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A118851
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Product of parts in n-th partition in Abramowitz and Stegun order.
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7
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1, 1, 2, 1, 3, 2, 1, 4, 3, 4, 2, 1, 5, 4, 6, 3, 4, 2, 1, 6, 5, 8, 9, 4, 6, 8, 3, 4, 2, 1, 7, 6, 10, 12, 5, 8, 9, 12, 4, 6, 8, 3, 4, 2, 1, 8, 7, 12, 15, 16, 6, 10, 12, 16, 18, 5, 8, 9, 12, 16, 4, 6, 8, 3, 4, 2, 1, 9, 8, 14, 18, 20, 7, 12, 15, 16, 20, 24, 27, 6, 10, 12, 16, 18, 24, 5, 8, 9, 12, 16, 4
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OFFSET
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0,3
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COMMENTS
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Let Theta(n) denote the set of norm values corresponding to all the partitions of n. The following results hold regarding this set: (i) Theta(n) is a subset of Theta(n+1); (ii) A prime p will appear as a norm only for partitions of n>=p; (iii) There exists a prime p not in Theta(n) for all n>=6; (iv) Let h(k) be the prime floor function which gives the greatest prime less than or equal to the k, then the prime p=h(n+1) does not belong to Theta(n); and (v) The primes not in the set Theta(n) are A000720(A000792(n)) - A000720(n). - Abhimanyu Kumar, Nov 25 2020
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REFERENCES
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Abramowitz and Stegun, Handbook (1964) page 831.
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LINKS
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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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EXAMPLE
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a(9) = 4 because the 9th partition is [2,2] and 2*2 = 4.
Table T(n,k) starts:
1;
1;
2, 1;
3, 2, 1;
4, 3, 4, 2, 1;
5, 4, 6, 3, 4, 2, 1;
6, 5, 8, 9, 4, 6, 8, 3, 4, 2, 1;
7, 6, 10, 12, 5, 8, 9, 12, 4, 6, 8, 3, 4, 2, 1;
8, 7, 12, 15, 16, 6, 10, 12, 16, 18, 5, 8, 9, 12, 16, 4, 6, 8, 3, 4, 2, 1;
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PROG
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(PARI)
C(sig)={vecprod(sig)}
Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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