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 A239929 Numbers n with the property that the symmetric representation of sigma(n) has two parts. 20
 3, 5, 7, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 41, 43, 44, 46, 47, 52, 53, 58, 59, 61, 62, 67, 68, 71, 73, 74, 76, 78, 79, 82, 83, 86, 89, 92, 94, 97, 101, 102, 103, 106, 107, 109, 113, 114, 116, 118, 122, 124, 127, 131, 134, 136, 137, 138 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. There are no odd composite numbers in this sequence. First differs from A173708 at a(13). 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 From Hartmut F. W. Hoft, Sep 16 2015: (Start) The following two statements are equivalent:   (1) The symmetric representation of sigma(n) has two parts, and   (2) n = q * p where q is in A174973, p is prime, and 2 * q < p. For a proof see the link and also the link in A071561. 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) From Hartmut F. W. Hoft, Dec 06 2016: (Start) For the respective columns of the irregular triangle with fixed m: k = 2^m * p, m >= 1, 2^(m+1) < p and p prime: (a) each number k is representable as the sum of 2^(m+1) but no fewer consecutive positive integers [since 2^(m+1) < p]. (b) each number k has 2^m as largest divisor <= sqrt(k) [since 2^m < sqrt(k) < p]. (c) each number k is of the form 2^m * p with p prime [by definition]. m = 1: (a) A100484 even semiprimes (except 4 and 6)        (b) A161344 (except 4, 6 and 8)        (c) A001747 (except 2, 4 and 6) m = 2: (a) A270298        (b) A161424 (except 16, 20, 24, 28 and 32)        (c) A001749 (except 8, 12, 20 and 28) m = 3: (a) A270301        (b) A162528 (except 64, 72, 80, 88, 96, 104, 112 and 128)        (c) sequence not in OEIS b(i,j) = A174973(j) * {1,5) mod 6 * A174973(j), for all i,j >= 1; see A091999 for j=2. (End) LINKS Hartmut F. W. Hoft, Proof of Characterization Theorem FORMULA 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): b(i,j) = A000040(i+1) * A174973(j) where A000040(i+1) > 2 * A174973(j). - Hartmut F. W. Hoft, Dec 06 2016 EXAMPLE From Hartmut F. W. Hoft, Sep 16 2015: (Start) a(23) = 52 = 2^2 * 13 = q * p with q = 4 in A174973 and 8 < 13 = p. a(59) = 136 = 2^3 * 17 = q * p with q = 8 in A174973 and 16 < 17 = p. The first six columns of the irregular triangle through prime 37:    1    2    4    6    8   12 ...   -------------------------------    3    5   10    7   14   11   22   44   13   26   52   78   17   34   68  102  136   19   38   76  114  152   23   46   92  138  184   29   58  116  174  232  348   31   62  124  186  248  372   37   74  148  222  296  444   ... (End) MAPLE isA174973 := proc(n)     option remember;     local k, dvs;     dvs := sort(convert(numtheory[divisors](n), list)) ;     for k from 2 to nops(dvs) do         if op(k, dvs) > 2*op(k-1, dvs) then             return false;         end if;     end do:     true ; end proc: A174973 := proc(n)     if n = 1 then         1;     else         for a from procname(n-1)+1 do             if isA174973(a) then                 return a;             end if;         end do:     end if; end proc: isA239929 := proc(n)     local i, p, j, a73;     for i from 1 do         p := ithprime(i+1) ;         if p > n then             return false;         end if;         for j from 1 do             a73 := A174973(j) ;             if a73 > n then                 break;             end if;             if p > 2*a73 and n = p*a73 then                 return true;             end if;         end do:     end do: end proc: for n from 1 to 200 do     if isA239929(n) then         printf("%d, ", n) ;     end if; end do: # R. J. Mathar, Oct 04 2018 MATHEMATICA (* sequence of numbers k for m <= k <= n having exactly two parts *) (* Function a237270[] is defined in A237270 *) a239929[m_, n_]:=Select[Range[m, n], Length[a237270[#]]==2&] a239929[1, 260] (* data *) (* Hartmut F. W. Hoft, Jul 07 2014 *) (* test for membership in A174973 *) a174973Q[n_]:=Module[{d=Divisors[n]}, Select[Rest[d] - 2 Most[d], #>0&]=={}] a174973[n_]:=Select[Range[n], a174973Q] (* compute numbers satisfying the condition *) a239929Stalk[start_, bound_]:=Module[{p=NextPrime[2 start], list={}}, While[start p<=bound, AppendTo[list, start p]; p=NextPrime[p]]; list] a239929F[n_]:=Sort[Flatten[Map[a239929Stalk[#, n]&, a174973[n]]]] a239929F (* data *)(* Hartmut F. W. Hoft, Sep 16 2015 *) CROSSREFS Column 2 of A240062. Cf. A000203, A006254, A065091, A071561, A174973, A196020, A236104, A235791, A237591, A237593, A237270, A237271, A238443, A239660, A239663, A239665, A239931-A239934, A244050, A245062, A262626. Cf. A000040, A001747, A001749, A091999, A100484, A161344, A161424, A162528, A270298, A270301. - Hartmut F. W. Hoft, Dec 06 2016 Sequence in context: A241561 A241008 A173708 * A246955 A167907 A306639 Adjacent sequences:  A239926 A239927 A239928 * A239930 A239931 A239932 KEYWORD nonn AUTHOR Omar E. Pol, Apr 06 2014 EXTENSIONS Extended beyond a(56) by Michel Marcus, Apr 07 2014 STATUS approved

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Last modified March 30 09:49 EDT 2020. Contains 333125 sequences. (Running on oeis4.)