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Smallest k such that n = sigma_0(k) - ((bigomega(k)-1)*omega(k)), where sigma_0 = A000005, omega = A001221, bigomega = A001222.
11

%I #33 Nov 18 2019 10:43:48

%S 1,2,36,72,144,180,576,420,360,864,1296,720,36864,1080,1440,1260,5184,

%T 1800,2160,3360,5760,15552,4620,2520,150994944,6480,5400,13440,8640,

%U 6300,9663676416,5040,12960,9240,331776,7560,186624,248832,34560,10080,1327104,13860

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

%C a(n) = smallest k for which A328959(k) = n-2. a(31) > 2^28. - _Antti Karttunen_, Nov 17 2019

%C 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 The sequence of terms together with their prime signatures begins:

%e 1: ()

%e 2: (1)

%e 36: (2,2)

%e 72: (3,2)

%e 144: (4,2)

%e 180: (2,2,1)

%e 576: (6,2)

%e 420: (2,1,1,1)

%e 360: (3,2,1)

%e 864: (5,3)

%e 1296: (4,4)

%e 720: (4,2,1)

%e 36864: (12,2)

%e 1080: (3,3,1)

%e 1440: (5,2,1)

%e 1260: (2,2,1,1)

%e 5184: (6,4)

%e 1800: (3,2,2)

%e 2160: (4,3,1)

%e 3360: (5,1,1,1)

%e 5760: (7,2,1)

%e 15552: (6,5)

%e 4620: (2,1,1,1,1)

%e 2520: (3,2,1,1)

%e 150994944: (24,2)

%t dat=Table[DivisorSigma[0,n]-(PrimeOmega[n]-1)*PrimeNu[n],{n,1000}];

%t Table[Position[dat,i][[1,1]],{i,First[Split[Union[dat],#2==#1+1&]]}]

%o (PARI)

%o search_up_to = 2^28;

%o A307408(n) = 2+((bigomega(n)-1)*omega(n));

%o A328959(n) = (numdiv(n) - A307408(n));

%o A328963(search_up_to) = { my(m=Map(),t,lista=List([])); for(n=1,search_up_to,t =

%o 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 v328963 = A328963(search_up_to);

%o A328963(n) = v328963[n]; \\ _Antti Karttunen_, Nov 17 2019

%Y Positions of first appearances in A328959.

%Y All terms are in A025487.

%Y Cf. A000005, A001221, A001222, A113901, A124010, A307409, A320632, A323023, A328956, A328958, A328963, A328964, A328965.

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 02 2019

%E Definition corrected and terms a(25) - a(30) added by _Antti Karttunen_, Nov 17 2019

%E a(31)-a(42) from _Giovanni Resta_, Nov 18 2019