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

1, 2, 36, 72, 144, 180, 576, 420, 360, 864, 1296, 720, 36864, 1080, 1440, 1260, 5184, 1800, 2160, 3360, 5760, 15552, 4620, 2520, 150994944, 6480, 5400, 13440, 8640, 6300, 9663676416, 5040, 12960, 9240, 331776, 7560, 186624, 248832, 34560, 10080, 1327104, 13860

Offset: 1

Author

Gus Wiseman, Nov 02 2019

Keywords

• nonn

Comments

a(n) = smallest k for which A328959(k) = n-2. a(31) > 2^28. - Antti Karttunen, Nov 17 2019

a(n) <= 2^(n-1)*3^2, with equality for n = 3, 4, 5, 7, 13, 25, 31, 43,... . - Giovanni Resta, Nov 18 2019

Examples

			The sequence of terms together with their prime signatures begins:
        1: ()
        2: (1)
       36: (2,2)
       72: (3,2)
      144: (4,2)
      180: (2,2,1)
      576: (6,2)
      420: (2,1,1,1)
      360: (3,2,1)
      864: (5,3)
     1296: (4,4)
      720: (4,2,1)
    36864: (12,2)
     1080: (3,3,1)
     1440: (5,2,1)
     1260: (2,2,1,1)
     5184: (6,4)
     1800: (3,2,2)
     2160: (4,3,1)
     3360: (5,1,1,1)
     5760: (7,2,1)
    15552: (6,5)
     4620: (2,1,1,1,1)
     2520: (3,2,1,1)
150994944: (24,2)
		

Crossrefs

Positions of first appearances in A328959.

All terms are in A025487.

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

Programs

Mathematica

dat=Table[DivisorSigma[0,n]-(PrimeOmega[n]-1)*PrimeNu[n],{n,1000}];
Table[Position[dat,i][[1,1]],{i,First[Split[Union[dat],#2==#1+1&]]}]

PARI

search_up_to = 2^28;
A307408(n) = 2+((bigomega(n)-1)*omega(n));
A328959(n) = (numdiv(n) - A307408(n));
A328963(search_up_to) = { my(m=Map(),t,lista=List([])); for(n=1,search_up_to,t =
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))); };
v328963 = A328963(search_up_to);
A328963(n) = v328963[n]; \\ Antti Karttunen, Nov 17 2019

Extensions

Definition corrected and terms a(25) - a(30) added by Antti Karttunen, Nov 17 2019

a(31)-a(42) from Giovanni Resta, Nov 18 2019