print([k for k in range(740) if ok(k)]) # Michael S. Branicky, Jul 24 2022

A355770 a(n) is the number of terms of A333369 that divide n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 1, 2, 2, 2, 4, 1, 2, 3, 2, 2, 3, 2, 1, 2, 2, 2, 3, 2, 1, 4, 2, 1, 2, 2, 4, 3, 2, 2, 4, 2, 1, 3, 1, 3, 5, 1, 1, 2, 2, 2, 4, 2, 2, 3, 2, 2, 4, 1, 2, 4, 1, 2, 4, 1, 3, 4, 1, 2, 2, 4, 2, 3, 2, 2, 5, 2, 2, 4, 2, 2, 3, 1, 1, 3, 3, 1, 2

Offset: 1

Bernard Schott, Jul 16 2022

Cf. A333369, A353735, A355771, A355772, A355773.

Similar sequences: A083230, A087990, A087991, A332268, A355302.

q[n_] := AllTrue[Tally @ IntegerDigits[n], EvenQ[Plus @@ #] &]; a[n_] := DivisorSum[n, 1 &, q[#] &]; Array[a, 100] (* Amiram Eldar, Jul 16 2022 *)

issimber(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); \\ A333369
a(n) = sumdiv(n, d, issimber(d)); \\ Michel Marcus, Jul 18 2022

from sympy import divisors
def c(n): s = str(n); return all(s.count(d)%2 == int(d)%2 for d in set(s))
def a(n): return sum(1 for d in divisors(n, generator=True) if c(d))
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Jul 16 2022

More terms from Michael S. Branicky, Jul 16 2022

A355853 Primes in A333369.

Original entry on oeis.org

3, 5, 7, 13, 17, 19, 31, 37, 53, 59, 71, 73, 79, 97, 137, 139, 157, 173, 179, 193, 197, 223, 227, 229, 317, 359, 379, 397, 443, 449, 571, 593, 661, 719, 739, 751, 881, 883, 887, 937, 953, 971, 1009, 1117, 1151, 1171, 1223, 1229, 1447, 1511, 1579, 1597, 1663, 1667, 1669

Offset: 1

Bernard Schott, Jul 19 2022

			443 is prime and 443 has two 4's and one 3 in its decimal expansion, hence 443 is a term.

Intersection of A000040 and A333369.
Subsequence of A355773.
Supersequence of A155045.
Similar sequences: A002385, A004023.

simQ[n_] := AllTrue[Tally @ IntegerDigits[n], EvenQ[Plus @@ #] &]; Select[Prime[Range[300]], simQ] (* Amiram Eldar, Jul 19 2022 *)

issimber(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); \\ A333369
isok(m) = isprime(m) && issimber(m); \\ Michel Marcus, Jul 19 2022

from itertools import count, islice
from sympy import isprime
def A355853_gen(startvalue=1): # generator of terms
    return filter(lambda n:not any((str(n).count(d)^int(d))&1 for d in set(str(n))) and isprime(n),count(max(startvalue,1)))
A355853_list = list(islice(A355853_gen(),30)) # Chai Wah Wu, Jul 21 2022

Extended by Michel Marcus, Jul 19 2022

A356177 Palindromes in A333369.

Original entry on oeis.org

1, 3, 5, 7, 9, 22, 44, 66, 88, 111, 212, 232, 252, 272, 292, 333, 414, 434, 454, 474, 494, 555, 616, 636, 656, 676, 696, 777, 818, 838, 858, 878, 898, 999, 2002, 2222, 2442, 2662, 2882, 4004, 4224, 4444, 4664, 4884, 6006, 6226, 6446, 6666, 6886, 8008, 8228, 8448, 8668, 8888, 10101

Offset: 1

Bernard Schott, Jul 28 2022

If a term has a decimal digit that is odd, it must have an odd number of decimal digits and all odd digits are the same. - Chai Wah Wu, Jul 29 2022

If a term has an even number of decimal digits, then it must have only even decimal digits. - Bernard Schott, Jul 30 2022

			474 is palindrome and 474 has two 4's and one 7 in its decimal expansion, hence 474 is a term.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Intersection of A002113 and A333369.

