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

1,2

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).

Cf. A206287, A357299, A357300.

Sequence in context: A257795 A257950 A052009 * A357300 A248040 A034088

Adjacent sequences: A355589 A355590 A355591 * A355593 A355594 A355595

Keyword

nonn,base

Author

Bernard Schott, Sep 24 2022

Extensions

More terms from Michel Marcus, Sep 24 2022

Status

approved