A001222 -id:A001222 - cp's OEIS frontend

A076191 First differences of A001222.

1, 0, 1, -1, 1, -1, 2, -1, 0, -1, 2, -2, 1, 0, 2, -3, 2, -2, 2, -1, 0, -1, 3, -2, 0, 1, 0, -2, 2, -2, 4, -3, 0, 0, 2, -3, 1, 0, 2, -3, 2, -2, 2, 0, -1, -1, 4, -3, 1, -1, 1, -2, 3, -2, 2, -2, 0, -1, 3, -3, 1, 1, 3, -4, 1, -2, 2, -1, 1, -2, 4, -4, 1, 1, 0, -1, 1, -2, 4, -1, -2, -1, 3, -2, 0, 0, 2, -3, 3, -2, 1, -1, 0, 0, 4, -5, 2, 0, 1, -3

Offset: 1

Joseph L. Pe, Nov 03 2002

a(A045920(n)) = 0. - Reinhard Zumkeller, Mar 19 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001222, A045920.

a076191 n = a076191_list !! (n-1)
a076191_list = zipWith (-) (tail a001222_list) a001222_list
-- Reinhard Zumkeller, Mar 20 2012

Omega[n_] := Apply[Plus, Transpose[FactorInteger[n]][[2]]]; Flatten[Append[{1}, Table[Omega[n + 1] - Omega[n], {n, 2, 100}]]]

a(n) = bigomega(n + 1) - bigomega(n); \\ Indranil Ghosh, Mar 15 2017

a(n) = Omega(n+1)-Omega(n), where Omega(n) (A001222) denotes the number of prime factors of n, counting multiplicity.

G.f.: ((1 - x)/x)*Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k)). - Ilya Gutkovskiy, Mar 15 2017

Name changed by Arkadiusz Wesolowski, Jul 27 2012

A102467 Positive integers k such that d(k) <> Omega(k) + omega(k), where d = A000005, Omega = A001222 and omega = A001221.

Original entry on oeis.org

1, 12, 18, 20, 24, 28, 30, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 66, 68, 70, 72, 75, 76, 78, 80, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 126, 130, 132, 135, 136, 138, 140, 144, 147, 148, 150, 152, 153, 154, 156

Offset: 1

Reinhard Zumkeller, Jan 09 2005

nonn

These are the numbers which are neither prime powers (>1) nor semiprimes. - M. F. Hasler, Jan 31 2008

For n > 1, positive integers k with a composite divisor, d < k, that is relatively prime to k/d. For example 12 is in the sequence since 4 (composite) is coprime to 12/4 = 3. - Wesley Ivan Hurt, Apr 25 2020

			10 is not in the sequence since d(10) = 4 is equal to Omega(10) + omega(10) = 2 + 2 = 4.
12 is in the sequence since d(12) = 6 is not equal to Omega(12) + omega(12) = 3 + 2 = 5. - _Wesley Ivan Hurt_, Apr 25 2020

T. D. Noe, Table of n, a(n) for n = 1..1000
Joerg Arndt, Matters Computational (The Fxtbook), section 40.4.1.3 "Testing for irreducibility without GCD computations", pp. 839-840.

Cf. A000005 (tau), A001221 (omega), A001222 (Omega).

Cf. A102466, A135767.

a102467 n = a102467_list !! (n-1)
a102467_list = [x | x <- [1..], a000005 x /= a001221 x + a001222 x]
-- Reinhard Zumkeller, Dec 14 2012

with(numtheory):
q:= n-> is(tau(n)<>bigomega(n)+nops(factorset(n))):
select(q, [$1..200])[];  # Alois P. Heinz, Jul 14 2023

Select[Range[200], DivisorSigma[0, #] != PrimeOmega[#] + PrimeNu[#]&] (* Jean-François Alcover, Jun 22 2018 *)

is(n)=my(f=factor(n)[,2]); #f!=1 && f!=[1,1]~ \\ Charles R Greathouse IV, Oct 19 2015

def is_A102467(n) :
    return sloane.A001221(n) != 1 and sloane.A001222(n) != 2
def A102467_list(n) :
    return [k for k in (1..n) if is_A102467(k)]
A102467_list(156)  # Peter Luschny, Feb 07 2012

Complement of A102466; A000005(a(n)) <> A001221(a(n)) + A001222(a(n)).

