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 A082548 a(n) is the number of values of k such that k can be expressed as the sum of distinct primes with largest prime in the sum equal to prime(n). 3
 1, 2, 4, 7, 12, 23, 36, 53, 72, 95, 124, 155, 192, 233, 276, 323, 376, 435, 496, 563, 634, 707, 786, 869, 958, 1055, 1156, 1259, 1366, 1475, 1588, 1715, 1846, 1983, 2122, 2271, 2422, 2579, 2742, 2909, 3082, 3261, 3442, 3633, 3826, 4023, 4222, 4433, 4656, 4883 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Surprisingly, except for the initial term, the first differences of this sequence is the sequence of primes with 7 omitted. [John W. Layman, Feb 25 2012] Also number of k that can be expressed as a signed sum of the first n-1 primes. - Seiichi Manyama, Oct 01 2019 LINKS Seiichi Manyama, Table of n, a(n) for n = 1..10000 FORMULA a(n) = A007504(n-1) - 5 for n > 4. - Seiichi Manyama, Oct 02 2019 EXAMPLE For n=4; 7 is the 4th prime. 7 = 7, 9 = 2+7, 10 = 3+7, 12 = 5+7 = 2+3+7, 14 = 2+5+7, 15 = 3+5+7, 17 = 2+3+5+7. Values of m are 7 and 9,10,12,14,15,17. so a(4)=7. From Seiichi Manyama, Oct 01 2019: (Start) 7       =  7, so 7*2         = 14 = 24-10 = 24+(-2-3-5). 2+7     =  9, so (2+7)*2     = 18 = 24- 6 = 24+( 2-3-5). 3+7     = 10, so (3+7)*2     = 20 = 24- 4 = 24+(-2+3-5). 5+7     = 12, so (5+7)*2     = 24 = 24+ 0 = 24+(-2-3+5). 2+5+7   = 14, so (2+5+7)*2   = 28 = 24+ 4 = 24+( 2-3+5). 3+5+7   = 15, so (3+5+7)*2   = 30 = 24+ 6 = 24+(-2+3+5). 2+3+5+7 = 17. so (2+3+5+7)*2 = 34 = 24+10 = 24+( 2+3+5). (End) From Seiichi Manyama, Oct 02 2019: (Start) Let b(n) be the number of k (>=0) that can be expressed as the sum of distinct primes with largest prime in the sum not greater than prime(n). n |b(n)|            | --+----+------------+-------------------------------------- 4 | 12 |  0         | 11   |    |  2         | 13 =  2+11   |    |  3         | 14 =  3+11   |    |  5         | 16 =  5+11   |    |  7         | 18 =  7+11   |    |  8 = 3+5   | 19 =  8+11 = (3+5)+11   |    |  9 = 17-8  | 20 =  9+11 = (2+3+5+7)-(3+5)+11   |    | 10 = 17-7  | 21 = 10+11 = (2+3+5+7)-7    +11   |    | 12 = 17-5  | 23 = 12+11 = (2+3+5+7)-5    +11   |    | 14 = 17-3  | 25 = 14+11 = (2+3+5+7)-3    +11   |    | 15 = 17-2  | 26 = 15+11 = (2+3+5+7)-2    +11   |    | 17 = 17-0  | 28 = 17+11 = (2+3+5+7)      +11 --+----+------------+-------------------------------------- 5 | 23 |  0         | 13   |    |  2         | 15 =  2+13   |    |  3         | 16 =  3+13   |    |  5         | 18 =  5+13   |    |  7         | 20 =  7+13   |    |  8 = 3+5   | 21 =  8+13 = (3+5)  +13   |    |  9 = 2+7   | 22 =  9+13 = (2+7)  +13   |    | 10 = 2+3+5 | 23 = 10+13 = (2+3+5)+13   |    | 11         | 24 = 11+13   |    | ...        | ...   |    | 17 = 28-11 | 30 = 17+13 = (2+3+5+7+11)-11     +13   |    | 18 = 28-10 | 31 = 18+13 = (2+3+5+7+11)-(2+3+5)+13   |    | 19 = 28- 9 | 32 = 19+13 = (2+3+5+7+11)-(2+7)  +13   |    | 20 = 28- 8 | 33 = 20+13 = (2+3+5+7+11)-(3+5)  +13   |    | 21 = 28- 7 | 34 = 21+13 = (2+3+5+7+11)- 7     +13   |    | 23 = 28- 5 | 36 = 23+13 = (2+3+5+7+11)- 5     +13   |    | 25 = 28- 3 | 38 = 25+13 = (2+3+5+7+11)- 3     +13   |    | 26 = 28- 2 | 39 = 26+13 = (2+3+5+7+11)- 2     +13   |    | 28 = 28- 0 | 41 = 28+13 = (2+3+5+7+11)        +13 --+----+------------+------------------------------------- ... b(n) = Sum_{k=1..n} prime(k) + 1 - 3*2 = A007504(n) - 5 for n>3. So a(n) = b(n-1) = A007504(n-1) - 5 for n>4. PROG (PARI) limit = 70; M = sum(i = 1, limit, prime(i)); v = vector(M); primeSum = 0; forprime (n = 1, prime(limit), count = 1; forstep (i = primeSum, 1, -1, if (v[i], count = count + 1; v[i + n] = 1)); v[n] = 1; print(count); primeSum = primeSum + n) CROSSREFS Cf. A007504, A082533, A082534, A327467. Sequence in context: A072641 A280352 A135360 * A270995 A299023 A007323 Adjacent sequences:  A082545 A082546 A082547 * A082549 A082550 A082551 KEYWORD easy,nonn AUTHOR Naohiro Nomoto, May 02 2003 EXTENSIONS More terms from David Wasserman, Sep 16 2004 STATUS approved

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Last modified October 17 11:47 EDT 2019. Contains 328108 sequences. (Running on oeis4.)