A328956 -id:A328956 - cp's OEIS frontend

A113901 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.

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

Offset: 1

Cino Hilliard, Jan 29 2006

Positions of first appearances are A328964. - Gus Wiseman, Nov 05 2019

G. C. Greubel, Table of n, a(n) for n = 1..1000

A307409(n) is (bigomega(n) - 1) * omega(n).
A328958(n) is sigma_0(n) - bigomega(n) * omega(n).
Cf. A000005, A001221, A001222, A060687, A066921, A066922, A068993, A070175, A071625, A124010, A303555, A320632, A323023, A328956, A328957, A328964, A328965.

Table[PrimeNu[n]*PrimeOmega[n], {n,1,50}] (* G. C. Greubel, Apr 23 2017 *)

PARI
```
a(n) = omega(n)*bigomega(n);
```

a(n) = 1 iff n is prime.
A068993(a(n)) = 4. - Reinhard Zumkeller, Mar 13 2011
a(n) = A066921(n)*A066922(n). - Amiram Eldar, May 07 2025

A307409 a(n) = (A001222(n) - 1)*A001221(n).

Original entry on oeis.org

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

Offset: 1

Mats Granvik, Apr 07 2019

nonn

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
Positions of first appearances are A328965. - Gus Wiseman, Nov 05 2019
Regarding Neder's comment above, see also my comments in A322437. - Antti Karttunen, Feb 17 2021

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Two less than A307408.
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.

a[n_] := (PrimeOmega[n] - 1)*PrimeNu[n];
aa = Table[a[n], {n, 1, 104}];

a(n) = (bigomega(n) - 1)*omega(n); \\ Michel Marcus, May 15 2019

a(n) = (A001222(n) - 1)*A001221(n).
a(n) = binomial(A001222(n) - 1, 1)*binomial(A001221(n), 1).
a(n) = A307408(n) - 2.

A320632 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.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 09 2018

nonn

Positions of nonzero terms in A322437 or A322438.
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

			An example of such a pair for 36 is (4*9)|(6*6).

Antti Karttunen, Table of n, a(n) for n = 1..23437
Christophe Cordero, Factorizations of Cyclic Groups and Bayonet Codes, arXiv:2301.13566 [math.CO], 2023, p. 20.

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).
Cf. A060687, A070175, A124010, A323023, A328956, A328957, A328960, A328961, A328963.

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[#]]&]!={}&]

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

A328958 a(n) = d(n) - (omega(n) * bigomega(n)), where d (number of divisors) = A000005, omega = A001221, bigomega = A001222.

Original entry on oeis.org

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, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, -1, 1, 0, -1

Offset: 1

Gus Wiseman, Nov 02 2019

a(n) = sigma_0(n) - omega(n) * nu(n), where sigma_0 = A000005, nu = A001221, omega = A001222. - The original name of the sequence.

			a(144) = sigma_0(144) - omega(144) * nu(144) = 15 - 6 * 2 = 3.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from exponents in factorization of n.

Positions of first appearances are A328962.
Zeros are A328956.
Nonzeros are A328957.
omega(n) * nu(n) is A113901(n).
(omega(n) - 1) * nu(n) is A307409(n).
sigma_0(n) - 2 - (omega(n) - 1) * nu(n) is A328959(n).
Cf. A000005, A001221, A001222, A112798, A124010, A323023, A328960, A328961, A328963, A328964.

Table[DivisorSigma[0,n]-PrimeOmega[n]*PrimeNu[n],{n,100}]

A328958(n) = (numdiv(n)-(omega(n)*bigomega(n))); \\ Antti Karttunen, Jan 27 2025

a(n) = A000005(n) - A001222(n) * A001221(n) = A000005(n) - A113901(n).

More terms added and the function names in the definition replaced with standard OEIS ones - Antti Karttunen, Jan 27 2025

A328959 a(n) = sigma_0(n) - 2 - (omega(n) - 1) * nu(n), where sigma_0 = A000005, nu = A001221, omega = A001222.

Original entry on oeis.org

-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 02 2019. The idea for this sequence came from Mats Granvik

sign

Conjecture: All terms are nonnegative except for a(1) = -1.

			a(72) = sigma_0(72) - 2 - (omega(72) - 1) * nu(72) = 12 - 2 - (5 - 1) * 2 = 2.

