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Numbers n with the property that the symmetric representation of sigma(n) has two parts.
27

%I #71 Oct 04 2018 18:18:41

%S 3,5,7,10,11,13,14,17,19,22,23,26,29,31,34,37,38,41,43,44,46,47,52,53,

%T 58,59,61,62,67,68,71,73,74,76,78,79,82,83,86,89,92,94,97,101,102,103,

%U 106,107,109,113,114,116,118,122,124,127,131,134,136,137,138

%N Numbers n with the property that the symmetric representation of sigma(n) has two parts.

%C All odd primes are in the sequence because the parts of the symmetric representation of sigma(prime(i)) are [m, m], where m = (1 + prime(i))/2, for i >= 2.

%C There are no odd composite numbers in this sequence.

%C First differs from A173708 at a(13).

%C Since sigma(p*q) >= 1 + p + q + p*q for odd p and q, the symmetric representation of sigma(p*q) has more parts than the two extremal ones of size (p*q + 1)/2; therefore, the above comments are true. - _Hartmut F. W. Hoft_, Jul 16 2014

%C From _Hartmut F. W. Hoft_, Sep 16 2015: (Start)

%C The following two statements are equivalent:

%C (1) The symmetric representation of sigma(n) has two parts, and

%C (2) n = q * p where q is in A174973, p is prime, and 2 * q < p.

%C For a proof see the link and also the link in A071561.

%C This characterization allows for much faster computation of numbers in the sequence - function a239929F[] in the Mathematica section - than computations based on Dyck paths. The function a239929Stalk[] gives rise to the associated irregular triangle whose columns are indexed by A174973 and whose rows are indexed by A065091, the odd primes. (End)

%C From _Hartmut F. W. Hoft_, Dec 06 2016: (Start)

%C For the respective columns of the irregular triangle with fixed m: k = 2^m * p, m >= 1, 2^(m+1) < p and p prime:

%C (a) each number k is representable as the sum of 2^(m+1) but no fewer consecutive positive integers [since 2^(m+1) < p].

%C (b) each number k has 2^m as largest divisor <= sqrt(k) [since 2^m < sqrt(k) < p].

%C (c) each number k is of the form 2^m * p with p prime [by definition].

%C m = 1: (a) A100484 even semiprimes (except 4 and 6)

%C (b) A161344 (except 4, 6 and 8)

%C (c) A001747 (except 2, 4 and 6)

%C m = 2: (a) A270298

%C (b) A161424 (except 16, 20, 24, 28 and 32)

%C (c) A001749 (except 8, 12, 20 and 28)

%C m = 3: (a) A270301

%C (b) A162528 (except 64, 72, 80, 88, 96, 104, 112 and 128)

%C (c) sequence not in OEIS

%C b(i,j) = A174973(j) * {1,5) mod 6 * A174973(j), for all i,j >= 1; see A091999 for j=2. (End)

%H Hartmut F. W. Hoft, <a href="/A239929/a239929.pdf">Proof of Characterization Theorem</a>

%F Entries b(i, j) in the irregular triangle with rows indexed by i>=1 and columns indexed by j>=1 (alternate indexing of the example):

%F b(i,j) = A000040(i+1) * A174973(j) where A000040(i+1) > 2 * A174973(j). - _Hartmut F. W. Hoft_, Dec 06 2016

%e From _Hartmut F. W. Hoft_, Sep 16 2015: (Start)

%e a(23) = 52 = 2^2 * 13 = q * p with q = 4 in A174973 and 8 < 13 = p.

%e a(59) = 136 = 2^3 * 17 = q * p with q = 8 in A174973 and 16 < 17 = p.

%e The first six columns of the irregular triangle through prime 37:

%e 1 2 4 6 8 12 ...

%e -------------------------------

%e 3

%e 5 10

%e 7 14

%e 11 22 44

%e 13 26 52 78

%e 17 34 68 102 136

%e 19 38 76 114 152

%e 23 46 92 138 184

%e 29 58 116 174 232 348

%e 31 62 124 186 248 372

%e 37 74 148 222 296 444

%e ...

%e (End)

%p isA174973 := proc(n)

%p option remember;

%p local k,dvs;

%p dvs := sort(convert(numtheory[divisors](n),list)) ;

%p for k from 2 to nops(dvs) do

%p if op(k,dvs) > 2*op(k-1,dvs) then

%p return false;

%p end if;

%p end do:

%p true ;

%p end proc:

%p A174973 := proc(n)

%p if n = 1 then

%p 1;

%p else

%p for a from procname(n-1)+1 do

%p if isA174973(a) then

%p return a;

%p end if;

%p end do:

%p end if;

%p end proc:

%p isA239929 := proc(n)

%p local i,p,j,a73;

%p for i from 1 do

%p p := ithprime(i+1) ;

%p if p > n then

%p return false;

%p end if;

%p for j from 1 do

%p a73 := A174973(j) ;

%p if a73 > n then

%p break;

%p end if;

%p if p > 2*a73 and n = p*a73 then

%p return true;

%p end if;

%p end do:

%p end do:

%p end proc:

%p for n from 1 to 200 do

%p if isA239929(n) then

%p printf("%d,",n) ;

%p end if;

%p end do: # _R. J. Mathar_, Oct 04 2018

%t (* sequence of numbers k for m <= k <= n having exactly two parts *)

%t (* Function a237270[] is defined in A237270 *)

%t a239929[m_, n_]:=Select[Range[m, n], Length[a237270[#]]==2&]

%t a239929[1, 260] (* data *)

%t (* _Hartmut F. W. Hoft_, Jul 07 2014 *)

%t (* test for membership in A174973 *)

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

%t a174973[n_]:=Select[Range[n], a174973Q]

%t (* compute numbers satisfying the condition *)

%t a239929Stalk[start_, bound_]:=Module[{p=NextPrime[2 start], list={}}, While[start p<=bound, AppendTo[list, start p]; p=NextPrime[p]]; list]

%t a239929F[n_]:=Sort[Flatten[Map[a239929Stalk[#, n]&, a174973[n]]]]

%t a239929F[138] (* data *)(* _Hartmut F. W. Hoft_, Sep 16 2015 *)

%Y Column 2 of A240062.

%Y Cf. A000203, A006254, A065091, A071561, A174973, A196020, A236104, A235791, A237591, A237593, A237270, A237271, A238443, A239660, A239663, A239665, A239931-A239934, A244050, A245062, A262626.

%Y Cf. A000040, A001747, A001749, A091999, A100484, A161344, A161424, A162528, A270298, A270301. - _Hartmut F. W. Hoft_, Dec 06 2016

%K nonn

%O 1,1

%A _Omar E. Pol_, Apr 06 2014

%E Extended beyond a(56) by _Michel Marcus_, Apr 07 2014