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%I #69 Dec 25 2023 22:51:41
%S 1,2,4,7,12,23,36,53,72,95,124,155,192,233,276,323,376,435,496,563,
%T 634,707,786,869,958,1055,1156,1259,1366,1475,1588,1715,1846,1983,
%U 2122,2271,2422,2579,2742,2909,3082,3261,3442,3633,3826,4023,4222,4433,4656,4883
%N 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).
%C 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]
%C Also number of k that can be expressed as a signed sum of the first n-1 primes. - _Seiichi Manyama_, Oct 01 2019
%H Seiichi Manyama, <a href="/A082548/b082548.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A007504(n-1) - 5 for n > 4. - _Seiichi Manyama_, Oct 02 2019
%e 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.
%e From _Seiichi Manyama_, Oct 01 2019: (Start)
%e 7 = 7, so 7*2 = 14 = 24-10 = 24+(-2-3-5).
%e 2+7 = 9, so (2+7)*2 = 18 = 24- 6 = 24+( 2-3-5).
%e 3+7 = 10, so (3+7)*2 = 20 = 24- 4 = 24+(-2+3-5).
%e 5+7 = 12, so (5+7)*2 = 24 = 24+ 0 = 24+(-2-3+5).
%e 2+5+7 = 14, so (2+5+7)*2 = 28 = 24+ 4 = 24+( 2-3+5).
%e 3+5+7 = 15, so (3+5+7)*2 = 30 = 24+ 6 = 24+(-2+3+5).
%e 2+3+5+7 = 17. so (2+3+5+7)*2 = 34 = 24+10 = 24+( 2+3+5). (End)
%e From _Seiichi Manyama_, Oct 02 2019: (Start)
%e 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).
%e n |b(n)| |
%e --+----+------------+--------------------------------------
%e 4 | 12 | 0 | 11
%e | | 2 | 13 = 2+11
%e | | 3 | 14 = 3+11
%e | | 5 | 16 = 5+11
%e | | 7 | 18 = 7+11
%e | | 8 = 3+5 | 19 = 8+11 = (3+5)+11
%e | | 9 = 17-8 | 20 = 9+11 = (2+3+5+7)-(3+5)+11
%e | | 10 = 17-7 | 21 = 10+11 = (2+3+5+7)-7 +11
%e | | 12 = 17-5 | 23 = 12+11 = (2+3+5+7)-5 +11
%e | | 14 = 17-3 | 25 = 14+11 = (2+3+5+7)-3 +11
%e | | 15 = 17-2 | 26 = 15+11 = (2+3+5+7)-2 +11
%e | | 17 = 17-0 | 28 = 17+11 = (2+3+5+7) +11
%e --+----+------------+--------------------------------------
%e 5 | 23 | 0 | 13
%e | | 2 | 15 = 2+13
%e | | 3 | 16 = 3+13
%e | | 5 | 18 = 5+13
%e | | 7 | 20 = 7+13
%e | | 8 = 3+5 | 21 = 8+13 = (3+5) +13
%e | | 9 = 2+7 | 22 = 9+13 = (2+7) +13
%e | | 10 = 2+3+5 | 23 = 10+13 = (2+3+5)+13
%e | | 11 | 24 = 11+13
%e | | ... | ...
%e | | 17 = 28-11 | 30 = 17+13 = (2+3+5+7+11)-11 +13
%e | | 18 = 28-10 | 31 = 18+13 = (2+3+5+7+11)-(2+3+5)+13
%e | | 19 = 28- 9 | 32 = 19+13 = (2+3+5+7+11)-(2+7) +13
%e | | 20 = 28- 8 | 33 = 20+13 = (2+3+5+7+11)-(3+5) +13
%e | | 21 = 28- 7 | 34 = 21+13 = (2+3+5+7+11)- 7 +13
%e | | 23 = 28- 5 | 36 = 23+13 = (2+3+5+7+11)- 5 +13
%e | | 25 = 28- 3 | 38 = 25+13 = (2+3+5+7+11)- 3 +13
%e | | 26 = 28- 2 | 39 = 26+13 = (2+3+5+7+11)- 2 +13
%e | | 28 = 28- 0 | 41 = 28+13 = (2+3+5+7+11) +13
%e --+----+------------+-------------------------------------
%e ...
%e b(n) = Sum_{k=1..n} prime(k) + 1 - 3*2 = A007504(n) - 5 for n>3.
%e So a(n) = b(n-1) = A007504(n-1) - 5 for n>4.
%o (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)
%Y Cf. A007504, A082533, A082534, A327467.
%K easy,nonn
%O 1,2
%A _Naohiro Nomoto_, May 02 2003
%E More terms from _David Wasserman_, Sep 16 2004
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