A001222 -id:A001222 - cp's OEIS frontend

A076136 Numbers n such that Omega(n) = Omega(n-1) + Omega(n-2), where Omega(n) (A001222) denotes the number of prime factors of n, counting multiplicity.

3, 4, 8, 12, 16, 36, 40, 54, 63, 75, 88, 104, 112, 132, 135, 140, 150, 195, 200, 204, 208, 220, 252, 279, 280, 294, 328, 375, 390, 399, 405, 408, 416, 423, 444, 456, 464, 486, 510, 516, 520, 525, 558, 560, 592, 612, 615, 616, 620, 630, 636, 644, 656, 663, 680

Offset: 1

Joseph L. Pe, Oct 30 2002

nonn

			E.g. Omega(3) = 1 + 0 = Omega(2) + Omega(1). Omega(4) = 1 + 1 = Omega(3) + Omega(2).
8 is a term because Omega(8)=3=Omega(7)+Omega(6)=1+2=3

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A076137, A001222.

Omega[n_] := Apply[Plus, Transpose[FactorInteger[n]][[2]]]; l = {3}; Do[If[Omega[n] == Omega[n - 1] + Omega[n - 2], l = Append[l, n]], {n, 4, 1000}]; l
Flatten[Position[Partition[PrimeOmega[Range[700]],3,1],?(#[[1]]+#[[2]]==#[[3]]&),1,Heads->False]]+2 (* _Harvey P. Dale, Aug 24 2019 *)

j=[]; for(n=1,1000,if(bigomega(n)==bigomega(n-1)+bigomega(n-2),j=concat(j,n))); j

A079148 Primes p such that p-1 has at most 2 prime factors, counted with multiplicity; i.e., primes p such that bigomega(p-1) = A001222(p-1) <= 2.

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2459, 2579, 2819, 2879

Offset: 1

Cino Hilliard, Dec 27 2002

Sum of reciprocals ~ 1.477.

			83 is in the sequence because 83 - 1 = 2*41 has 2 prime factors.

Except for 2 and 3, this is identical to A005385.

Cf. A079147, A079149, A079151.

Select[Prime[Range[500]],PrimeOmega[#-1]<3&] (* Harvey P. Dale, May 17 2011 *)

s(n) = {sr=0; forprime(x=2,n, if(bigomega(x-1) < 3, print1(x" "); sr+=1.0/x; ); ); print(); print(sr); } \\ Lists primes p<=n such that p-1 has at most 2 prime factors.

A261256 Let S_k denote the sequence of numbers j such that A001222(j) - A001221(j) = k. Then a(n) is the n-th term of S_n.

Original entry on oeis.org

4, 24, 72, 160, 432, 896, 2592, 5632, 12800, 26624, 61440, 124416, 278528, 622592, 1376256, 2949120, 5971968, 12058624, 25690112, 60817408, 130023424, 262144000, 528482304, 1107296256, 2264924160, 4586471424, 9395240960, 19864223744, 40265318400, 83751862272

Offset: 1

Carlos Eduardo Olivieri, Aug 12 2015

nonn

S_0 would correspond to the squarefree numbers (A005117), that is, numbers j such that A001222(j) = A001221(j). Note that S_0 is excluded from the scheme. - Michel Marcus, Sep 21 2015

			For n = 1, S_1 = {4, 9, 12, 18, 20, 25, ...}, so a(1) = S_1(1) = 4.
For n = 2, S_2 = {8, 24, 27, 36, 40, 54, ...}, so a(2) = S_2(2) = 24.
For n = 3, S_3 = {16, 48, 72, 80, 81, 108, ...}, so a(3) = S_3(3) = 72.
For n = 4, S_4 = {32, 96, 144, 160, 216, 224, ...}, so a(4) = S_4(4) = 160.
For n = 5, S_5 = {64, 192, 288, 320, 432, 448, ...}, so a(5) = S_5(5) = 432.

