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

Offset: 1

Author

Bernard Schott, Sep 24 2022

Keywords

• nonn
• base

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A342833 (with last digit).

Cf. A206287, A357299, A357300.

Programs

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 *)

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

Extensions

More terms from Michel Marcus, Sep 24 2022