A357299 -id:A357299 - cp's OEIS frontend

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

Bernard Schott, Sep 24 2022

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

			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.

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

Cf. A342833 (with last digit).

Cf. A206287, A357299, A357300.

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 *)

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

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

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

More terms from Michel Marcus, Sep 24 2022

A206287 Numbers with all divisors starting with digit 1.

Original entry on oeis.org

1, 11, 13, 17, 19, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1111

Offset: 1

Jaroslav Krizek, Feb 12 2012

Equivalently, integers m with all divisors starting with the same first digit of m; in fact, as 1 divides all the integers, this digit is necessarily 1; also, for these terms m: A357299(m) = A000005(m). - Bernard Schott, Sep 25 2022

			All divisors of 187 (1, 11, 17, 187) start with digit 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Disjoint union of A045707 and A206288.
Cf. A000005, A355592, A357299, A357300.
Cf. A004615 (with last digit)

filter:= proc(n) andmap(t -> floor(t/10^ilog10(t)) = 1, numtheory:-divisors(n)) end proc:
select(filter, [seq($10^d .. 2*10^d-1, d=0..3)]); # Robert Israel, Dec 25 2024

fQ[n_] := Module[{d = Divisors[n]}, Union[IntegerDigits[#][[1]] & /@ d] == {1}]; Select[Range[1111], fQ] (* T. D. Noe, Feb 13 2012 *)

A357300 a(n) is the smallest number m with exactly n divisors whose first digit equals the first digit of m.

Original entry on oeis.org

1, 10, 100, 108, 120, 180, 1040, 1020, 1170, 1008, 1260, 1680, 10010, 10530, 10200, 10260, 10560, 10800, 11340, 10920, 12600, 10080, 15840, 18480, 15120, 102060, 104400, 101640, 100320, 102600, 100980, 117600, 114660, 107100, 174240, 113400, 105840, 100800, 120120, 143640

Offset: 1

Bernard Schott, Sep 23 2022

a(m) <= a(551) = 18681062400 for m < 555. All terms with values up to 2*10^10 start with 1. Do there exist a(n) starting with any other digit? - Charles R Greathouse IV, Sep 25 2022

			Of the twelve divisors of 108, four have their first digit equals to the first digit of 108: 1, 12, 18 and 108, and there is no such smaller number, hence a(4) = 108.

Cf. A335491 (with last digit), A206287, A355592, A357299.

Similar, but with: A333456 (Niven numbers), A335038 (Zuckerman numbers).

f[n_] := IntegerDigits[n][[1]]; s[n_] := Module[{fn = f[n]}, DivisorSum[n, 1 &, f[#] == fn &]]; seq[len_, nmax_] := Module[{v = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = s[n]; If[i <= len && v[[i]] == 0, c++; v[[i]] = n]; n++]; v]; seq[40, 10^6] (* Amiram Eldar, Sep 23 2022 *)

f(n) = my(fd=digits(n)[1]); sumdiv(n, d, digits(d)[1] == fd); \\ A357299
a(n) = my(k=1); while (f(k)!=n, k++); k; \\ Michel Marcus, Sep 23 2022

v=vector(1000); v[1]=r=1; forfactored(n=2, 10^11, t=a(n[1],n[2],r); if(t>r && v[t]==0, v[t]=n[1]; print(t" "n[1]" = "n[2]); while(v[r],r++); r--)) \\ Charles R Greathouse IV, Sep 25 2022

More terms from Michel Marcus, Sep 23 2022

A356549 a(n) is the number of divisors of 10^n whose first digit is 1.

Original entry on oeis.org

1, 2, 3, 5, 8, 11, 15, 20, 25, 31, 38, 45, 52, 60, 69, 78, 88, 99, 110, 122, 135, 148, 161, 175, 190, 205, 221, 238, 255, 273, 292, 311, 330, 350, 371, 392, 414, 437, 460, 484, 509, 534, 559, 585, 612, 639, 667, 696, 725, 755, 786, 817, 848, 880, 913, 946, 980, 1015, 1050, 1086

Offset: 0

Michel Marcus, Sep 23 2022

			The divisors of 1000 with initial digit 1 are: 1, 10, 100, 125 and 1000, so a(3)=5.

Michel Marcus, Table of n, a(n) for n = 0..500

Cf. A011557, A357299.

a:= n-> add(`if`((""||d)[1]="1", 1, 0), d=numtheory[divisors](10^n)):
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 23 2022

a[n_] := DivisorSum[10^n, 1 &, IntegerDigits[#][[1]] == 1 &]; Array[a, 60, 0] (* Amiram Eldar, Sep 23 2022 *)

a(n) = sumdiv(10^n, d, digits(d)[1] == 1);

from sympy import divisors
def a(n): return sum(1 for d in divisors(10**n, generator=True) if str(d)[0]=="1")
print([a(n) for n in range(60)]) # Michael S. Branicky, Sep 23 2022

def A356549(n): return n+1+sum(n-m+1 for m in range(1,n+2) for d in (2,5) if str(d**m).startswith('1')) # Chai Wah Wu, Sep 23 2022

a(n) = A357299(A011557(n)).

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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.

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A206287 Numbers with all divisors starting with digit 1.

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A357300 a(n) is the smallest number m with exactly n divisors whose first digit equals the first digit of m.

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A356549 a(n) is the number of divisors of 10^n whose first digit is 1.

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