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).

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

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: A280352 A135360 A347017 * 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