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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.
11
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
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.
Sequence in context: A037418 A239343 A058517 * A081310 A187298 A069067
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