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A299196 Number of partitions of n into distinct parts that are lesser of twin primes (A001359).

%I A299196 #20 Feb 16 2025 08:33:53
%S A299196 1,0,0,1,0,1,0,0,1,0,0,1,0,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,1,0,1,1,1,
%T A299196 1,0,1,1,0,0,1,1,0,1,1,1,2,0,1,2,0,1,1,0,1,1,0,2,1,1,2,1,2,1,1,1,1,1,
%U A299196 0,1,2,1,1,2,2,2,2,1,2,2,0,2,1,0,2,1,1,3,2,1,3,2,2,2,0,2,2,0,1,2,2,2,2,3,3,3
%N A299196 Number of partitions of n into distinct parts that are lesser of twin primes (A001359).
%C A299196 For n > 0 let b(n) be the inverse Euler transform of a(n). It appears that, if p is the lesser of twin primes, then b(p) = 1 and b(2*p) = -1; otherwise b(n) = 0. - _Georg Fischer_, Aug 15 2020
%H A299196 Robert Israel, <a href="/A299196/b299196.txt">Table of n, a(n) for n = 0..10000</a>
%H A299196 Ilya Gutkovskiy, <a href="/A299196/a299196.pdf">Scatter plot of a(n) - A299197(n)</a>
%H A299196 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TwinPrimes.html">Twin Primes</a>
%H A299196 <a href="/index/Par#part">Index entries for related partition-counting sequences</a>
%F A299196 G.f.: Product_{k>=1} (1 + x^A001359(k)).
%e A299196 a(46) = 2 because we have [41, 5] and [29, 17].
%p A299196 P:= select(isprime,{seq(i,i=3..201,2)}):
%p A299196 TP:= P intersect map(`-`,P,2):
%p A299196 G:= mul(1+x^p,p=TP):
%p A299196 seq(coeff(G,x,i),i=0..200); # _Robert Israel_, Dec 15 2024
%t A299196 nmax = 105; CoefficientList[Series[Product[1 + Boole[PrimeQ[k] && PrimeQ[k + 2]] x^k, {k, 1, nmax}], {x, 0, nmax}], x]
%Y A299196 Cf. A000586, A000607, A001097, A001359, A077608, A129363, A283875, A283876, A299197.
%K A299196 nonn
%O A299196 0,47
%A A299196 _Ilya Gutkovskiy_, Feb 04 2018