A355592 - OEIS

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.

1, 10, 100, 108, 120, 180, 1008, 1260, 1680, 10010, 10080, 15120, 100320, 100800, 110880, 166320, 196560, 1003200, 1004640, 1005480, 1028160, 1053360, 1081080, 1441440, 1884960, 10024560, 10090080, 10533600, 10810800, 12252240, 17297280, 100069200, 100124640, 100212840, 100245600

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OFFSET

1,2

COMMENTS

Observation: all terms start with the digit 1.

The corresponding records are: 1, 2, 3, 4, 5, 6, 10, 11, 12, ...

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

LINKS

David A. Corneth, Table of n, a(n) for n = 1..101

EXAMPLE

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.

MATHEMATICA

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 *)

PROG

(PARI) f(n) = my(fd=digits(n)[1]); sumdiv(n, d, digits(d)[1] == fd); \\ A357299

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

(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

(Python)

from sympy import divisors

from itertools import count, islice

def b(n): f = str(n)[0]; return sum(1 for d in divisors(n) if str(d)[0]==f)

def agen(): # generator of terms

record = -1

for m in count(1):

v = b(m)

if v > record: yield m; record = v

print(list(islice(agen(), 17))) # Michael S. Branicky, Sep 24 2022

CROSSREFS

Cf. A342833 (with last digit).