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

The positions of positive terms are conjectured to be A320632.
Positions of first appearances are A328963.
omega(n) * nu(n) is A113901(n).
(omega(n) - 1) * nu(n) is A307409.
sigma_0(n) - omega(n) * nu(n) is A328958(n).
Cf. A000005, A001221, A001222, A112798, A124010, A307408, A323023, A328956, A328960, A328961, A328962, A328965.

Table[DivisorSigma[0,n]-2-(PrimeOmega[n]-1)*PrimeNu[n],{n,100}]

A307408(n) = 2+((bigomega(n)-1)*omega(n));
A328959(n) = (numdiv(n) - A307408(n)); \\ Antti Karttunen, Nov 17 2019

a(n) = A000005(n) - A307408(n). - Antti Karttunen, Nov 17 2019

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

Original entry on oeis.org

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

Gus Wiseman, Nov 02 2019

nonn

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

			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)

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.

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&]]}]

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

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

A328965 Smallest k such that (bigomega(k) - 1) * omega(k) = n, and 0 if none exists, where omega = A001221, bigomega = A001222.

Original entry on oeis.org

1, 4, 6, 16, 12, 64, 24, 256, 48, 60, 96, 4096, 120, 16384, 384, 240, 420, 262144, 480, 1048576, 840, 960, 6144, 16777216, 1680, 4620, 24576, 3840, 3360, 1073741824, 7680, 4294967296, 6720, 15360, 393216, 18480, 13440, 274877906944, 1572864, 61440, 26880, 4398046511104

Offset: 0

Gus Wiseman, Nov 02 2019

nonn

For n > 0, a(n) is of the form 2^k*primorial(d) where d is a divisor of n and k = n / d - d + 1. a(n) is never 0 since A307409(2^(n+1)) = n. - Andrew Howroyd, Nov 04 2019

			The sequence of terms together with their prime signatures begins:
      1: ()
      4: (2)
      6: (1,1)
     16: (4)
     12: (2,1)
     64: (6)
     24: (3,1)
    256: (8)
     48: (4,1)
     60: (2,1,1)
     96: (5,1)
   4096: (12)
    120: (3,1,1)
  16384: (14)
    384: (7,1)
    240: (4,1,1)
    420: (2,1,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Positions of first appearances in A307409.

Cf. A001221, A001222, A002110, A113901, A124010, A320632, A323023, A328956, A328958, A328959, A328962, A328963, A328964.

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

a(n)={if(n<1, 1, my(m=oo); fordiv(n, d, if(d<=n/d+1, m=min(m, 2^(n/d-d+1)*vecprod(primes(d))))); m)} \\ Andrew Howroyd, Nov 04 2019

From Andrew Howroyd, Nov 03 2019: (Start)
a(p) = 2^(p + 1) for odd prime p.
a(n) = min_{d|n, d<=n/d+1} 2^(n/d-d+1)*A002110(d) for n > 0. (End)

Terms a(23) and beyond from Andrew Howroyd, Nov 03 2019

A328964 Smallest k such that omega(k) * bigomega(k) = n, where omega = A001221, bigomega = A001222.

Original entry on oeis.org

1, 2, 4, 8, 6, 32, 12, 128, 24, 30, 48, 2048, 60, 8192, 192, 120, 210, 131072, 240, 524288, 420, 480, 3072, 8388608, 840, 2310, 12288, 1920, 1680, 536870912, 3840, 2147483648, 3360, 7680, 196608, 9240, 6720, 137438953472, 786432, 30720, 13440, 2199023255552, 60060, 8796093022208

Offset: 0

Gus Wiseman, Nov 02 2019

nonn

			The sequence of terms together with their prime signatures begins:
     1: ()
     2: (1)
     4: (2)
     8: (3)
     6: (1,1)
    32: (5)
    12: (2,1)
   128: (7)
    24: (3,1)
    30: (1,1,1)
    48: (4,1)
  2048: (11)
    60: (2,1,1)
  8192: (13)
   192: (6,1)
   120: (3,1,1)
   210: (1,1,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Positions of first appearances in A113901.

Cf. A001221, A001222, A002110, A025487, A124010, A307409, A320632, A323023, A328956, A328958, A328959, A328962, A328963, A328965.

