A118851 - OEIS

%I #49 Aug 02 2023 07:09:38

%S 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,

%T 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,

%U 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

%N Product of parts in n-th partition in Abramowitz and Stegun order.

%C 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

%D Abramowitz and Stegun, Handbook (1964) page 831.

%H Andrew Howroyd, <a href="/A118851/b118851.txt">Table of n, a(n) for n = 0..2713</a> (rows 0..20)

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Walter Bridges and William Craig, <a href="https://arxiv.org/abs/2308.00123">On the distribution of the norm of partitions</a>, arXiv:2308.00123 [math.CO], 2023.

%H Abhimanyu Kumar and Meenakshi Rana, <a href="http://www.mathjournals.org/jrms/2020-035-003/2020-035-003-005.html">On the treatment of partitions as factorization and further analysis</a>, Journal of the Ramanujan Mathematical Society 35(3), 263-276 (2020).

%H Wolfdieter Lang, <a href="/A118851/a118851_1.pdf">Rows n=1..10.</a>

%H Robert Schneider and Andrew V. Sills, <a href="http://math.colgate.edu/~integers/uproc13/uproc13.pdf">The Product of Parts or "Norm" of a Partition</a>, INTEGERS, Volume 20A (2020) Paper #A13, 16 pp.

%H Andrew V. Sills and Robert Schneider, <a href="https://arxiv.org/abs/1904.08004">The product of parts or "norm" of a partition</a>, arXiv:1904.08004 [math.NT], 2019-2021.

%F a(n) = A085643(n)/A048996(n).

%F T(n,k) = A005361(A036035(n,k)). - _Andrew Howroyd_, Oct 19 2020

%e a(9) = 4 because the 9th partition is [2,2] and 2*2 = 4.

%e Table T(n,k) starts:

%e   1;

%e   1;

%e   2, 1;

%e   3, 2,  1;

%e   4, 3,  4,  2,  1;

%e   5, 4,  6,  3,  4, 2,  1;

%e   6, 5,  8,  9,  4, 6,  8,  3,  4,  2, 1;

%e   7, 6, 10, 12,  5, 8,  9, 12,  4,  6, 8, 3, 4,  2,  1;

%e   8, 7, 12, 15, 16, 6, 10, 12, 16, 18, 5, 8, 9, 12, 16, 4, 6, 8, 3, 4, 2, 1;

%o (PARI)

%o C(sig)={vecprod(sig)}

%o Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}

%o { for(n=0, 7, print(Row(n))) } \\ _Andrew Howroyd_, Oct 19 2020

%Y Cf. A000041 (row lengths), A006906 (row sums).

%Y Cf. A005361, A036035, A036036, A048996, A078812, A085643, A103921, A111786.

%K nonn,tabf

%O 0,3

%A _Alford Arnold_, May 01 2006

%E Corrected and extended by _Franklin T. Adams-Watters_, May 26 2006