Cf. A355770, A355771, A355772, A100706 (subsequence of repunits).

simQ[n_] := AllTrue[Tally @ IntegerDigits[n], EvenQ[Plus @@ #] &]; Select[Range[10^4], PalindromeQ[#] && simQ[#] &] (* Amiram Eldar, Jul 28 2022 *)

from itertools import count, islice, product
def simb(n): s = str(n); return all(s.count(d)%2==int(d)%2 for d in set(s))
def pals(): # generator of palindromes
    digits = "0123456789"
    for d in count(1):
        for p in product(digits, repeat=d//2):
            if d > 1 and p[0] == "0": continue
            left = "".join(p); right = left[::-1]
            for mid in [[""], digits][d%2]:
                yield int(left + mid + right)
def agen(): yield from filter(simb, pals())
print(list(islice(agen(), 55))) # Michael S. Branicky, Jul 28 2022

# faster version based on Comments
from itertools import count, islice, product
def odgen(d): yield from [1, 3, 5, 7, 9] if d == 1 else sorted(int(f+"".join(p)+o+"".join(p[::-1])+f) for o in "13579" for f in o + "2468" for p in product(o+"02468", repeat=d//2-1))
def evgen(d): yield from (int(f+"".join(p)+"".join(p[::-1])+f) for f in "2468" for p in product("02468", repeat=d//2-1))
def A356177gen():
    for d in count(1, step=2): yield from odgen(d); yield from evgen(d+1)
print(list(islice(A356177gen(), 55))) # Michael S. Branicky, Jul 30 2022

A353007 Nonnegative integers in which any odd digit, if present, occurs an even number of times, and any even digit, if present, occurs an odd number of times.

Original entry on oeis.org

0, 2, 4, 6, 8, 11, 20, 24, 26, 28, 33, 40, 42, 46, 48, 55, 60, 62, 64, 68, 77, 80, 82, 84, 86, 99, 101, 110, 112, 114, 116, 118, 121, 141, 161, 181, 204, 206, 208, 211, 222, 233, 240, 246, 248, 255, 260, 264, 268, 277, 280, 284, 286, 299, 303, 323, 330, 332, 334

Offset: 1

Bernard Schott, Apr 15 2022

Like the converse of A333369.

			181 is a 3-digit term because it has two 1's and one 8.

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

Cf. A333369.

filter:= proc(n) local L;
    L:= map(rhs-lhs,Statistics:-Tally(convert(n,base,10)));
    andmap(type,L,odd)
end proc:
select(filter, [$0..1000]); # Robert Israel, Jul 31 2024

q[n_] := AllTrue[Tally @ IntegerDigits[n], OddQ[Plus @@ #] &]; Select[Range[0, 300], q] (* Amiram Eldar, Apr 15 2022 *)

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, Apr 15 2022

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(335) if ok(k)]) # Michael S. Branicky, Apr 15 2022

A355772 Positions of records in A355770.

Original entry on oeis.org

1, 3, 9, 15, 45, 105, 195, 315, 900, 1575, 2100, 3900, 6300, 18900, 25200, 27300, 31500, 44100, 81900, 220500, 245700, 333900, 409500, 491400, 573300, 600600, 1201200, 2402400, 3603600, 4804800, 7207200, 10810800, 14414400, 20420400, 21621600, 40840800, 43243200

Offset: 1

Bernard Schott, Jul 18 2022

Corresponding records are 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 17, ...

			a(5) = 45 is in the sequence because A355770(45) = 5 is larger than any earlier value in A355770.

Cf. A333369, A353735, A355770, A355771, A355773.

Similar sequences: A093036, A093037, A340549, A350424.

q[n_] := AllTrue[Tally @ IntegerDigits[n], EvenQ[Plus @@ #] &]; f[n_] := DivisorSum[n, 1 &, q[#] &]; fm = -1; s = {}; Do[If[(fn = f[n]) > fm, fm = fn; AppendTo[s, n]], {n, 1, 10^5}]; s (* Amiram Eldar, Jul 18 2022 *)

from sympy import divisors
from itertools import count, islice
def c(n): s = str(n); return all(s.count(d)%2 == int(d)%2 for d in set(s))
def f(n): return sum(1 for d in divisors(n, generator=True) if c(d))
def agen(record=-1):
    for k in count(1):
        if f(k) > record: record = f(k); yield k
print(list(islice(agen(), 20))) # Michael S. Branicky, Jul 25 2022

a(21)-a(31) from Michel Marcus, Jul 18 2022

a(32)-a(37) from Amiram Eldar, Jul 18 2022

A353736 Number of n-digit terms in A353007.

Original entry on oeis.org

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

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A353735 Number of n-digit terms in A333369.

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A355771 a(n) is the smallest integer that has exactly n divisors from A333369.

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A355773 Numbers all of whose divisors are members of A333369.

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A355770 a(n) is the number of terms of A333369 that divide n.

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A355853 Primes in A333369.

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A356177 Palindromes in A333369.

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A353007 Nonnegative integers in which any odd digit, if present, occurs an even number of times, and any even digit, if present, occurs an odd number of times.

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