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

a(n)={if(n<1, 1, my(m=oo); fordiv(n, d, if(d<=n/d, m=min(m, 2^(n/d-d)*vecprod(primes(d))))); m)} \\ Andrew Howroyd, Nov 04 2019

a(p) = 2^p, for p prime. - Daniel Suteu, Nov 03 2019

a(n) = min_{d|n, d<=n/d} 2^(n/d-d)*A002110(d) for n > 0. - Andrew Howroyd, Nov 04 2019

More terms from Daniel Suteu, Nov 03 2019

A328957 Numbers k such that sigma_0(k) != omega(k) * Omega(k), where sigma_0 = A000005, omega = A001221, Omega = A001222.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 36, 37, 41, 42, 43, 47, 49, 53, 59, 61, 64, 66, 67, 70, 71, 72, 73, 78, 79, 81, 83, 89, 97, 100, 101, 102, 103, 105, 107, 108, 109, 110, 113, 114, 120, 121, 125, 127, 128, 130, 131, 137

Offset: 1

Gus Wiseman, Nov 01 2019

nonn

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   30: {1,2,3}
   31: {11}
   32: {1,1,1,1,1}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Nonzeros of A328958.
The complement is A328956.
Prime signature is A124010.
Omega-sequence is A323023.
omega(n) * Omega(n) is A113901(n).
(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. A060687, A070175, A090858, A112798, A303555, A320632, A328960, A328961, A328962, A328963, A328964, A328965.

Select[Range[100],DivisorSigma[0,#]!=PrimeOmega[#]*PrimeNu[#]&]

is(k) = {my(f = factor(k)); numdiv(f) != omega(f) * bigomega(f);} \\ Amiram Eldar, Jul 28 2024

A000005(a(n)) != A001222(a(n)) * A001221(a(n)).

A328960 Number of integer partitions of n whose number of nontrivial submultisets is greater than their number of distinct parts times their number of parts minus 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 6, 10, 18, 28, 45, 63, 93, 129, 178, 238, 321, 419, 551, 708, 911, 1158, 1472, 1845, 2316, 2883, 3583, 4421, 5453, 6680, 8180, 9964, 12122, 14687, 17771, 21418, 25788, 30949, 37092, 44324, 52906, 62980, 74885, 88832, 105243, 124429

Offset: 0

Gus Wiseman, Nov 02 2019

nonn

These partitions are conjectured to be precisely those that have a pair of multiset partitions such that no part of one is a submultiset of any part of the other (see A320632). For example, such a pair of partitions of {1,1,2,2} is ({{1,1},{2,2}}, {{1,2},{1,2}}).

			The a(6) = 1 through a(10) = 18 partitions:
  (2211)  (3211)   (3221)    (3321)     (3322)
          (22111)  (3311)    (4221)     (4321)
                   (4211)    (4311)     (4411)
                   (22211)   (5211)     (5221)
                   (32111)   (32211)    (5311)
                   (221111)  (33111)    (6211)
                             (42111)    (32221)
                             (222111)   (33211)
                             (321111)   (42211)
                             (2211111)  (43111)
                                        (52111)
                                        (222211)
                                        (322111)
                                        (331111)
                                        (421111)
                                        (2221111)
                                        (3211111)
                                        (22111111)
For example, the partition (4,2,2,1,1) has 16 nontrivial submultisets: {(1), (2), (4), (11), (21), ..., (2211), (4211), (4221)}, and 5 parts, 3 of which are distinct. Since 16 > (5 - 1) * 3 = 12, the partition (42211) is counted under a(10)

The Heinz numbers of these partitions are conjectured to be A320632.
A307409(n) is (omega(n) - 1) * nu(n).
A328958(n) is sigma_0(n) - omega(n) * nu(n).
A328959(n) is sigma_0(n) - 2 - (omega(n) - 1) * nu(n).
Cf. A008284, A032741, A116608, A328956, A328961, A328963.

Table[Length[Select[IntegerPartitions[n],0

cp's OEIS Frontend

A113901 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.

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A307409 a(n) = (A001222(n) - 1)*A001221(n).

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A320632 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.

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A328958 a(n) = d(n) - (omega(n) * bigomega(n)), where d (number of divisors) = A000005, omega = A001221, bigomega = A001222.

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A328959 a(n) = sigma_0(n) - 2 - (omega(n) - 1) * nu(n), where sigma_0 = A000005, nu = A001221, omega = A001222.

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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.

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PARI

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A328965 Smallest k such that (bigomega(k) - 1) * omega(k) = n, and 0 if none exists, where omega = A001221, bigomega = A001222.

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