A357842 a(n) is the smallest number k for which k and the arithmetic derivative k' (A003415) have exactly n triangular divisors (A000217).
2, 27, 18, 72, 612, 1764, 756, 8100, 27000, 97200, 66528, 175500, 93600, 280800, 1731600, 661500, 680400, 3704400, 34177500, 11107800, 16581600, 20065500, 108486000, 102910500, 108353700, 181912500, 314874000, 462672000, 4408236000, 229975200, 2297786400, 672348600, 925041600, 1344697200, 158230800
Offset: 1
Keywords
Examples
2 has only the divisor 1 = A000217(1) and 2' = 1 = A000217(1), so a(1) = 2. 27 and 27' = 27 have the divisors 1 = A000217(1), 3 = A000217(2) triangular numbers, so a(2) = 27.
Programs
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Magma
tr:=func
; a:=[]; for n in [1..30] do k:=2 ; while tr(k) ne n or tr(Floor(f(k))) ne n do k:=k+1; end while; Append(~a,k); end for; a; -
Mathematica
d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); tridiv[n_] := DivisorSum[n, 1 &, IntegerQ[Sqrt[8*# + 1]] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 2, i}, While[c < len && n < nmax, i = tridiv[n]; If[i <= len && s[[i]] == 0 && tridiv[d[n]] == i, c++; s[[i]] = n]; n++]; s]; seq[10, 10^6] (* Amiram Eldar, Oct 21 2022 *) -
PARI
f(n) = sumdiv(n, d, ispolygonal(d, 3)); \\ A007862 ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415 a(n) = my(k=2); while((f(k)!=n) || (f(ad(k))!=n), k++); k; \\ Michel Marcus, Oct 23 2022