This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
binPalQ[n_] := PalindromeQ @ IntegerDigits[n, 2];
binAllDivPalQ[n_] := binPalQ[n] && AllTrue[Most @ Divisors[n], binPalQ];
divNumMax = 0; seq={}; Do[If[binAllDivPalQ[n] && (divNum = DivisorSigma[0, n]) > divNumMax, divNumMax = divNum; AppendTo[seq, n]],{n, 1, 2*10^5}]; seq
6 is a term since all the divisors of 6, i.e., 1, 2, 3 and 6, are Niven numbers.
nivenQ[n_] := Divisible[n, Plus @@ IntegerDigits[n]]; allQ[n_] := AllTrue[Divisors[n], nivenQ]; p = {1, 2, 3, 5, 7}; s = {1}; n = 0; While[Length[s] != n, n = Length[s]; s = Select[Union @ Flatten @ Outer[Times, s, p], allQ]]; s
111 is a term since all the divisors of 111, i.e., 1, 3, 37 and 111, are in A333369.
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
6 is a term since all the divisors of 6, i.e., 1, 2, 3 and 6, are Zuckerman numbers.
zuckQ[n_] := (prod = Times @@ IntegerDigits[n]) > 0 && Divisible[n, prod]; Select[Range[24], AllTrue[Divisors[#], zuckQ] &] (* Amiram Eldar, Oct 01 2020 *)
isok(m) = {fordiv(m, d, my(p=vecprod(digits(d))); if (!p || (d % p), return (0))); return (1);} \\ Michel Marcus, Oct 05 2020
9 is a term since it has 3 binary palindromic divisors, 1, 3 and 9, whose binary representations are 1, 11 and 1001. All the numbers below 9 have less than 3 binary palindromic divisors.
binPalDiv[n_] := DivisorSum[n, 1 &, PalindromeQ @ IntegerDigits[#, 2] &]; bmax = 0; seq = {}; Do[b = binPalDiv[n]; If[b > bmax, bmax = b; AppendTo[seq, n]], {n, 1, 10^5}]; seq
9 is a term since the binary representations of its divisors, 1, 3 and 9, are palindromic: 1, 11 and 1001, i.e., the binary reversals of themselves. 95 is a term since the binary representations of its divisors, 1, 5, 19 and 95, are 1, 101, 10011 and 1011111, and their binary reversals, 1, 101, 11001, 1111101, or 1, 5, 25 and 125 in decimal representation, are the divisors of 125, which is the binary reversal of 95.
Select[Range[1, 2000, 2], CompositeQ[#] && (Divisors @ IntegerReverse[#, 2]) == IntegerReverse[Divisors[#], 2] &]
95 is a term since the binary representations of its divisors, 1, 5, 19, and 95, are 1, 101, 10011 and 1011111, and their binary reversals, 1, 101, 11001 and 1111101, or 1, 5, 25 and 125 in decimal representation, are the divisors of 125, which is the binary reversal of 95, and 19 and 95 are not binary palindromes.
binPalQ[n_] := PalindromeQ @ IntegerDigits[n, 2]; Select[Range[1, 4000, 2], CompositeQ[#] && (Divisors @ IntegerReverse[#, 2]) == IntegerReverse[(d = Divisors[#]), 2] && !AllTrue[Rest[d], binPalQ] &]
4847 is a term since the binary representations of its divisors, 1, 37, 131 and 4847, are 1, 100101, 10000011 and 1001011101111, and their binary reversals, 1, 101001, 11000001 and 1111011101001, or 1, 41, 193 and 7913 in decimal representation, are the divisors of 7913, and none of the divisors of 4847 except 1 are binary palindromes.
binPalQ[n_] := PalindromeQ @ IntegerDigits[n, 2]; Select[Range[1, 5*10^5, 2], CompositeQ[#] && (Divisors@IntegerReverse[#, 2]) == IntegerReverse[(d = Divisors[#]), 2] && !AnyTrue[Rest[d], binPalQ] &]
32 is a term since all the divisors of 32, i.e., 1, 2, 4, 8, 16 and 32, are alternating numbers
q[n_] := AllTrue[Divisors[n], !MemberQ[Differences[Mod[IntegerDigits[#], 2]], 0] &]; Select[Range[300], q] (* Amiram Eldar, Jul 12 2022 *)
isokd(n, d=digits(n))=for(i=2, #d, if((d[i]-d[i-1])%2==0, return(0))); 1; \\ A030141 isok(m) = sumdiv(m, d, isokd(d)) == numdiv(m); \\ Michel Marcus, Jul 12 2022
from sympy import divisors
def p(d): return 0 if d in "02468" else 1
def c(n):
if n < 10: return True
s = str(n)
return all(p(s[i]) != p(s[i+1]) for i in range(len(s)-1))
def ok(n):
return c(n) and all(c(d) for d in divisors(n, generator=True))
print([k for k in range(1, 200) if ok(k)]) # Michael S. Branicky, Jul 12 2022
21 is a term since all the divisors of 21, {1, 3, 7, 21}, are palindromes in negabinary representation: {1, 111, 11011, 10101}.
negabin[n_] := negabin[n] = If[n==0, 0, negabin[Quotient[n-1, -2]]*10 + Mod[n, 2]]; nbPalinQ[n_] := PalindromeQ @ negabin[n]; negaBinAllDivPalQ[n_] := nbPalinQ[n] && AllTrue[Most @ Divisors[n], nbPalinQ]; Select[Range[5000], negaBinAllDivPalQ]
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