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A355303 a(n) is the smallest integer that has n normal undulating divisors.

%I A355303 #27 Jul 11 2022 16:10:04
%S A355303 1,2,4,6,16,12,64,24,36,48,126,60,320,144,168,120,252,180,560,240,630,
%T A355303 420,780,360,1890,960,1920,720,1560,1080,1260,1440,1680,4368,2160,
%U A355303 3240,3120,3360,4320,2520,6300,6120,8640,6240,13104,5040,12480,9360,12240,7560
%N A355303 a(n) is the smallest integer that has n normal undulating divisors.
%C A355303 Normal undulating numbers are in A355301.
%C A355303 The first ten terms are the same as A005179, then A005179(11) = 1024 while a(11) = 126 (see example); also, a(n) = A005179(n) for n = 12, 16, 18, 20, 24 (up to n = 50).
%H A355303 David A. Corneth, <a href="/A355303/b355303.txt">Table of n, a(n) for n = 1..402</a>
%e A355303 16 has 5 divisors: {1, 2, 4, 8, 16}, all of which are normal undulating integers; no positive integer smaller than 16 has five normal undulating divisors, hence a(5) = 16.
%e A355303 126 has 12 divisors: {1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126}; only 126 is not normal undulating; no positive integer smaller than 126 has eleven normal undulating divisors, hence a(11) = 126.
%t A355303 nuQ[n_] := AllTrue[(s = Sign[Differences[IntegerDigits[n]]]), # != 0 &] && AllTrue[Differences[s], # != 0 &]; f[n_] := DivisorSum[n, 1 &, nuQ[#] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n]; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[50, 10^5] (* _Amiram Eldar_, Jun 29 2022 *)
%o A355303 (PARI) isok(m) = if (m<10, return(1)); my(d=digits(m), dd = vector(#d-1, k, sign(d[k+1]-d[k]))); if (#select(x->(x==0), dd), return(0)); my(pdd = vector(#dd-1, k, dd[k+1]*dd[k])); #select(x->(x>0), pdd) == 0; \\ A355301
%o A355303 a(n) = my(k=1); while (sumdiv(k, d, isok(d)) != n, k++); k; \\ _Michel Marcus_, Jun 30 2022
%Y A355303 Cf. A005179, A355301, A355302, A355304.
%K A355303 nonn,base
%O A355303 1,2
%A A355303 _Bernard Schott_, Jun 29 2022
%E A355303 Terms a(11) and beyond from _Amiram Eldar_, Jun 29 2022