For n > 1, A086971(a(n)) > 1. - Reinhard Zumkeller, Dec 14 2012

Name changed by Wesley Ivan Hurt, Apr 25 2020

A144494 a(n) = 0 if n is prime, otherwise A001222(n).

Original entry on oeis.org

0, 0, 0, 2, 0, 2, 0, 3, 2, 2, 0, 3, 0, 2, 2, 4, 0, 3, 0, 3, 2, 2, 0, 4, 2, 2, 3, 3, 0, 3, 0, 5, 2, 2, 2, 4, 0, 2, 2, 4, 0, 3, 0, 3, 3, 2, 0, 5, 2, 3, 2, 3, 0, 4, 2, 4, 2, 2, 0, 4, 0, 2, 3, 6, 2, 3, 0, 3, 2, 3, 0, 5, 0, 2, 3, 3, 2, 3, 0, 5, 4, 2, 0, 4, 2, 2, 2, 4, 0, 4, 2, 3, 2, 2, 2, 6, 0, 3, 3, 4, 0, 3, 0, 4, 3

Offset: 1

N. J. A. Sloane, Dec 12 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A001221, A001222, A087624, A055636.

with(numtheory); f:=n->if isprime(n) then 0 else bigomega(n); fi;

Table[If[PrimeQ[n], 0, PrimeOmega[n]], {n,1,100}] (* G. C. Greubel, Apr 25 2017 *)

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

A333484 Sort all positive integers, first by sum of prime indices (A056239), then by decreasing number of prime indices (A001222).

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 5, 16, 12, 9, 10, 7, 32, 24, 18, 20, 14, 15, 11, 64, 48, 36, 40, 27, 28, 30, 21, 22, 25, 13, 128, 96, 72, 80, 54, 56, 60, 42, 44, 45, 50, 26, 33, 35, 17, 256, 192, 144, 160, 108, 112, 120, 81, 84, 88, 90, 100, 52, 63, 66, 70, 75, 34, 39, 49, 55, 19

Offset: 0

Gus Wiseman, May 10 2020

A refinement of A215366.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			Triangle begins:
    1
    2
    4   3
    8   6   5
   16  12   9  10   7
   32  24  18  20  14  15  11
   64  48  36  40  27  28  30  21  22  25  13
  128  96  72  80  54  56  60  42  44  45  50  26  33  35  17

OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
Ignoring length gives A215366 (graded Heinz numbers).
Sorting by increasing length gives A333483.
Number of prime indices is A001222.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in (sum/length/colex) order are A036037.
Sum of prime indices is A056239.
Reverse-lexicographically ordered partitions are A080577.
Sorting reversed partitions by Heinz number gives A112798.
Lexicographically ordered partitions are A193073.
Sorting partitions by Heinz number gives A296150.
Cf. A124734, A129129, A211992, A228100, A333219, A334301, A334433, A334434, A334439, A334441, A334442.

Join@@@Table[Sort[Times@@Prime/@#&/@IntegerPartitions[n,{k}]],{n,0,8},{k,n,0,-1}]

A360617 Half the number of prime factors of n (counted with multiplicity, A001222), rounded up.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 1, 2, 1, 3, 2, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Mar 08 2023

nonn

			The prime indices of 378 are {1,2,2,2,4}, so a(378) = ceiling(5/2) = 3.

Positions of 0's and 1's are 1 and A037143.
Positions of first appearances are A081294.
Rounding down instead of up gives A360616.
A112798 lists prime indices, length A001222, sum A056239, median* A360005.
A360673 counts multisets by right sum (exclusive), inclusive A360671.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A000040, A000302, A001248, A026424, A168645, A359912, A360006, A360457.

