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A328963 Smallest k such that n = sigma_0(k) - ((bigomega(k)-1)*omega(k)), where sigma_0 = A000005, omega = A001221, bigomega = A001222.

%I A328963 #33 Nov 18 2019 10:43:48
%S A328963 1,2,36,72,144,180,576,420,360,864,1296,720,36864,1080,1440,1260,5184,
%T A328963 1800,2160,3360,5760,15552,4620,2520,150994944,6480,5400,13440,8640,
%U A328963 6300,9663676416,5040,12960,9240,331776,7560,186624,248832,34560,10080,1327104,13860
%N A328963 Smallest k such that n = sigma_0(k) - ((bigomega(k)-1)*omega(k)), where sigma_0 = A000005, omega = A001221, bigomega = A001222.
%C A328963 a(n) = smallest k for which A328959(k) = n-2. a(31) > 2^28. - _Antti Karttunen_, Nov 17 2019
%C A328963 a(n) <= 2^(n-1)*3^2, with equality for n = 3, 4, 5, 7, 13, 25, 31, 43,... . - _Giovanni Resta_, Nov 18 2019
%e A328963 The sequence of terms together with their prime signatures begins:
%e A328963         1: ()
%e A328963         2: (1)
%e A328963        36: (2,2)
%e A328963        72: (3,2)
%e A328963       144: (4,2)
%e A328963       180: (2,2,1)
%e A328963       576: (6,2)
%e A328963       420: (2,1,1,1)
%e A328963       360: (3,2,1)
%e A328963       864: (5,3)
%e A328963      1296: (4,4)
%e A328963       720: (4,2,1)
%e A328963     36864: (12,2)
%e A328963      1080: (3,3,1)
%e A328963      1440: (5,2,1)
%e A328963      1260: (2,2,1,1)
%e A328963      5184: (6,4)
%e A328963      1800: (3,2,2)
%e A328963      2160: (4,3,1)
%e A328963      3360: (5,1,1,1)
%e A328963      5760: (7,2,1)
%e A328963     15552: (6,5)
%e A328963      4620: (2,1,1,1,1)
%e A328963      2520: (3,2,1,1)
%e A328963 150994944: (24,2)
%t A328963 dat=Table[DivisorSigma[0,n]-(PrimeOmega[n]-1)*PrimeNu[n],{n,1000}];
%t A328963 Table[Position[dat,i][[1,1]],{i,First[Split[Union[dat],#2==#1+1&]]}]
%o A328963 (PARI)
%o A328963 search_up_to = 2^28;
%o A328963 A307408(n) = 2+((bigomega(n)-1)*omega(n));
%o A328963 A328959(n) = (numdiv(n) - A307408(n));
%o A328963 A328963(search_up_to) = { my(m=Map(),t,lista=List([])); for(n=1,search_up_to,t =
%o A328963 A328959(n); if(!mapisdefined(m,t+2), mapput(m,t+2,n))); for(u=1,oo,if(!mapisdefined(m,u,&t),return(Vec(lista)), listput(lista,t))); };
%o A328963 v328963 = A328963(search_up_to);
%o A328963 A328963(n) = v328963[n]; \\ _Antti Karttunen_, Nov 17 2019
%Y A328963 Positions of first appearances in A328959.
%Y A328963 All terms are in A025487.
%Y A328963 Cf. A000005, A001221, A001222, A113901, A124010, A307409, A320632, A323023, A328956, A328958, A328963, A328964, A328965.
%K A328963 nonn
%O A328963 1,2
%A A328963 _Gus Wiseman_, Nov 02 2019
%E A328963 Definition corrected and terms a(25) - a(30) added by _Antti Karttunen_, Nov 17 2019
%E A328963 a(31)-a(42) from _Giovanni Resta_, Nov 18 2019