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A185974
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Partitions in Abramowitz-Stegun order A036036 mapped one-to-one to positive integers.
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2
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1, 2, 3, 4, 5, 6, 8, 7, 10, 9, 12, 16, 11, 14, 15, 20, 18, 24, 32, 13, 22, 21, 25, 28, 30, 27, 40, 36, 48, 64, 17, 26, 33, 35, 44, 42, 50, 45, 56, 60, 54, 80, 72, 96, 128, 19, 34, 39, 55, 49, 52, 66, 70, 63, 75, 88, 84, 100, 90, 81, 112, 120, 108, 160, 144, 192, 256, 23, 38, 51, 65, 77, 68, 78, 110, 98, 99, 105, 125, 104, 132, 140, 126, 150, 135, 176, 168, 200, 180, 162, 224, 240, 216, 320, 288, 384, 512, 29, 46, 57, 85, 91, 121, 76, 102, 130, 154, 117, 165, 147, 175, 136, 156, 220, 196, 198, 210, 250, 189, 225, 208, 264, 280, 252, 300, 270, 243, 352, 336, 400, 360, 324, 448, 480, 432, 640, 576, 768, 1024
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OFFSET
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0,2
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COMMENTS
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This mapping of the set of all partitions of N>=1 to {2,3,...} (set of natural numbers without 1) is one to one (bijective). The empty partition for N=0 maps to 1.
A129129 seems to be analogous, except that the partition ordering A080577 is used. This ordering does, however, not care about the number of parts: e.g., 1^2 ,4 = 4,1^2 comes before 3^2, so a(23)=28 and a(22)=25 are interchanged.
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LINKS
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Table of n, a(n) for n=0..138.
M. Abramowitz and I. A. Stegun, eds.,
Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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FORMULA
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a(n)=product(p(j)^e(j),j=1..N(n)), with p(j):=A000040(j) (j-th prime), and the exponent e(j)>=0 of the part j in the n-th partition written in Abramowitz-Stegun (A-St) order, indicated in A036036. Note that j^0 is not 1 but has to be omitted in the partition. N(n) is the index (argument)of the smallest A026905-number greater or equal to n (the index of the A026905-ceiling of n).
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EXAMPLE
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a(22)=25 because the 22nd partition in A-St order is the 2-part partition 3^2 = 3,3 with N=6 because A026905(5)=18 and A026905(6)=29, so ceiling(A026905,22)=29.
a(23)=28 relates to the partition 1^2 4 = 4 1^2 with three parts, also belonging to N=6.
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CROSSREFS
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Sequence in context: A215501 A117333 A078840 * A129129 A114622 A125624
Adjacent sequences: A185971 A185972 A185973 * A185975 A185976 A185977
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang, Feb 10 2011
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STATUS
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approved
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