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A283530
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The number of reduced phi-partitions of n.
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4
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0, 0, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 2, 3, 3, 1, 1, 2, 3, 3, 3, 1, 0, 1, 5, 4, 4, 3, 2, 1, 5, 4, 4, 1, 1, 1, 6, 6, 7, 1, 3, 2, 6, 5, 8, 1, 3, 4, 8, 6, 10, 1, 1, 1, 11, 9, 12, 5, 2, 1, 12, 8, 5, 1, 5, 1, 14, 13, 14, 5, 3, 1, 13, 9, 16, 1, 1
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OFFSET
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1,8
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COMMENTS
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The reduced phi-partitions of n are partitions n= a_1 +a_2 +a_3 +... +a_k into at least 2 parts such that each part is simple (i.e. each part in A002110, as required in A283529) and such that in addition phi(n) = sum_i phi(a_i), as required in A283528. phi(.) = A000010(.) is Euler's totient.
Numbers n where a(n)=1 are called semisimple. 3, 4, 5, 7, 9, 10, 11, 12, 13, 17, 18, 19, 23, 24,... are semisimple (see A283320). In this list of semisimple numbers there are no odd numbers besides 9 and the odd primes.
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LINKS
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FORMULA
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EXAMPLE
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a(15)=2 counts 1+2+2+2+2+2+2= 1+1+1+2+2+2+6.
a(16)=3 counts 2+2+2+2+2+2+2+2 = 1+1+2+2+2+2+6 = 1+1+1+1+6+6.
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MAPLE
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isA002110 := proc(n)
member(n, [1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, 1922760350154212639070]) ;
end proc:
local a, k, issimp, p ;
a := 0 ;
for k in combinat[partition](n) do
issimp := true ;
for p in k do
if not isA002110(p) then
issimp := false;
break;
end if;
end do:
if issimp and nops(k) > 1 then
phip := add(numtheory[phi](p), p=k) ;
if phip = numtheory[phi](n) then
a := a+1 ;
end if;
end if;
end do:
a ;
end proc:
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MATHEMATICA
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v={1, 2, 6, 30, 210}; e=10^9 v + EulerPhi@v; a[n_] := Length@ IntegerPartitions[ 10^9 n + EulerPhi[n], {2, Infinity}, e]; Array[a, 100] (* suitable for n <= 1000, Giovanni Resta, Mar 10 2017 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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