A329419 -id:A329419 - cp's OEIS frontend

A342572 Positive numbers all of whose prime factors are binary palindromes.

1, 3, 5, 7, 9, 15, 17, 21, 25, 27, 31, 35, 45, 49, 51, 63, 73, 75, 81, 85, 93, 105, 107, 119, 125, 127, 135, 147, 153, 155, 175, 189, 217, 219, 225, 243, 245, 255, 257, 279, 289, 313, 315, 321, 343, 357, 365, 375, 381, 405, 425, 441, 443, 459, 465, 511, 525, 527

Offset: 1

Amiram Eldar, Mar 27 2021

			15 is a term since the binary representation of its prime factors, 3 and 5, are both palindromes: 11 and 101.
1 is a term because it has no prime factors, and "the empty set has every property". - _N. J. A. Sloane_, Jan 16 2022

Amiram Eldar, Table of n, a(n) for n = 1..10010 (terms below 10^7)

The binary version of A033620.
Subsequences: A016041, A329419.
Cf. A006995.

seq[max_] := Module[{ps = Select[Range[max], PalindromeQ @ IntegerDigits[#, 2] && PrimeQ[#] &], s = {1}, s1, s2}, Do[p = ps[[k]]; emax = Floor@Log[p, max]; s1 = Join[{1}, p^Range[emax]]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= max &]; s = Union[s, s2], {k, 1, Length[ps]}]; s]; seq[1000]
Join[{1},Module[{bps=Select[Prime[Range[200]],IntegerDigits[#,2] == Reverse[ IntegerDigits[ #,2]]&]},Select[ Range[Max[ bps]],SubsetQ[ bps,FactorInteger[#][[All,1]]]&]]] (* Harvey P. Dale, Jan 16 2022 *)

from sympy import factorint
def ispal(s): return s == s[::-1]
def ok(n): return n > 0 and all(ispal(bin(f)[2:]) for f in factorint(n))
print([k for k in range(528) if ok(k)]) # Michael S. Branicky, Jan 17 2022

Sum_{n>=1} 1/a(n) = Product_{p in A016041} p/(p-1) = 2.52136...

"Positive" added to definition by N. J. A. Sloane, Jan 16 2022

A355716 a(n) is the smallest number that has exactly n binary palindrome divisors (A006995).

Original entry on oeis.org

1, 3, 9, 15, 99, 45, 135, 189, 315, 495, 945, 765, 2079, 6237, 3465, 5355, 4095, 8415, 31185, 20475, 25245, 12285, 85995, 58905, 61425, 45045, 69615, 176715, 446985, 225225, 328185, 208845, 135135, 405405, 528255, 1396395, 675675, 2027025, 765765, 5360355, 2993445, 3968055, 3828825

Offset: 1

Bernard Schott, Jul 15 2022

			a(4) = 15 since 15 has 4 divisors {1, 3, 5, 15} that are all palindromes when written in binary: 1, 11, 101 and 1111; no positive integer smaller than 15 has four divisors that are binary palindromes, hence a(4) = 15.
a(5) = 99 since 99 has 6 divisors {1, 3, 9, 11, 33, 99} of which only 11 is not a palindrome when written in binary: 11_10 = 1011_2; no positive integer smaller than 99 has five divisors that are binary palindromes, hence a(5) = 99.

Michael S. Branicky, Table of n, a(n) for n = 1..67

Cf. A006995, A175242, A329419, A329420.

f[n_] := DivisorSum[n, 1 &, PalindromeQ[IntegerDigits[#, 2]] &]; 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[25, 10^5] (* Amiram Eldar, Jul 15 2022 *)

is(n) = my(d=binary(n)); d==Vecrev(d); \\ A006995
a(n) = my(k=1); while (sumdiv(k, d, is(d)) != n, k++); k; \\ Michel Marcus, Jul 15 2022

from sympy import divisors
from itertools import count, islice
def c(n): b = bin(n)[2:]; return b == b[::-1]
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(), 20))) # Michael S. Branicky, Jul 23 2022

More terms from Michael S. Branicky, Jul 15 2022

cp's OEIS Frontend

A342572 Positive numbers all of whose prime factors are binary palindromes.

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