A093037 -id:A093037 - cp's OEIS frontend

A355969 Positions of records in A227872, i.e., integers whose number of odious divisors sets a new record.

1, 2, 4, 8, 16, 28, 56, 84, 112, 168, 336, 672, 1344, 2184, 4368, 8736, 17472, 30576, 34944, 41664, 48048, 61152, 80080, 83328, 96096, 122304, 160160, 192192, 240240, 320320, 336336, 480480, 672672, 960960, 1345344, 1681680, 1921920, 2489760, 2690688, 2738736

Offset: 1

Bernard Schott, Jul 22 2022

Corresponding records of number of odious divisors are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, ...

			a(7) = 56 is in the sequence because A227872(56) = 8 is larger than any earlier value in A227872.

Cf. A000069, A093696, A227872, A330289, A355968.

Similar sequences: A093036, A093037, A330815, A340549.

f[n_] := DivisorSum[n, 1 &, OddQ[DigitCount[#, 2, 1]] &]; fm = -1; s = {}; Do[If[(fn = f[n]) > fm, fm = fn; AppendTo[s, n]], {n, 1, 10^5}]; s (* Amiram Eldar, Jul 22 2022 *)

lista(nn)= my(list = List(), m=0, new); for (n=1, nn, new = sumdiv(n, d, isod(d)); if (new > m, listput(list, n); m = new);); Vec(list); \\ Michel Marcus, Jul 22 2022

from sympy import divisors
from itertools import count, islice
def c(n): return bin(n).count("1")&1
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(), 30))) # Michael S. Branicky, Jul 23 2022

More terms from Amiram Eldar, Jul 22 2022

A356020 Positions of records in A356018, i.e., integers whose number of evil divisors sets a new record.

Original entry on oeis.org

1, 3, 6, 12, 18, 30, 60, 90, 120, 180, 360, 540, 720, 1080, 1440, 2160, 3780, 4320, 6120, 7560, 8640, 12240, 15120, 24480, 27720, 30240, 36720, 48960, 50400, 55440, 73440, 83160, 110880, 128520, 138600, 166320, 221760, 257040, 277200, 332640, 471240, 514080, 554400

Offset: 1

Bernard Schott, Jul 24 2022

Corresponding records of number of evil divisors are 0, 1, 2, 3, 4, 6, 9, 10, 12, 15, ...

			60 is in the sequence because A356018(60) = 9 is larger than any earlier value in A356018.

David A. Corneth, Table of n, a(n) for n = 1..189

Cf. A001969, A093688, A093696, A356018, A356019.

Similar sequences: A093036, A093037, A330815, A350756, A355969.

f[n_] := DivisorSum[n, 1 &, EvenQ[DigitCount[#, 2, 1]] &]; fm = -1; s = {}; Do[If[(fn = f[n]) > fm, fm = fn; AppendTo[s, n]], {n, 1, 10^5}]; s (* Amiram Eldar, Jul 24 2022 *)

upto(n) = my(res = List(), r=-1); forfactored(i=1, n, if(numdiv(i[2]) > r, d = divisors(i[2]); c=sum(j=1, #d, isevil(d[j])); if(c>r, r=c; listput(res,i[1])))); res
isevil(n) = bitand(hammingweight(n), 1)==0 \\ David A. Corneth, Jul 24 2022

from sympy import divisors
from itertools import count, islice
def c(n): return bin(n).count("1")&1 == 0
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(), 40))) # Michael S. Branicky, Jul 24 2022

More terms from Amiram Eldar, Jul 24 2022

A334391 Numbers whose only palindromic divisor is 1.

Original entry on oeis.org

1, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 103, 107, 109, 113, 127, 137, 139, 149, 157, 163, 167, 169, 173, 179, 193, 197, 199, 211, 221, 223, 227, 229, 233, 239, 241, 247, 251, 257, 263, 269, 271, 277, 281, 283, 289, 293, 299, 307

