OFFSET
1,2
COMMENTS
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).
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.
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.
See A340506 for another way to look at this data. - N. J. A. Sloane, Jan 23 2021
FORMULA
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.
a(p) = 2^ceiling((p-1)*(log_2(3)) - 1) * 3^(p-1) for primes p.
EXAMPLE
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).
MATHEMATICA
(* function a2[ ] is defined in A249223 *)
smallQ[n_] := Module[{x=2^IntegerExponent[n, 2]}, n/x<2x]
ndWidth[{m_, n_}] := Select[Range[m, n], smallQ]
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]
(* returns (argument, result) data pairs since sequence is non-monotonic *)
Sort[a250071[ndWidth[{1, 100000000}]]] (* computed in steps *)
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Hartmut F. W. Hoft, Nov 11 2014
STATUS
approved