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
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.
Similar, but with:
A333456 (Niven numbers),
A335038 (Zuckerman numbers).
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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 *)
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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
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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
A335491
a(n) is the smallest number m with exactly n divisors whose last digit equals the last digit of m.
Original entry on oeis.org
1, 11, 40, 60, 160, 120, 640, 240, 360, 480, 8064, 600, 18144, 1920, 1440, 1200, 72576, 1800, 52416, 2400, 5760, 30720, 183456, 3600, 12960, 122880, 9000, 9600, 602784, 7200, 445536, 8400, 92160, 798336, 51840, 12600, 2159136, 576576, 368640, 16800, 2935296, 28800
Offset: 1
Of the twelve divisors of 60, four have their last digit equals to the last digit of 60: 10, 20, 30 and 60, and there is no smaller number k with four divisors whose last digit equals the last digit of k, hence a(4) = 60.
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a:=[]; for n in [1..30] do k:=1; while #[d:d in Divisors(k)|k mod 10 eq d mod 10] ne n do k:=k+1; end while; Append(~a,k); end for; a; // Marius A. Burtea, Jun 12 2020
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d[n_] := DivisorSum[n, 1 &, Mod[# - n, 10] == 0 &]; mx = 20; c = 0; n = 1; s = Table[0, {mx}]; While[c < mx, i = d[n]; If[i <= mx && s[[i]] == 0, c++; s[[i]] = n]; n++]; s (* Amiram Eldar, Jun 12 2020 *)
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f(n) = my(u=n%10); sumdiv(n, d, (d%10) == u);
a(n) = my(k=1); while(f(k) != n, k++); k; \\ Michel Marcus, Jun 12 2020
A333457
a(n) is the smallest number with exactly n divisors that are Moran numbers, or -1 if no such number exists.
Original entry on oeis.org
18, 42, 84, 126, 252, 756, 1998, 1596, 2394, 4662, 4788, 9324, 18648, 23940, 46620, 93240, 139860, 177156, 559440, 354312, 708624, 1062936, 885780, 4606056, 1771560, 3543120, 5314680, 10629360, 38974320, 23030280, 46060560, 69090840, 138181680, 506666160
Offset: 1
Of the divisors of 18 (1, 2, 3, 6, 9, 18), only 18 is a Moran number: 18 / digsum (18) = 2.
Of the divisors of 84 (1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84), only 21, 42 and 84 are Moran numbers: 21 / digsum (21) = 7, 42 / digsum (42) = 7 and 84 / digsum (84) = 7.
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a:=[]; for n in [1..20] do m:=1; while #[d:d in Divisors(m)|d mod &+Intseq(d) eq 0 and IsPrime(d div &+Intseq(d))] ne n do m:=m+1; end while; Append(~a,m); end for; a;
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numDiv[n_] := DivisorSum[n, 1 &, PrimeQ[#/Plus @@ IntegerDigits[#]] &]; a[n_] := Module[{k = 1}, While[numDiv[k] != n, k++]; k]; Array[a, 20] (* Amiram Eldar, May 11 2020 *)
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