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A034960 Divide odd numbers into groups with prime(n) elements and add together.

%I A034960 #60 Jun 23 2025 14:32:30
%S A034960 4,21,75,189,495,897,1683,2565,4071,6641,8959,13209,17835,22317,28623,
%T A034960 37577,48439,57401,71623,85697,98623,118737,138195,163493,196231,
%U A034960 224321,249775,281945,310759,347249,420751,467801,525943,571985,656047
%N A034960 Divide odd numbers into groups with prime(n) elements and add together.
%H A034960 Hieronymus Fischer, <a href="/A034960/b034960.txt">Table of n, a(n) for n = 1..10000</a>
%F A034960 From _Hieronymus Fischer_, Sep 26 2012: (Start)
%F A034960 a(n) = Sum_{k=A007504(n-1)+1..A007504(n)} (2*k-1).
%F A034960 a(n) = A007504(n)^2 - A007504(n-1)^2.
%F A034960 a(n) = 2*A034957(n) + A000040(n).
%F A034960 a(n) = 2*A034956(n) - A000040(n).
%F A034960 a(n) = A034959(n) + A000040(n). (End)
%F A034960 a(n) = A061802(n)*A000040(n). - _Marco Zárate_, May 12 2023
%e A034960 {1,3} #2 S=4;
%e A034960 {5,7,9} #3 S=21;
%e A034960 {11,13,15,17,19} #5 S=75;
%e A034960 {21,23,25,27,29,31,33} #7 S=189.
%p A034960 S:= n-> sum(ithprime(k), k=1..n): seq(S(n+1)^2-S(n)^2, n=0..40); # _Gary Detlefs_, Dec 20 2011
%t A034960 Accumulate[Join[{2}, ListConvolve[{1, 1}, #]]]*# & [Prime[Range[50]]] (* _Paolo Xausa_, Jun 23 2025 *)
%o A034960 (Python)
%o A034960 from itertools import islice
%o A034960 from sympy import nextprime
%o A034960 def A034960_gen(): # generator of terms
%o A034960     a, p = 0, 2
%o A034960     while True:
%o A034960         yield p*((a<<1)+p)
%o A034960         a, p = a+p, nextprime(p)
%o A034960 A034960_list = list(islice(A034960_gen(),20)) # _Chai Wah Wu_, Mar 22 2023
%o A034960 (PARI) a0(n) = vecsum(primes(n))^2 - vecsum(primes(n-1))^2; \\ _Michel Marcus_, Jun 16 2024
%Y A034960 Cf. A006003, A027441, A034959.
%Y A034960 Cf. A007504.
%Y A034960 Cf. A000040, A034956, A034957, A034958, A046992, A061802.
%K A034960 nonn
%O A034960 1,1
%A A034960 _Patrick De Geest_, Oct 15 1998