Charlie Neder, Table of n, a(n) for n = 1..500

Cf. A001221, A001222.
Cf. A060687, A195086, A195087, A195088, A195089, A195090, A195091, A195092, A195093.
Cf. A046660, A257851, A264959.

a261256 n = a257851 n (n - 1)  -- Reinhard Zumkeller, Nov 29 2015

OutSeq = {}; For[i = 1, i <= 16, i++, l = Select[Range[10^2*2^i], PrimeOmega[#] - PrimeNu[#] == i &]; AppendTo[OutSeq, l[[i]]]]; OutSeq

a(n) = {ik = 1; nbk = 0; while (nbk != n, ik++; if (bigomega(ik) == omega(ik) + n, nbk++);); ik;} \\ Michel Marcus, Oct 06 2015

a(n+1) > 2*a(n).
a(n) >= 2^prime(n) for n < 5.
a(n) = A257851(n,n-1). - Reinhard Zumkeller, Nov 29 2015
a(n) = b(n)*2^(n+1), where b(n) consists of the values of k/2^excess(k) over odd k, sorted in ascending order. In particular, a(n) <= prime(n)*2^(n+1), with equality only when n = 2. - Charlie Neder, Jan 31 2019

a(17)-a(21) from Jon E. Schoenfield, Sep 12 2015

More terms from Charlie Neder, Jan 31 2019

A324570 Numbers where the sum of distinct prime indices (A066328) is equal to the number of prime factors counted with multiplicity (A001222).

Original entry on oeis.org

1, 2, 9, 12, 18, 40, 100, 112, 125, 240, 250, 352, 360, 392, 405, 540, 600, 672, 675, 810, 832, 900, 1008, 1125, 1350, 1372, 1500, 1512, 1701, 1875, 1936, 2112, 2176, 2240, 2250, 2268, 2352, 2401, 3168, 3402, 3528, 3750, 3969, 4752, 4802, 4864, 4992, 5292

Offset: 1

Gus Wiseman, Mar 07 2019

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. For example, 540 = prime(1)^2 * prime(2)^3 * prime(3)^1 has sum of distinct prime indices 1 + 2 + 3 = 6, while the number of prime factors counted with multiplicity is 2 + 3 + 1 = 6, so 540 belongs to the sequence.

Also Heinz numbers of the integer partitions counted by A114638. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    9: {2,2}
   12: {1,1,2}
   18: {1,2,2}
   40: {1,1,1,3}
  100: {1,1,3,3}
  112: {1,1,1,1,4}
  125: {3,3,3}
  240: {1,1,1,1,2,3}
  250: {1,3,3,3}
  352: {1,1,1,1,1,5}
  360: {1,1,1,2,2,3}
  392: {1,1,1,4,4}
  405: {2,2,2,2,3}
  540: {1,1,2,2,2,3}
  600: {1,1,1,2,3,3}
  672: {1,1,1,1,1,2,4}

Cf. A001221, A001222, A056239, A066328, A112798, A114638, A117144, A276078.

Cf. A109298, A324524, A324525, A324570, A324571, A324572.

with(numtheory):
q:= n-> is(add(pi(p), p=factorset(n))=bigomega(n)):
select(q, [$1..5600])[];  # Alois P. Heinz, Mar 07 2019

Select[Range[1000],Total[PrimePi/@First/@FactorInteger[#]]==PrimeOmega[#]&]

A066328(a(n)) = A001222(a(n)).

A328956 Numbers k such that sigma_0(k) = omega(k) * Omega(k), where sigma_0 = A000005, omega = A001221, Omega = A001222.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 01 2019

nonn

First differs from A084227 in having 60.

			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}

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

Zeros of A328958.
The complement is A328957.
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. A000040, A005117, 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)).

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

A277892 a(n) = A001222(A048675(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 2, 1, 4, 2, 5, 2, 2, 2, 6, 1, 7, 2, 2, 1, 8, 1, 3, 2, 2, 2, 9, 1, 10, 1, 3, 2, 3, 2, 11, 2, 2, 1, 12, 1, 13, 3, 3, 1, 14, 2, 4, 2, 3, 2, 15, 1, 3, 1, 3, 4, 16, 3, 17, 3, 3, 2, 4, 1, 18, 3, 3, 1, 19, 1, 20, 2, 2, 3, 4, 2, 21, 3, 3, 2, 22, 3, 3, 2, 2, 1, 23, 2, 4, 3, 5, 3, 4, 1, 24, 1, 3, 2, 25, 1, 26, 2, 2

Offset: 2

Antti Karttunen, Nov 08 2016

nonn

For n >= 3, a(n) = index of the row where n is located in array A277898.

