A279105 a(n), n>1, is the smallest number k whose symmetric representation of sigma(k) has two parts and has a larger number of legs in its two parts than a(n-1); a(1)=3.

3, 10, 44, 78, 136, 348, 592, 666, 820, 1272, 1652, 1830, 2144, 2628, 3320, 3738, 4656, 5886, 6328, 7620, 8384, 9042, 10728, 13040, 14532, 15752, 16290, 18528, 21100, 21944, 24084, 25424, 28920, 32382, 32896, 35508, 39340, 42192, 46050, 48828

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.

Links

Table of n, a(n) for n=1..40.

Formula

a(n) = A174973(n) * A007918(2 * A174973(n) + 1).

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

Right border of A239929.

Supersequence of A246956 and A262259.

Cf. A007918, A174973, A237048, A237270, A237271, A237593, A246956, A249223, A279105.

Sequence in context: A032269 A179501 A041737 * A246956 A026682 A335635

Adjacent sequences: A279102 A279103 A279104 * A279106 A279107 A279108

Keyword

nonn

Author

Hartmut F. W. Hoft, Dec 06 2016

Status

approved