Mathematica
```
Table[Ceiling[PrimeOmega[n]/2],{n,100}]
```

A371170 Positive integers with at most as many prime factors (A001222) as distinct divisors of prime indices (A370820).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 92

Offset: 1

Gus Wiseman, Mar 16 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 105 are {2,3,4}, and there are 3 prime factors (3,5,7) and 4 distinct divisors of prime indices (1,2,3,4), so 105 is in the sequence.
The terms together with their prime indices begin:
     1: {}       22: {1,5}      42: {1,2,4}    63: {2,2,4}
     2: {1}      23: {9}        43: {14}       65: {3,6}
     3: {2}      25: {3,3}      45: {2,2,3}    66: {1,2,5}
     5: {3}      26: {1,6}      46: {1,9}      67: {19}
     6: {1,2}    28: {1,1,4}    47: {15}       69: {2,9}
     7: {4}      29: {10}       49: {4,4}      70: {1,3,4}
     9: {2,2}    30: {1,2,3}    51: {2,7}      71: {20}
    10: {1,3}    31: {11}       52: {1,1,6}    73: {21}
    11: {5}      33: {2,5}      53: {16}       74: {1,12}
    13: {6}      34: {1,7}      55: {3,5}      75: {2,3,3}
    14: {1,4}    35: {3,4}      57: {2,8}      76: {1,1,8}
    15: {2,3}    37: {12}       58: {1,10}     77: {4,5}
    17: {7}      38: {1,8}      59: {17}       78: {1,2,6}
    19: {8}      39: {2,6}      61: {18}       79: {22}
    21: {2,4}    41: {13}       62: {1,11}     82: {1,13}

The complement is A370348, counted by A371171.
The case of equality is A370802, counted by A371130, strict A371128.
The RHS is A370820, for prime factors instead of divisors A303975.
The strict version is A371168 counted by A371173.
The opposite version is A371169.
Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, indices A112798, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A003963, A319899, A355737, A355739, A355741, A370808, A371127.

Select[Range[100],PrimeOmega[#]<=Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]

A374246 Number of prime factors of n counted with multiplicity (A001222) minus the greatest number of runs possible in a permutation of them (A373957).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jul 07 2024

nonn

a(n) = 0 iff n has separable prime factors (A335433). A multiset is separable iff it has a permutation that is an anti-run (meaning there are no adjacent equal parts).

			The runs of the 4 permutations of the prime factors of 24 are:
  ((2,2,2),(3))
  ((2,2),(3),(2))
  ((2),(3),(2,2))
  ((3),(2,2,2))
The longest have length 3, so a(24) = 4 - 3 = 1.

Using the minimum instead of maximum number of runs gives A046660.
Positions of first appearances are A151821 (powers of 2 except 2 itself).
Positions of positive terms are A335448, complement A335433.
This is an opposite version of A373957.
The sister-sequence A374247 uses A001221 instead of A001222.
This is the number of zeros at the end of row n of A374252.
A001221 counts distinct prime factors, A001222 with multiplicity.
A008480 counts permutations of prime factors.
A027746 lists prime factors, row-sums A001414.
A027748 is run-compression of prime factors, row-sums A008472.
A304038 is run-compression of prime indices, row-sums A066328.
A374250 maximizes sum of run-compression, for indices A373956.
Cf. A003242, A007947, A026549, A056239, A124767, A246655, A316413, A333755.

prifacs[n_]:=If[n==1,{}, Flatten[ConstantArray@@@FactorInteger[n]]];
Table[PrimeOmega[n]-Max@@Table[Length[Split[y]], {y,Permutations[prifacs[n]]}],{n,100}]

a(n) = A001222(n) - A373957(n).

A068936 Numbers having the sum of distinct prime factors not greater than the sum of exponents in prime factorization, A008472(k) <= A001222(k).

