OFFSET

1,1

COMMENTS

A number k with two parts in its symmetric representation of sigma(k) [ssrs(k) = 2] has the form k = q*p with q in A174973, p prime and 2*q < p. This implies that 2*q <= row(k) < p and the first 0 in the k-th row of A249223 (having row(k) = floor((sqrt(8*k+1)-1)/2) entries) occurs at position 2*q so that 2*q-1 is the number of legs in each of the two parts. Therefore, the numbers 2*q-1 with q in A174973 are the only possible leg counts when ssrs(k) = 2, and for given q in A174973 and smallest prime p(q) > 2*q the number k = q*p(q) is the smallest with a leg count of 2*q-1. Consequently, each number q*p in the column of the irregular triangle A239929 labeled by q in A174793 with p prime satisfies ssrs(q*p) = 2*q-1.

a(1) = 3 is the only odd number since 1 is the only odd number in A174973.

Every number n = 2^m * p, m >= 0, 2^(m+1) < p and p prime, in this sequence is the sum of 2^(m+1) consecutive positive integers which includes every number in A246956.

EXAMPLE

a(3)=44 is the smallest number whose symmetric representation has 2 parts and 7 legs in each part.

a(4)=78 is the smallest number whose symmetric representation has 2 parts and 11 legs in each part.

No number k whose symmetric representation of sigma(k) has 2 parts can have 21 legs in its parts since there is no q in A174973 such that 2*q - 1 = 21.

MATHEMATICA

a174973Q[n_] := Module[{d=Divisors[n]}, Select[Rest[d] - 2*Most[d], #>0&]=={}]

a279105[n_] := Map[# * NextPrime[2*#]&, Select[Range[n], a174973Q]]

a279105[150] (* sequence data *)

CROSSREFS

KEYWORD

nonn

AUTHOR

Hartmut F. W. Hoft, Dec 06 2016

STATUS

approved