A001222 -id:A001222 - cp's OEIS frontend

A275812 Sum of exponents larger than one in the prime factorization of n: A001222(n) - A056169(n).

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

Offset: 1

Antti Karttunen, Aug 11 2016

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A001222, A001694, A005117, A028234, A046660, A056169, A056170, A057521, A067029, A247180.
Differs from A212172 for the first time at n=36, where a(36)=4, while A212172(36)=2.
Cf. A085548, A136141.

Table[Total@ Map[Last, Select[FactorInteger@ n, Last@ # > 1 &] /. {} -> {{0, 0}}], {n, 120}] (* Michael De Vlieger, Aug 11 2016 *)

a(n) = my(f = factor(n)); sum(k=1, #f~, if (f[k,2] > 1, f[k,2])); \\ Michel Marcus, Jul 19 2017

sub a275812 { vecsum( grep {$> 1} map {$->[1]} factor_exp(shift) ); } # Dana Jacobsen, Aug 15 2016

from sympy import factorint, primefactors
def a001222(n):
    return 0 if n==1 else a001222(n//primefactors(n)[0]) + 1
def a056169(n):
    f=factorint(n)
    return 0 if n==1 else sum(1 for i in f if f[i]==1)
def a(n):
    return a001222(n) - a056169(n)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 19 2017

a(1) = 0, and for n > 1, if A067029(n)=1 [when n is one of the terms of A247180], a(n) = a(A028234(n)), otherwise a(n) = A067029(n)+a(A028234(n)).
a(n) = A001222(n) - A056169(n).
a(n) = A001222(A057521(n)). - Antti Karttunen, Jul 19 2017
From Amiram Eldar, Sep 28 2023: (Start)
Additive with a(p) = 0, and a(p^e) = e for e >= 2.
a(n) >= 0, with equality if and only if n is squarefree (A005117).
a(n) <= A001222(n), with equality if and only if n is powerful (A001694).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (1/p^2 + 1/(p*(p-1))) = A085548 + A136141 = 1.22540408909086062637... . (End)
a(n) = A046660(n) + A056170(n). - Amiram Eldar, Jan 09 2024

A242424 Bulgarian solitaire operation on partition list A112798: a(1) = 1, a(n) = A000040(A001222(n)) * A064989(n).

Original entry on oeis.org

1, 2, 4, 3, 6, 6, 10, 5, 12, 9, 14, 10, 22, 15, 18, 7, 26, 20, 34, 15, 30, 21, 38, 14, 27, 33, 40, 25, 46, 30, 58, 11, 42, 39, 45, 28, 62, 51, 66, 21, 74, 50, 82, 35, 60, 57, 86, 22, 75, 45, 78, 55, 94, 56, 63, 35, 102, 69, 106, 42, 118, 87, 100, 13, 99, 70, 122, 65

Offset: 1

Antti Karttunen, May 13 2014

nonn

In "Bulgarian solitaire" a deck of cards or another finite set of objects is divided into one or more piles, and the "Bulgarian operation" is performed by taking one card from each pile, and making a new pile of them, which is added to the remaining set of piles. Essentially, this operation is a function whose domain and range are unordered integer partitions (cf. A000041) and which preserves the total size of a partition (the sum of its parts). This sequence is induced when the operation is implemented on the partitions as ordered by the list A112798.
Please compare to the definition of A122111, which conjugates the partitions encoded with the same system.
a(n) is even if and only if n is either a prime or a multiple of three.
Conversely, a(n) is odd if and only if n is a nonprime not divisible by three.

Martin Gardner, Colossal Book of Mathematics, Chapter 34, Bulgarian Solitaire and Other Seemingly Endless Tasks, pp. 455-467, W. W. Norton & Company, 2001.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Ethan Akin and Morton Davis, "Bulgarian solitaire", American Mathematical Monthly 92 (4): 237-250. (1985).
Wikipedia, Bulgarian solitaire
Index entries for sequences computed from indices in prime factorization

Row 1 of A243070 (table which gives successive "recursive iterates" of this sequence and converges towards A122111).
Fixed points: A002110 (primorial numbers).
Cf. A000041, A243051, A243072, A243073, A226062, A242422, A242423, A122111, A105560, A000040, A001222, A064989, A056239.

(define (A242424 n) (if (<= n 1) n (* (A000040 (A001222 n)) (A064989 n))))

a(1) = 1, a(n) = A000040(A001222(n)) * A064989(n) = A105560(n) * A064989(n).
a(n) = A241909(A243051(A241909(n))).
a(n) = A243353(A226062(A243354(n))).
a(A000079(n)) = A000040(n) for all n.
A056239(a(n)) = A056239(n) for all n.

