A355592 -id:A355592 - cp's OEIS frontend

A206287 Numbers with all divisors starting with digit 1.

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

cp's OEIS Frontend

A206287 Numbers with all divisors starting with digit 1.

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