| Displaying 1-5 of 5 results found.
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0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 4, 0, 2, 2, 3, 0, 4, 0, 4, 2, 2, 0, 6, 1, 2, 2, 4, 0, 6, 0, 4, 2, 2, 2, 6, 0, 2, 2, 6, 0, 6, 0, 4, 4, 2, 0, 8, 1, 4, 2, 4, 0, 6, 2, 6, 2, 2, 0, 9, 0, 2, 4, 5, 2, 6, 0, 4, 2, 6, 0, 8, 0, 2, 4, 4, 2, 6, 0, 8, 3, 2, 0, 9, 2, 2, 2, 6, 0, 9, 2, 4, 2, 2, 2, 10, 0, 4, 4, 6, 0, 6, 0, 6
COMMENTS
a(n) + 2 appears to differ from A000005 at n=1 and when n is a term of A320632. Verified up to n=3000. If A320632 contains the numbers such that A001222(n) - A051903(n) > 1, then this sequence contains precisely the numbers p^k and p^k*q for distinct primes p and q. The comment follows, since d(p^k) = k+1 = (k-1)*1 + 2 and d(p^k*q) = 2k+2 = ((k+1)-1)*2 + 2. - Charlie Neder, May 14 2019
MATHEMATICA
a[n_] := (PrimeOmega[n] - 1)*PrimeNu[n];
aa = Table[a[n], {n, 1, 104}];
PROG
(PARI) a(n) = (bigomega(n) - 1)*omega(n); \\ Michel Marcus, May 15 2019
CROSSREFS
A113901(n) is bigomega(n) * omega(n).
A328958(n) is sigma_0(n) - bigomega(n) * omega(n).
Cf. A000005, A001221, A001222, A060687, A070175, A071625, A113901, A124010, A303555, A320632, A323023, A328956, A328957, A328964, A328965, A322437.
Product of omega(n) and bigomega(n) = A001221(n)* A001222(n), where omega(x): number of distinct prime divisors of x. bigomega(x): number of prime divisors of x, counted with multiplicity.
+10
15
0, 1, 1, 2, 1, 4, 1, 3, 2, 4, 1, 6, 1, 4, 4, 4, 1, 6, 1, 6, 4, 4, 1, 8, 2, 4, 3, 6, 1, 9, 1, 5, 4, 4, 4, 8, 1, 4, 4, 8, 1, 9, 1, 6, 6, 4, 1, 10, 2, 6, 4, 6, 1, 8, 4, 8, 4, 4, 1, 12, 1, 4, 6, 6, 4, 9, 1, 6, 4, 9, 1, 10, 1, 4, 6, 6, 4, 9, 1, 10, 4, 4, 1, 12, 4, 4, 4, 8, 1, 12, 4, 6, 4, 4, 4, 12, 1, 6, 6, 8, 1, 9
MATHEMATICA
Table[PrimeNu[n]*PrimeOmega[n], {n, 1, 50}] (* G. C. Greubel, Apr 23 2017 *)
PROG
(PARI) list(n) = { local(x); for(x=1, n, print1(omega(x)*bigomega(x)", ") ) }
CROSSREFS
A307409(n) is (bigomega(n) - 1) * omega(n).
A328958(n) is sigma_0(n) - bigomega(n) * omega(n).
Cf. A000005, A001221, A001222, A060687, A070175, A071625, A124010, A303555, A320632, A323023, A328956, A328957, A328964, A328965.
Numbers k such that there exists a pair of factorizations of k into factors > 1 where no factor of one divides any factor of the other.
+10
13
36, 60, 72, 84, 90, 100, 108, 120, 126, 132, 140, 144, 150, 156, 168, 180, 196, 198, 200, 204, 210, 216, 220, 225, 228, 234, 240, 252, 260, 264, 270, 276, 280, 288, 294, 300, 306, 308, 312, 315, 324, 330, 336, 340, 342, 348, 350, 360, 364, 372, 378, 380, 390
COMMENTS
Mats Granvik has conjectured that these are all the positive integers k such that sigma_0(k) - 2 > (bigomega(k) - 1) * omega(k), where sigma_0 = A000005, omega = A001221, and bigomega = A001222. - Gus Wiseman, Nov 12 2019
Numbers with more semiprime divisors than prime divisors. - Wesley Ivan Hurt, Jun 10 2021
EXAMPLE
An example of such a pair for 36 is (4*9)|(6*6).