A105560 a(1) = 1, and for n >= 2, a(n) = prime(bigomega(n)), where prime(n) = A000040(n) and bigomega(n) = A001222(n).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 5, 3, 3, 2, 5, 2, 3, 3, 7, 2, 5, 2, 5, 3, 3, 2, 7, 3, 3, 5, 5, 2, 5, 2, 11, 3, 3, 3, 7, 2, 3, 3, 7, 2, 5, 2, 5, 5, 3, 2, 11, 3, 5, 3, 5, 2, 7, 3, 7, 3, 3, 2, 7, 2, 3, 5, 13, 3, 5, 2, 5, 3, 5, 2, 11, 2, 3, 5, 5, 3, 5, 2, 11, 7, 3, 2, 7, 3, 3, 3, 7, 2, 7, 3, 5, 3, 3, 3, 13, 2, 5, 5, 7

Offset: 1

Cino Hilliard, May 03 2005

From Antti Karttunen, Jul 21 2014: (Start)
a(n) divides A122111(n), A242424(n), A243072(n), A243073(n) because a(n) divides all the terms in column n of A243070.
a(2n-1) divides A243505(n) and a(2n-1)^2 divides A122111(2n-1).
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000040, A001222, A006530, A008578, A243070, A242424, A243072, A243073, A122111, A243505.

Table[Prime[Sum[FactorInteger[n][[i,2]],{i,1,Length[FactorInteger[n]]}]],{n,2,40}] (* Stefan Steinerberger, May 16 2007 *)

d(n) = for(x=2,n,print1(prime(bigomega(x))","))

from sympy import prime, primefactors
def a001222(n): return 0 if n==1 else a001222(n/primefactors(n)[0]) + 1
def a(n): return 1 if n==1 else prime(a001222(n)) # Indranil Ghosh, Jun 15 2017

a(1) = 1, and for n >= 2, a(n) = A000040(A001222(n)).
From Antti Karttunen, Jul 21 2014: (Start)
a(n) = A008578(1 + A001222(n)).
a(n) = A006530(A122111(n)).
a(n) = A122111(n) / A122111(A064989(n)).
a(2n-1) = A122111(2n-1) / A243505(n).
a(n) = A242424(n) / A064989(n).
(End)

a(1) = 1 prepended by Antti Karttunen, Jul 21 2014

A268375 Numbers k for which A001222(k) = A267116(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 25, 27, 28, 29, 31, 32, 37, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 59, 61, 63, 64, 67, 68, 71, 73, 75, 76, 79, 80, 81, 83, 89, 92, 97, 98, 99, 101, 103, 107, 109, 112, 113, 116, 117, 121, 124, 125, 127, 128, 131, 137, 139, 144, 147, 148, 149, 151, 153

Offset: 1

Antti Karttunen, Feb 03 2016

Numbers k whose prime factorization k = p_1^e_1 * ... * p_m^e_m contains no pair of exponents e_i and e_j (i and j distinct) whose base-2 representations have at least one shared digit-position in which both exponents have a 1-bit.
Equivalently, numbers k such that the factors in the (unique) factorization of k into powers of squarefree numbers with distinct exponents that are powers of two, are prime powers. For example, this factorization of 90 is 10^1 * 3^2, so 90 is not included, as 10 is not prime; whereas this factorization of 320 is 5^1 * 2^2 * 2^4, so 320 is included as 5 and 2 are both prime. - Peter Munn, Jan 16 2020
A225546 maps the set of terms 1:1 onto A138302. - Peter Munn, Jan 26 2020
Equivalently, numbers k for which A064547(k) = A331591(k). - Amiram Eldar, Dec 23 2023

			12 = 2^2 * 3^1 is included in the sequence as the exponents 2 ("10" in binary) and 1 ("01" in binary) have no 1-bits in the same position, and 18 = 2^1 * 3^2 is included for the same reason.
On the other hand, 24 = 2^3 * 3^1 is NOT included in the sequence as the exponents 3 ("11" in binary) and 1 ("01" in binary) have 1-bit in the same position 0.
720 = 2^4 * 3^2 * 5^1 is included as the exponents 1, 2 and 4 ("001", "010" and "100" in binary) have no 1-bits in shared positions.
Likewise, 10! = 3628800 = 2^8 * 3^4 * 5^2 * 7^1 is included as the exponents 1, 2, 4 and 8 ("0001", "0010", "0100" and "1000" in binary) have no 1-bits in shared positions. And similarly for any term of A191555.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Indices of zeros in A268374, also in A289618.
Cf. A001222, A064547, A267116, A046645, A289617, A331591.
Cf. A091862 (characteristic function), A268376 (complement).
Cf. A000961, A054753, A191555 (subsequences).
Related to A138302 via A225546.
Cf. also A318363 (a permutation).