Antti Karttunen (terms 2..3465) and Hans Havermann, Table of n, a(n) for n = 2..10000

Left inverse of A065091.
Cf. A001222, A048675.
Cf. A277319 (positions of ones).
Cf. A000040 (positions of records), A277900.
Cf. A277895 (ordinal transform from a(3) onward).
Cf. A277893, A277894, A277898.

A048675[n_] := If[n == 1, 0, Total[#[[2]]*2^(PrimePi[#[[1]]] - 1)& /@ FactorInteger[n]]];
a[n_] := PrimeOmega[A048675[n]];
Table[a[n], {n, 2, 105}] (* Jean-François Alcover, Jan 11 2022 *)

A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2;
A277892(n) = if(1==n,0,bigomega(A048675(n)));
for(n=1, 3465, write("b277892.txt", n, " ", A277892(n)));

from sympy import factorint, primepi, primefactors
def a001222(n): return 0 if n==1 else a001222(n//primefactors(n)[0]) + 1
def a048675(n):
    if n==1: return 0
    f=factorint(n)
    return sum(f[i]*2**(primepi(i) - 1) for i in f)
def a(n): return a001222(a048675(n))
print([a(n) for n in range(2, 101)]) # Indranil Ghosh, Jun 19 2017

(define (A277892 n) (if (= 1 n) 0 (A001222 (A048675 n))))

a(A019565(n)) = a(A260443(n)) = A001222(n).

For all n >= 2, a(A065091(n)) = n.

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

A340606 Numbers whose prime indices (A112798) are all divisors of the number of prime factors (A001222).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 16, 20, 24, 32, 36, 50, 54, 56, 64, 81, 84, 96, 125, 126, 128, 144, 160, 176, 189, 196, 216, 240, 256, 294, 324, 360, 384, 400, 416, 441, 486, 512, 540, 576, 600, 624, 686, 729, 810, 864, 896, 900, 936, 968, 1000, 1024, 1029, 1040, 1088, 1215

Offset: 1

Gus Wiseman, Jan 24 2021

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 sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
   9: {2,2}
  16: {1,1,1,1}
  20: {1,1,3}
  24: {1,1,1,2}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  50: {1,3,3}
  54: {1,2,2,2}
  56: {1,1,1,4}
  64: {1,1,1,1,1,1}
  81: {2,2,2,2}
  84: {1,1,2,4}
  96: {1,1,1,1,1,2}

Note: Heinz numbers are given in parentheses below.
The reciprocal version is A143773 (A316428).
These partitions are counted by A340693.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A003963 multiplies together the prime indices of n.
A018818 counts partitions of n into divisors of n (A326841).
A047993 counts balanced partitions (A106529).
A067538 counts partitions of n whose length divides n (A316413).
A056239 adds up the prime indices of n.
A061395 selects the maximum prime index.
A067538 counts partitions of n whose maximum divides n (A326836).
A072233 counts partitions by sum and length.
A112798 lists the prime indices of each positive integer.
A168659 = partitions whose length is divisible by their maximum (A340609).
A168659 = partitions whose maximum is divisible by their length (A340610).
A289509 lists numbers with relatively prime prime indices.
A326842 = partitions of n whose length and parts all divide n (A326847).
A326843 = partitions of n whose length and maximum both divide n (A326837).
A340852 have a factorization with factors dividing length.
Cf. A000720, A001222, A006141, A067539, A200750, A298423, A324925, A326149/A326155, A340608, A340827.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And@@IntegerQ/@(PrimeOmega[#]/primeMS[#])&]

cp's OEIS Frontend

A076136 Numbers n such that Omega(n) = Omega(n-1) + Omega(n-2), where Omega(n) (A001222) denotes the number of prime factors of n, counting multiplicity.

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A079148 Primes p such that p-1 has at most 2 prime factors, counted with multiplicity; i.e., primes p such that bigomega(p-1) = A001222(p-1) <= 2.

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A261256 Let S_k denote the sequence of numbers j such that A001222(j) - A001221(j) = k. Then a(n) is the n-th term of S_n.

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A324570 Numbers where the sum of distinct prime indices (A066328) is equal to the number of prime factors counted with multiplicity (A001222).

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A328956 Numbers k such that sigma_0(k) = omega(k) * Omega(k), where sigma_0 = A000005, omega = A001221, Omega = A001222.

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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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A277892 a(n) = A001222(A048675(n)).

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