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Select[Range[100], Select[Subsets[facs[#], {2}], And[!Or@@Divisible@@@Tuples[#], !Or@@Divisible@@@Reverse/@Tuples[#]]&]!={}&]
PROG
(PARI)
factorizations(n, m=n, f=List([]), z=List([])) = if(1==n, listput(z, Vec(f)); z, my(newf); fordiv(n, d, if((d>1)&&(d<=m), newf = List(f); listput(newf, d); z = factorizations(n/d, d, newf, z))); (z));
is_ndf_pair(fac1, fac2) = { for(i=1, #fac1, for(j=1, #fac2, if(!(fac1[i]%fac2[j])||!(fac2[j]%fac1[i]), return(0)))); (1); };
has_at_least_one_ndfpair(z) = { for(i=1, #z, for(j=i+1, #z, if(is_ndf_pair(z[i], z[j]), return(1)))); (0); };
isA320632(n) = has_at_least_one_ndfpair(Vec(factorizations(n))); \\ Antti Karttunen, Dec 10 2020
CROSSREFS
Cf. A001055, A050336, A285572, A303362, A305149, A305193, A317144, A322435, A322437, A322439, A322440, A322441, A322442. The following are additional cross-references relating to Granvik's conjecture.
bigomega(n) * omega(n) is A113901(n). (bigomega(n) - 1) * omega(n) is A307409(n). sigma_0(n) - bigomega(n) * omega(n) is A328958(n). sigma_0(n) - 2 - (omega(n) - 1) * nu(n) is A328959(n).
Numbers k such that sigma_0(k) = omega(k) * Omega(k), where sigma_0 = A000005, omega = A001221, Omega = A001222.
+10
13
6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 68, 69, 74, 75, 76, 77, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 104, 106, 111, 112, 115, 116, 117
COMMENTS
First differs from A084227 in having 60.
EXAMPLE
The sequence of terms together with their prime indices begins:
6: {1,2}
10: {1,3}
12: {1,1,2}
14: {1,4}
15: {2,3}
18: {1,2,2}
20: {1,1,3}
21: {2,4}
22: {1,5}
24: {1,1,1,2}
26: {1,6}
28: {1,1,4}
33: {2,5}
34: {1,7}
35: {3,4}
38: {1,8}
39: {2,6}
40: {1,1,1,3}
44: {1,1,5}
45: {2,2,3}
MATHEMATICA
Select[Range[100], DivisorSigma[0, #]==PrimeOmega[#]*PrimeNu[#]&]
PROG
(PARI) is(k) = {my(f = factor(k)); numdiv(f) == omega(f) * bigomega(f); } \\ Amiram Eldar, Jul 28 2024
CROSSREFS
(Omega(n) - 1) * omega(n) is A307409(n). sigma_0(n) - omega(n) * Omega(n) is A328958(n). sigma_0(n) - 2 - (Omega(n) - 1) * omega(n) is A328959(n). Cf. A000040, A005117, A060687, A070175, A090858, A112798, A303555, A320632, A328960, A328961, A328962, A328963, A328964, A328965.
1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, -1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, -1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, -1, 1, 0, 0, -1, 1, 2, 1, 0, 0, 0, 0, -1, 1, 0, 1, 0, 1, 0, 0
EXAMPLE
a(144) = sigma_0(144) - omega(144) * nu(144) = 15 - 6 * 2 = 3.
MATHEMATICA
Table[DivisorSigma[0, n]-PrimeOmega[n]*PrimeNu[n], {n, 100}]
CROSSREFS
Positions of first appearances are A328962. (omega(n) - 1) * nu(n) is A307409(n). sigma_0(n) - 2 - (omega(n) - 1) * nu(n) is A328959(n).
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