{1}~Join~Select[Range@ 160, PrimeOmega@ # == BitOr @@ Map[Last, FactorInteger@ #] &] (* Michael De Vlieger, Feb 04 2016 *)

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.

A340608 The number of prime factors of n (A001222) is relatively prime to the maximum prime index of n (A061395).

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 19, 22, 23, 25, 27, 28, 29, 31, 32, 33, 34, 37, 40, 41, 42, 43, 44, 46, 47, 48, 51, 53, 55, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 76, 77, 79, 80, 82, 83, 85, 88, 89, 90, 93, 94, 97, 98, 99

Offset: 1

Gus Wiseman, Jan 27 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:
     2: {1}          22: {1,5}          44: {1,1,5}
     3: {2}          23: {9}            46: {1,9}
     4: {1,1}        25: {3,3}          47: {15}
     5: {3}          27: {2,2,2}        48: {1,1,1,1,2}
     7: {4}          28: {1,1,4}        51: {2,7}
     8: {1,1,1}      29: {10}           53: {16}
    10: {1,3}        31: {11}           55: {3,5}
    11: {5}          32: {1,1,1,1,1}    59: {17}
    12: {1,1,2}      33: {2,5}          60: {1,1,2,3}
    13: {6}          34: {1,7}          61: {18}
    15: {2,3}        37: {12}           62: {1,11}
    16: {1,1,1,1}    40: {1,1,1,3}      63: {2,2,4}
    17: {7}          41: {13}           64: {1,1,1,1,1,1}
    18: {1,2,2}      42: {1,2,4}        66: {1,2,5}
    19: {8}          43: {14}           67: {19}

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

Note: Heinz numbers are given in parentheses below.
These are the Heinz numbers of the partitions counted by A200750.
The case of equality is A047993 (A106529).
The divisible instead of coprime version is A168659 (A340609).
The dividing instead of coprime version is A168659 (A340610), with strict case A340828 (A340856).
A001222 counts prime factors.
A006141 counts partitions whose length equals their minimum (A324522).
A051424 counts singleton or pairwise coprime partitions (A302569).
A056239 adds up prime indices.
A061395 selects the maximum prime index.
A067538 counts partitions whose length divides their sum (A316413).
A067538 counts partitions whose maximum divides their sum (A326836).
A112798 lists the prime indices of each positive integer.
A259936 counts singleton or pairwise coprime factorizations.
A325134 = A001222 + A061395.
A326845 = A056239 * A061395.
A326849 counts partitions whose sum divides length times maximum (A326848).
A327516 counts pairwise coprime partitions (A302696).
Cf. A003114, A039900, A096401, A244990/A244991, A340606, A340653, A340691, A340787/A340788.

Select[Range[100],GCD[PrimeOmega[#],PrimePi[FactorInteger[#][[-1,1]]]]==1&]

A181591 a(n) = binomial(bigOmega(n),omega(n)), where omega = A001221 and bigOmega = A001222.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1, 4, 1, 3, 1, 3, 1, 1, 1, 6, 2, 1, 3, 3, 1, 1, 1, 5, 1, 1, 1, 6, 1, 1, 1, 6, 1, 1, 1, 3, 3, 1, 1, 10, 2, 3, 1, 3, 1, 6, 1, 6, 1, 1, 1, 4, 1, 1, 3, 6, 1, 1, 1, 3, 1, 1, 1, 10, 1, 1, 3, 3, 1, 1, 1, 10, 4, 1, 1, 4, 1, 1, 1, 6, 1, 4, 1, 3, 1, 1, 1, 15, 1, 3, 3, 6

Offset: 1

Reinhard Zumkeller, Nov 01 2010

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A001221, A001222, A007318, A136277.

Cf. A088529, A136565 (equal for n = 2..23).

a[n_] := Binomial[PrimeOmega[n], PrimeNu[n]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 29 2013 *)

a(n) = binomial(bigomega(n), omega(n)); \\ Michel Marcus, Jul 10 2022

a(n) = A007318(A001222(n),A001221(n)).
a(n) = A088529(n) = A136565(n) for 2 <= n < 24.
a(n) = A136277(A156552(n)). - Antti Karttunen, May 29 2017

A319899 Numbers whose number of prime factors with multiplicity (A001222) is the number of distinct prime factors (A001221) in the product of the prime indices (A003963).

