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Search: id:A134622
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| A134622 |
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Numbers which can be written as a sum of squares of primes. |
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+0
8
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| 4, 8, 9, 12, 13, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82
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OFFSET
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1,1
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COMMENT
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Complementary to A078135. This sequence is infinite since numbers m>=24 are always sums of squares of primes. (see A078135 for the proof; also see reasoning below).
Also numbers which can be represented as a sum of squares >1.
Equivalently, numbers which can be written as a sum of squares of 2 and 3. Proof for numbers m>=24: if m=4*(k+6), k>=0, then m=(k+6)*2^2; if m=4*(k+6)+1 than m=(k+4)*2^2+3^2; if m=4*(k+6)+2 then m=(k+2)*2^2+2*3^2; if m=4*(k+6)+3 then m=k*2^2+3*3^2. Clearly, the numbers a(n)<24 can also be written as sums of squares of 2 and 3.
Explicit representation as a sum of squares of 2 and 3 for numbers m>23: m=c*2^2+d*3^2, where c:=((floor(m/4) - 2*(m mod 4))>=0 and d:=m mod 4.
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FORMULA
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a(n)=12+n for n>=12.
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EXAMPLE
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a(4)=12 because 12=3*2^2, a(5)=13 because 13=2^2+3^2,
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CROSSREFS
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Cf. A001597, A025475, A078134, A078135, A078136, A078139, A090677.
Cf. A134600, A134605, A134608, A134612, A134616, A134618, A134620.
Adjacent sequences: A134619 A134620 A134621 this_sequence A134623 A134624 A134625
Sequence in context: A118715 A104623 A078137 this_sequence A010453 A078136 A037973
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KEYWORD
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nonn
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AUTHOR
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Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Nov 11 2007
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