Offset: 1

Bernard Schott, Apr 26 2020

Equivalent: Numbers such that the LCM of their palindromic divisors (A087999) is 1, or,
Numbers such that the number of palindromic divisors (A087990) is 1.
All terms are odd.
The 1st family consists of non-palindromic primes that form the subsequence A334321.
The 2nd family consists of {p^k, p prime, k >= 2} such that p^j for 1 <= j <= k is not a palindrome {169 = 13^2, 289 = 17^2, 361 = 19^2, ..., 2197 = 13^3, ...} (see examples).
The 3rd family consists of products p_1^q_1 * ... * p_k^q_k with k >= 2, all of whose divisors are nonpalindromic {221 = 13 * 27, 247 = 13 * 19, 299 = 13 * 23, 377 = 13 * 29, 391 = 17 * 23, 403 = 13 * 31, 481 = 13 * 37, ...}.
Also, equivalent: numbers all of whose divisors > 1 are nonpalindromic (A029742). - Bernard Schott, Jul 14 2022

			49 = 7^2, the divisor 7 is a palindrome so 49 is not a term.
169 = 13^2, divisors of 169 are {1, 13, 169} and 169 is a term.
391 = 17*23, divisors of 391 are {1,17,23,391} and 391 is a term.
307^2 = 94249 that is palindrome, so 94249 is not a term.

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

A334321 is a subsequence.
Cf. A008365, A087990, A087999, A334139.
Cf. A029742, A087991, A093037, A355695.

notpali:= proc(n) local L;
  L:= convert(n,base,10);
  L <> ListTools:-Reverse(L)
end proc:
filter:= proc(n) option remember; andmap(notpali,numtheory:-divisors(n) minus {1}) end proc:
select(filter, [seq(i,i=1..400,2)]); # Robert Israel, Apr 28 2020

Select[Range[300], !AnyTrue[Rest @ Divisors[#], PalindromeQ] &] (* Amiram Eldar, Apr 26 2020 *)

ispal(n) = my(d=digits(n)); d == Vecrev(d);
isok(n) = fordiv(n, d, if (d>1 && ispal(d), return(0))); return(1); \\ Michel Marcus, Apr 26 2020

from sympy.ntheory import divisors, is_palindromic
def ok(n): return not any(is_palindromic(d) for d in divisors(n)[1:])
print(list(filter(ok, range(1, 308, 2)))) # Michael S. Branicky, May 08 2021

A087990(a(n)) = 1.

A087999(a(n)) = 1.

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

A355695 a(n) is the smallest number that has exactly n nonpalindromic divisors (A029742).

Original entry on oeis.org

1, 10, 20, 30, 48, 72, 60, 140, 144, 120, 210, 180, 300, 240, 560, 504, 360, 420, 780, 1764, 900, 960, 720, 1200, 840, 1560, 2640, 1260, 1440, 2400, 3900, 3024, 1680, 3120, 2880, 4800, 7056, 3600, 2520, 3780, 3360, 5460, 6480, 16848, 6300, 8820, 7200, 9240, 6720, 12480, 5040

Offset: 0

Bernard Schott, Jul 14 2022

			48 has 10 divisors: {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}, only 12, 16, 24 and 48 are nonpalindromic; no positive integer smaller than 48 has four nonpalindromic divisors, hence a(4) = 48.

Michael S. Branicky, Table of n, a(n) for n = 0..710

Cf. A029742, A087991, A093037, A334391.

Similar sequences: A087997, A333456, A355303, A355594.

f[n_] := DivisorSum[n, 1 &, ! PalindromeQ[#] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[50, 10^5] (* Amiram Eldar, Jul 14 2022 *)

isnp(n) = my(d=digits(n)); d!=Vecrev(d); \\ A029742
a(n) = my(k=1); while (sumdiv(k, d, isnp(d)) != n, k++); k; \\ Michel Marcus, Jul 14 2022

from sympy import divisors
from itertools import count, islice
def c(n): s = str(n); return s != s[::-1]
def f(n): return sum(1 for d in divisors(n, generator=True) if c(d))
def agen():
    n, adict = 0, 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(), 51))) # Michael S. Branicky, Jul 27 2022

More terms from Michel Marcus, Jul 14 2022

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A355969 Positions of records in A227872, i.e., integers whose number of odious divisors sets a new record.

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A356020 Positions of records in A356018, i.e., integers whose number of evil divisors sets a new record.

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A334391 Numbers whose only palindromic divisor is 1.

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A355772 Positions of records in A355770.

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A355695 a(n) is the smallest number that has exactly n nonpalindromic divisors (A029742).

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