Original entry on oeis.org

1, 4, 8, 16, 27, 32, 48, 64, 72, 81, 96, 108, 128, 144, 162, 192, 216, 243, 256, 288, 320, 324, 384, 432, 486, 512, 576, 640, 648, 729, 768, 800, 864, 972, 1024, 1152, 1280, 1296, 1458, 1536, 1600, 1728, 1792, 1944, 2000, 2048, 2187, 2304, 2560, 2592, 2916

Offset: 1

Reinhard Zumkeller, Mar 08 2002

nonn

The product of any two terms is also a term. - Amiram Eldar, May 14 2025

			a(5) = 27 = 3^3, 3 = 3.
a(10) = 81 = 3^4, 3 < 4.
a(100) = 16000 = 2^7 * 5^3,  2+5 < 7+3.
a(1000) = 10321920 = 2^15 * 3^2 * 5 * 7, 2+3+5+7 < 15+2+1+1.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)

Cf. A001222, A008472, A068935, A068937, A068938.

Subsequences: A054411, A100716, A257278.

a068936 n = a068936_list !! (n-1)
a068936_list = [x | x <- [1..], a008472 x <= a001222 x]
-- Reinhard Zumkeller, Nov 10 2013

fQ[n_] := Block[{f = FactorInteger@n}, Plus @@ Last /@ f >= Plus @@ First /@ f]; Select[ Range@3000, fQ@ # &] (* Robert G. Wilson v, Jan 16 2006 *)
Select[Range@ 3000, First@ Differences@ Map[Total, Transpose@ FactorInteger@ #] >= 0 &] (* Michael De Vlieger, Dec 08 2016 *)

isok(k) = {my(f = factor(k)); vecsum(f[,1]) <= bigomega(f);} \\ Amiram Eldar, May 14 2025

More terms from Robert G. Wilson v, Jan 16 2006

A124508 a(n) = 2^BigO(n) * 3^omega(n), where BigO = A001222 and omega = A001221, the numbers of prime factors of n with and without repetitions.

Original entry on oeis.org

1, 6, 6, 12, 6, 36, 6, 24, 12, 36, 6, 72, 6, 36, 36, 48, 6, 72, 6, 72, 36, 36, 6, 144, 12, 36, 24, 72, 6, 216, 6, 96, 36, 36, 36, 144, 6, 36, 36, 144, 6, 216, 6, 72, 72, 36, 6, 288, 12, 72, 36, 72, 6, 144, 36, 144, 36, 36, 6, 432, 6, 36, 72, 192, 36, 216, 6, 72, 36, 216, 6, 288, 6

Offset: 1

Reinhard Zumkeller, Nov 04 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Factorization.
Eric Weisstein's World of Mathematics, Smooth number.

Cf. A000079, A000244, A000400, A001221, A001222, A003586, A005117, A007283, A061142, A074816, A124509, A124510, A124511, A124512.

Table[2^PrimeOmega[n] 3^PrimeNu[n],{n,80}] (* Harvey P. Dale, Mar 26 2013 *)

a(n) = my(f = factor(n)); 2^bigomega(f) * 3^omega(f); \\ Amiram Eldar, Jul 11 2023

Multiplicative with p^e -> 3*2^e, p prime and e>0.
a(n) = A061142(n)*A074816(n) = A000079(A001222(n))*A000244(A001221(n)).
A124509 gives the range: A124509(n) = a(A124510(n)) and a(m) <> a(A124510(n)) for m < A124510(n).
For primes p, q with p <> q: a(p) = 6; a(p*q) = 36; a(p^k) = 3*2^k, k>0.
For squarefree numbers m: a(m) = 6^omega(m).
A001222(a(n)) = A001222(n)+1; A001221(a(n)) = 2 for n > 1.
A124511(n) = a(a(n)); A124512(n) = a(a(a(n))).

cp's OEIS Frontend

A076191 First differences of A001222.

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A102467 Positive integers k such that d(k) <> Omega(k) + omega(k), where d = A000005, Omega = A001222 and omega = A001221.

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A144494 a(n) = 0 if n is prime, otherwise A001222(n).

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

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A333484 Sort all positive integers, first by sum of prime indices (A056239), then by decreasing number of prime indices (A001222).

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A360617 Half the number of prime factors of n (counted with multiplicity, A001222), rounded up.

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A371170 Positive integers with at most as many prime factors (A001222) as distinct divisors of prime indices (A370820).

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A374246 Number of prime factors of n counted with multiplicity (A001222) minus the greatest number of runs possible in a permutation of them (A373957).

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