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A322043 Numbers k such that the coefficient of x^k in the expansion of Product_{m >= 1} (1-x^m)^15 is zero.

%I A322043 #38 Aug 04 2023 01:39:28
%S A322043 53,482,1340,2627,4343,6488,9062,12065,15497,19358,23648,28367,33515,
%T A322043 39092,45098,51533,58397,65690,73412,81563,90143,99152,108590,118457,
%U A322043 128753,139478,150632,162215,174227,186668,199538,212837,226565,240722,255308,270323,285767,301640,317942,334673,351833,369422,387440
%N A322043 Numbers k such that the coefficient of x^k in the expansion of Product_{m >= 1} (1-x^m)^15 is zero.
%C A322043 Van der Blij, discussing the conjecture that the Ramanujan numbers tau(k) (see A000594) are never zero, mentions that a certain "Ferguson" had shown that 52 is a member of the current sequence. No details were given, and the 52 appears to be a typo for 53.
%C A322043 The coefficients of the expansion of Product_{m >= 1} (1-x^m)^15 are given in A010822.
%D A322043 F. Van der Blij, "The function tau(n) of S. Ramanujan (an expository lecture)." Math. Student 18 (1950): 83-99. See page 85.
%H A322043 Joerg Arndt, <a href="/A322043/b322043.txt">Table of n, a(n) for n = 1..216</a>
%F A322043 Conjectures from _Colin Barker_, Dec 07 2018: (Start)
%F A322043 G.f.: x*(53 + 323*x + 53*x^2) / (1 - x)^3.
%F A322043 a(n) = (429*n^2 - 429*n + 106) / 2.
%F A322043 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
%F A322043 (End)
%F A322043 E.g.f.: (1/2)*exp(x)*(106 + 858*x + 429*x^2). - conjectured by _Stefano Spezia_, Dec 07 2018 after the conjectures of _Colin Barker_
%t A322043 sigma[k_] := sigma[k] = DivisorSigma[1, k];
%t A322043 a[0] = 1; a[n_] := a[n] = -15/n Sum[sigma[k] a[n-k], {k, 1, n}];
%t A322043 Reap[For[k = 1, k <= 200000, k++, If[a[k] == 0, Print[k]; Sow[k]]]][[2, 1]] (* _Jean-François Alcover_, Dec 20 2018 *)
%o A322043 (PARI) /* start with sufficient memory, e.g., gp -s16G */
%o A322043 x='x+O('x^1000000); v=Vec(eta(x)^15 - 1); for(k=1, #v, if(v[k]==0, print1(k, ", "))); \\ _Joerg Arndt_, Dec 20 2018
%Y A322043 Cf. A000594, A010822, A302057.
%K A322043 nonn
%O A322043 1,1
%A A322043 _N. J. A. Sloane_, Dec 07 2018
%E A322043 a(4)-a(7) supplied by _Rémy Sigrist_, Dec 07 2018, from the b-file for A010822.
%E A322043 a(8)-a(19) from _Seiichi Manyama_, Dec 07 2018
%E A322043 a(20)-a(31) from _Jean-François Alcover_, Dec 20 2018
%E A322043 More terms from _Joerg Arndt_, Dec 20 2018