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

1,2

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

Links

Table of n, a(n) for n=1..42.

Example

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)

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&]]}]

Prog

(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

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.

Sequence in context: A037418 A239343 A058517 * A081310 A187298 A069067

Adjacent sequences: A328960 A328961 A328962 * A328964 A328965 A328966

Keyword

nonn

Author

Gus Wiseman, Nov 02 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

Status

approved