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A239929
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Numbers n with the property that the symmetric representation of sigma(n) has two parts.
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24
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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
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OFFSET
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1,1
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COMMENTS
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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
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)
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) A161424 (except 16, 20, 24, 28 and 32)
(c) A001749 (except 8, 12, 20 and 28)
(b) A162528 (except 64, 72, 80, 88, 96, 104, 112 and 128)
(c) sequence not in OEIS
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LINKS
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FORMULA
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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):
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EXAMPLE
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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)
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MAPLE
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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:
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
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;
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MATHEMATICA
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(* 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 *)
(* 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]]]]
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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