Original entry on oeis.org

1, 3, 5, 7, 11, 15, 17, 19, 23, 26, 31, 33, 35, 39, 41, 51, 53, 55, 58, 59, 65, 67, 69, 74, 77, 83, 85, 86, 87, 91, 93, 94, 95, 97, 103, 109, 111, 119, 122, 123, 127, 129, 131, 142, 146, 155, 157, 158, 161, 165, 169, 177, 178, 179, 183, 185, 187, 191, 201, 202

Offset: 1

Gus Wiseman, Dec 17 2018

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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of square multiset multisystems, meaning the number of edges is equal to the number of distinct vertices.

			The sequence of multiset multisystems whose MM-numbers belong to the sequence begins:
   1: {}
   3: {{1}}
   5: {{2}}
   7: {{1,1}}
  11: {{3}}
  15: {{1},{2}}
  17: {{4}}
  19: {{1,1,1}}
  23: {{2,2}}
  26: {{},{1,2}}
  31: {{5}}
  33: {{1},{3}}
  35: {{2},{1,1}}
  39: {{1},{1,2}}
  41: {{6}}
  51: {{1},{4}}
  53: {{1,1,1,1}}
  55: {{2},{3}}
  58: {{},{1,3}}
  59: {{7}}
  65: {{2},{1,2}}
  67: {{8}}
  69: {{1},{2,2}}
  74: {{},{1,1,2}}

Cf. A003963, A057151, A064573, A120732, A319616, A319877, A320325, A322527, A322530.

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

A371168 Positive integers with fewer prime factors (A001222) than distinct divisors of prime indices (A370820).

Original entry on oeis.org

3, 5, 7, 11, 13, 14, 15, 17, 19, 21, 23, 26, 29, 31, 33, 35, 37, 38, 39, 41, 43, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 65, 67, 69, 70, 71, 73, 74, 76, 77, 78, 79, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 105, 106, 107, 109, 111, 113, 114, 115

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:
     3: {2}      35: {3,4}      59: {17}        86: {1,14}
     5: {3}      37: {12}       61: {18}        87: {2,10}
     7: {4}      38: {1,8}      65: {3,6}       89: {24}
    11: {5}      39: {2,6}      67: {19}        91: {4,6}
    13: {6}      41: {13}       69: {2,9}       93: {2,11}
    14: {1,4}    43: {14}       70: {1,3,4}     94: {1,15}
    15: {2,3}    46: {1,9}      71: {20}        95: {3,8}
    17: {7}      47: {15}       73: {21}        97: {25}
    19: {8}      49: {4,4}      74: {1,12}     101: {26}
    21: {2,4}    51: {2,7}      76: {1,1,8}    103: {27}
    23: {9}      52: {1,1,6}    77: {4,5}      105: {2,3,4}
    26: {1,6}    53: {16}       78: {1,2,6}    106: {1,16}
    29: {10}     55: {3,5}      79: {22}       107: {28}
    31: {11}     57: {2,8}      83: {23}       109: {29}
    33: {2,5}    58: {1,10}     85: {3,7}      111: {2,12}

The opposite version is A370348 counted by A371171.
The version for equality is A370802, counted by A371130, strict A371128.
The RHS is A370820, for prime factors instead of divisors A303975.
For divisors instead of prime factors on the LHS we get A371166.
The complement is counted by A371169.
The weak version is A371170.
Partitions of this type are counted by A371173.
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, A370814, A371127.

Mathematica
```
Select[Range[100],PrimeOmega[#]
				
```

A001222(a(n)) < A370820(a(n)).

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.

Original entry on oeis.org

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

cp's OEIS Frontend

A275812 Sum of exponents larger than one in the prime factorization of n: A001222(n) - A056169(n).

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Formula

A242424 Bulgarian solitaire operation on partition list A112798: a(1) = 1, a(n) = A000040(A001222(n)) * A064989(n).

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A105560 a(1) = 1, and for n >= 2, a(n) = prime(bigomega(n)), where prime(n) = A000040(n) and bigomega(n) = A001222(n).

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PARI

Python

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Extensions

A268375 Numbers k for which A001222(k) = A267116(k).

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

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Mathematica

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A340608 The number of prime factors of n (A001222) is relatively prime to the maximum prime index of n (A061395).

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A181591 a(n) = binomial(bigOmega(n),omega(n)), where omega = A001221 and bigOmega = A001222.

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A319899 Numbers whose number of prime factors with multiplicity (A001222) is the number of distinct prime factors (A001221) in the product of the prime indices (A003963).

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