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 A319895 a(n) is the number of partitions of n into consecutive parts, plus the total number of parts in those partitions. 1
 2, 2, 5, 2, 5, 6, 5, 2, 9, 7, 5, 6, 5, 7, 15, 2, 5, 11, 5, 8, 16, 7, 5, 6, 11, 7, 16, 10, 5, 17, 5, 2, 16, 7, 19, 15, 5, 7, 16, 8, 5, 19, 5, 11, 32, 7, 5, 6, 13, 13, 16, 11, 5, 21, 22, 10, 16, 7, 5, 21, 5, 7, 34, 2, 22, 23, 5, 11, 16, 21, 5, 16, 5, 7, 33, 11, 25, 24, 5, 8, 26, 7, 5, 23, 22, 7, 16, 14, 5 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS a(n) is also the total length of all pairs of orthogonal line segments whose horizontal and upper parts are in the n-th row of the diagram associated to partitions into consecutive parts as shown in the Example section. a(n) = 2 iff n is a power of 2. a(n) = 5 iff n is an odd prime. LINKS Antti Karttunen, Table of n, a(n) for n = 1..20000 FORMULA a(n) = A001227(n) + A204217(n). EXAMPLE Illustration of a diagram of partitions into consecutive parts (first 28 rows): . _ . _|1 . _|2 _ . _|3 |2 . _|4 _|1 . _|5 |3 _ . _|6 _|2|3 . _|7 |4 |2 . _|8 _|3 _|1 . _|9 |5 |4 _ . _|10 _|4 |3|4 . _|11 |6 _|2|3 . _|12 _|5 |5 |2 . _|13 |7 |4 _|1 . _|14 _|6 _|3|5 _ . _|15 |8 |6 |4|5 . _|16 _|7 |5 |3|4 . _|17 |9 _|4 _|2|3 . _|18 _|8 |7 |6 |2 . _|19 |10 |6 |5 _|1 . _|20 _|9 _|5 |4|6 _ . _|21 |11 |8 _|3|5|6 . _|22 _|10 |7 |7 |4|5 . _|23 |12 _|6 |6 |3|4 . _|24 _|11 |9 |5 _|2|3 . _|25 |13 |8 _|4|7 |2 . _|26 _|12 _|7 |8 |6 _|1 . _|27 |14 |10 |7 |5|7 _ . |28 |13 |9 |6 |4|6|7 ... For n = 21 we have that there are four partitions of 21 into consecutive parts, they are [21], [11, 10], [8, 7, 6], [6, 5, 4, 3, 2, 1]. The total number of parts is 1 + 2 + 3 + 6 = 12. Therefore the number of partitions plus the total number of parts is 4 + 12 = 16, so a(21) = 16. On the other hand, in the above diagram there are four pairs of orthogonal line segments whose horizontal upper part are located on the 21st row, as shown below: . _ _ _ _ . |21 |11 |8 |6 . |10 |7 |5 . |6 |4 . |3 . |2 . |1 . The four horizontal line segments have length 1, and the vertical line segments have lengths 1, 2, 3, 6 respectively. Therefore the total length of the line segments is 1 + 1 + 1 + 1 + 1 + 2 + 3 + 6 = 16, so a(21) = 16. PROG (PARI) A001227(n) = numdiv(n>>valuation(n, 2)); A204217(n) = { my(i=2, t=1); n--; while(n>0, t += (i*(n%i==0)); n-=i; i++); t }; \\ From A204217 by David A. Corneth, Apr 28 2017 A319895(n) = (A001227(n)+A204217(n)); \\ Antti Karttunen, Dec 06 2021 CROSSREFS For tables of partitions into consecutive parts see A286000 and A286001. Cf. A000079, A001227, A065091, A204217, A237048, A237593, A285898, A285899, A285900, A285900, A285901, A285902, A288529, A288772, A288773, A288774, A299765. Sequence in context: A162784 A093660 A093663 * A323504 A011143 A240081 Adjacent sequences: A319892 A319893 A319894 * A319896 A319897 A319898 KEYWORD nonn AUTHOR Omar E. Pol, Sep 30 2018 EXTENSIONS Term a(87) corrected from 6 to 16 by Antti Karttunen, Dec 06 2021 STATUS approved

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Last modified February 24 05:54 EST 2024. Contains 370294 sequences. (Running on oeis4.)