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.
132 has 12 divisors: {1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132} of which 10 are repdigits: {1, 2, 3, 4, 6, 11, 22, 33, 44, 66}. No positive integer smaller than 132 has as many as ten repdigit divisors; hence 132 is a term.
repQ[n_] := Length @ Union @ IntegerDigits[n] == 1; s[n_] := DivisorSum[n, 1 &, repQ[#] &]; smax = 0; seq = {}; Do[s1 = s[n]; If[s1 > smax, smax = s1; AppendTo[seq, n]], {n, 1, 10^5}]; seq (* Amiram Eldar, Jan 11 2021 *)
isrd(n) = {1 == #Set(digits(n))}; \\ A010785
f(n) = sumdiv(n, d, isrd(d));
lista(nn) = {my(m = 0); for (n=1, nn, my(x = f(n)); if (x > m, print1(n, ", "); m = x););} \\ Michel Marcus, Jan 11 2021
a(7) = 56 is in the sequence because A227872(56) = 8 is larger than any earlier value in A227872.
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
60 is in the sequence because A356018(60) = 9 is larger than any earlier value in A356018.
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
Terms include: 4 (3 divisors); 6 (4 divisors); 44 (6 divisors); 66 (8 divisors); 252 (18 divisors).
pal=Union@Flatten[Table[r=IntegerDigits@n;FromDigits/@(Join[r,#]&/@{Reverse@r,Rest@Reverse@r}),{n,10^5}]];m=0;lst={};Do[s=DivisorSigma[0,k];If[s>m,AppendTo[lst,k];m=s],{k,pal}];lst (* Giorgos Kalogeropoulos, Dec 08 2021 *)
\\ See PARI link. David A. Corneth, May 18 2021
1260 has 36 divisors of which 12 are triangular numbers {1, 3, 6, 10, 15, 21, 28, 36, 45, 105, 210, 630}. No positive integer smaller than 1260 has as many as twelve triangular divisors; hence 1260 is a term.
max=0;Do[If[(d=Length@Select[Divisors@k,IntegerQ[(Sqrt[8#+1]-1)/2]&])>max,Print@k;max=d],{k,10^10}] (* Giorgos Kalogeropoulos, Jan 13 2022 *)
lista(nn) = {my(r=0); for (n=1, nn, my(m = sumdiv(n, d, ispolygonal(d,3))); if (m>r, r=m; print1(n", ")));} \\ Michel Marcus, Jan 14 2022
a(6) = 24 is in the sequence because A355302(24) is larger than any earlier value in A355302.
nuQ[n_] := AllTrue[(s = Sign[Differences[IntegerDigits[n]]]), # != 0 &] && AllTrue[Differences[s], # != 0 &]; dm = -1; seq = {}; Do[If[(d = DivisorSum[n, 1 &, nuQ[#] &]) > dm, dm = d; AppendTo[seq, n]], {n, 1, 10^5}]; seq (* Amiram Eldar, Jun 30 2022 *)
a(5) = 45 is in the sequence because A355770(45) = 5 is larger than any earlier value in A355770.
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
The 8 divisors of 24 are all Niven numbers, and also, 24 is the smallest integer that has at least 8 divisors that are Niven numbers, hence 24 is a term.
s[n_] := DivisorSum[n, 1 &, Divisible[#, Plus @@ IntegerDigits[#]] &]; smax = 0; seq = {}; Do[s1 = s[n]; If[s1 > smax, smax = s1; AppendTo[seq, n]], {n, 1, 10^6}]; seq (* Amiram Eldar, Jan 14 2021 *)
f(n) = sumdiv(n, d, !(d % sumdigits(d))); \\ A332268 lista(nn) = {my(m=0); for (n=1, nn, my(x = f(n)); if (x > m, m = x; print1(n, ", ")););} \\ Michel Marcus, Jan 14 2021
40 has 8 divisors: {1, 2, 4, 5, 8, 10, 20, 40} of which 4 are Brazilian: {8, 10, 20, 40}. No positive integer smaller than 40 has as many as four Brazilian divisors; hence 40 is a term.
brazQ[n_] := Module[{b = 2, found = False}, While[b < n - 1 && Length[Union[ IntegerDigits[n, b]]] > 1, b++]; b < n - 1]; d[n_] := DivisorSum[n, 1 &, brazQ[#] &]; dm = -1; s = {}; Do[d1 = d[n]; If[d1 > dm, dm = d1; AppendTo[s, n]], {n, 1, 1000}]; s (* Amiram Eldar, Jan 24 2021 *)
isb(n) = for(b=2, n-2, d=digits(n, b); if(vecmin(d)==vecmax(d), return(1))); \\ A125134 nbd(n) = sumdiv(n, d, isb(d)); \\ A340795 lista(nn) = {my(m=-1); for (n=1, nn, my(x = nbd(n)); if (x > m, print1(n, ", "); m = x););} \\ Michel Marcus, Jan 24 2021
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