A353735 -id:A353735 - cp's OEIS frontend

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

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

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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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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A355772 Positions of records in A355770.

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A353736 Number of n-digit terms in A353007.

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