cp's OEIS Frontend

A250071 Smallest number k such that the symmetric representation of sigma(k) has maximum width n for those k whose representation has nondecreasing width up to the diagonal.

%I A250071 #33 Dec 12 2024 23:25:27
%S A250071 1,6,72,120,5184,1440,373248,6720,28800,103680,1934917632,80640,
%T A250071 278628139008,7464960,2073600,483840,1444408272617472,1612800,
%U A250071 103997395628457984,5806080,298598400,77396705280,539122498937926189056,7096320,1658880000,5572562780160,90316800,418037760,402452788967166148425547776,116121600
%N A250071 Smallest number k such that the symmetric representation of sigma(k) has maximum width n for those k whose representation has nondecreasing width up to the diagonal.
%C A250071 The symmetric representation of sigma(k) has nondecreasing width to the diagonal precisely when all odd divisors counted in the k-th row of A237048 occur at odd indices. If we write k = 2^m * q with m >= 0 and q odd, this property is equivalent to q < 2^(m+1).
%C A250071 The values for a(11), a(13), a(17) and a(19) were computed directly using the formula k = 2^m * 3^(p-1) where p is one of the four primes and m the smallest exponent so that 3^(p-1) < 2^(m+1). Each of these numbers has a symmetric representation of nondecreasing width ending in a prime number width, and they are the first such numbers since the number of divisors of an odd number is a prime precisely when the number is a power of a prime.
%C A250071 The other numbers listed whose symmetric representations of sigma(k) have nondecreasing width are smaller than 7500000. The only additional numbers k <= 100000000 are a(24) = 7096320, a(27) = 90316800 and a(32) = 85155840.
%C A250071 See A340506 for another way to look at this data. - _N. J. A. Sloane_, Jan 23 2021
%F A250071 a(n) = min(2^m * q, m >= 0 & q odd & sigma_0(q) = n & q < 2^(m+1)) where sigma_0 is the number of divisors.
%F A250071 a(p) = 2^ceiling((p-1)*(log_2(3)) - 1) * 3^(p-1) for primes p.
%e A250071 a(6) = 1440 = 2^5 * 3^2 * 5 has 6 odd divisors. It is the smallest number of the form 2^m * q with m > 0, q odd and such that q < 2^(m+1).
%t A250071 (* function a2[ ] is defined in A249223 *)
%t A250071 smallQ[n_] := Module[{x=2^IntegerExponent[n,2]}, n/x<2x]
%t A250071 ndWidth[{m_,n_}] := Select[Range[m, n], smallQ]
%t A250071 a250071[x_List] := Module[{i, max, acc={{1, 1}}}, For[i=1, i<=Length[x], i++, max={Max[a2[x[[i]]]], x[[i]]}; If[!MemberQ[Transpose[acc][[1]], max[[1]]], AppendTo[acc, max]]]; acc]
%t A250071 (* returns (argument,result) data pairs since sequence is non-monotonic *)
%t A250071 Sort[a250071[ndWidth[{1,100000000}]]] (* computed in steps *)
%t A250071 (* alternate implementation using function f[ ] by _T. D. Noe_ in A162247 *)
%t A250071 sF[n_] := Min[Map[Apply[Times, Prime[Range[2, Length[#]+1]]^#]&, Map[Reverse[#-1]&, f[n]]]]
%t A250071 f1U[n_] := Module[{s=sF[n], k}, k=Floor[Log[2, s]]; 2^k s]
%t A250071 a250071[n_] := Map[f1U, Range[n]]
%t A250071 a250071[30] (* _Hartmut F. W. Hoft_, Nov 27 2024 *)
%Y A250071 Cf. A000203, A237048, A237270, A237271, A237593, A241008, A241010, A246955, A247687, A249223, A250068, A250070.
%Y A250071 Cf. also A340506.
%Y A250071 Cf. A162247.
%K A250071 nonn
%O A250071 1,2
%A A250071 _Hartmut F. W. Hoft_, Nov 11 2014
%E A250071 a(21)-a(30) from _Hartmut F. W. Hoft_, Nov 27 2024