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A355592 Positions of records in A357299: integers m such that the number of divisors whose first digit equals the first digit of m sets a new record.

%I A355592 #29 Sep 26 2022 17:47:48
%S A355592 1,10,100,108,120,180,1008,1260,1680,10010,10080,15120,100320,100800,
%T A355592 110880,166320,196560,1003200,1004640,1005480,1028160,1053360,1081080,
%U A355592 1441440,1884960,10024560,10090080,10533600,10810800,12252240,17297280,100069200,100124640,100212840,100245600
%N A355592 Positions of records in A357299: integers m such that the number of divisors whose first digit equals the first digit of m sets a new record.
%C A355592 Observation: all terms start with the digit 1.
%C A355592 The corresponding records are: 1, 2, 3, 4, 5, 6, 10, 11, 12, ...
%C A355592 For even terms k we have A000005(k) >= 2*A357299(k). For 3 <= n <= 101, A000005(k) >= 3*A357299(k). - _David A. Corneth_, Sep 26 2022
%H A355592 David A. Corneth, <a href="/A355592/b355592.txt">Table of n, a(n) for n = 1..101</a>
%e A355592 1008 is a term because A357299(1008) = 10, the ten corresponding divisors are {1, 12, 14, 16, 18, 112, 126, 144, 168, 1008} and 10 is larger than any earlier value in A357299.
%t A355592 f[n_] := IntegerDigits[n][[1]]; s[n_] := Module[{fn = f[n]}, DivisorSum[n, 1 &, f[#] == fn &]]; seq = {}; sm = 0; Do[If[(sn = s[n]) > sm, sm = sn; AppendTo[seq, n]], {n, 1, 200000}]; seq (* _Amiram Eldar_, Sep 24 2022 *)
%o A355592 (PARI) f(n) = my(fd=digits(n)[1]); sumdiv(n, d, digits(d)[1] == fd); \\ A357299
%o A355592 lista(nn) = my(r=0, x, list=List()); for (n=1, nn, if ((x=f(n)) > r, listput(list, n); r = x);); Vec(list); \\ _Michel Marcus_, Sep 24 2022
%o A355592 (PARI) upto(n) = { r = -1; res = List(); forfactored(i = 1, n, if(numdiv(i[2]) >= r, d = divisors(i[2]); t = i[1]\10^logint(i[1], 10); c = sum(j = 1, #d, d[j]\10^logint(d[j], 10) == t); if(c > r, r = c; listput(res, i[1]); ) ) ); res } \\ _David A. Corneth_, Sep 24 2022
%o A355592 (Python)
%o A355592 from sympy import divisors
%o A355592 from itertools import count, islice
%o A355592 def b(n): f = str(n)[0]; return sum(1 for d in divisors(n) if str(d)[0]==f)
%o A355592 def agen(): # generator of terms
%o A355592     record = -1
%o A355592     for m in count(1):
%o A355592         v = b(m)
%o A355592         if v > record: yield m; record = v
%o A355592 print(list(islice(agen(), 17))) # _Michael S. Branicky_, Sep 24 2022
%Y A355592 Cf. A342833 (with last digit).
%Y A355592 Cf. A206287, A357299, A357300.
%K A355592 nonn,base
%O A355592 1,2
%A A355592 _Bernard Schott_, Sep 24 2022
%E A355592 More terms from _Michel Marcus_, Sep 24 2022