A244990 -id:A244990 - cp's OEIS frontend

A244991 Numbers whose greatest prime factor is a prime with an odd index; n such that A006530(n) is in A031368.

2, 4, 5, 8, 10, 11, 15, 16, 17, 20, 22, 23, 25, 30, 31, 32, 33, 34, 40, 41, 44, 45, 46, 47, 50, 51, 55, 59, 60, 62, 64, 66, 67, 68, 69, 73, 75, 77, 80, 82, 83, 85, 88, 90, 92, 93, 94, 97, 99, 100, 102, 103, 109, 110, 115, 118, 119, 120, 121, 123, 124, 125, 127, 128

Offset: 1

Antti Karttunen, Jul 21 2014

nonn

Equally, numbers n for which A061395(n) is odd.
A122111 maps each one of these numbers to a unique term of A026424 and vice versa.
If the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), these are the Heinz numbers of partitions whose greatest part is odd, counted by A027193. - Gus Wiseman, Feb 08 2021

			From _Gus Wiseman_, Feb 08 2021: (Start)
The sequence of terms together with their prime indices begins:
      2: {1}           32: {1,1,1,1,1}     64: {1,1,1,1,1,1}
      4: {1,1}         33: {2,5}           66: {1,2,5}
      5: {3}           34: {1,7}           67: {19}
      8: {1,1,1}       40: {1,1,1,3}       68: {1,1,7}
     10: {1,3}         41: {13}            69: {2,9}
     11: {5}           44: {1,1,5}         73: {21}
     15: {2,3}         45: {2,2,3}         75: {2,3,3}
     16: {1,1,1,1}     46: {1,9}           77: {4,5}
     17: {7}           47: {15}            80: {1,1,1,1,3}
     20: {1,1,3}       50: {1,3,3}         82: {1,13}
     22: {1,5}         51: {2,7}           83: {23}
     23: {9}           55: {3,5}           85: {3,7}
     25: {3,3}         59: {17}            88: {1,1,1,5}
     30: {1,2,3}       60: {1,1,2,3}       90: {1,2,2,3}
     31: {11}          62: {1,11}          92: {1,1,9}
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10001

Complement: A244990.
Cf. A006530, A026424, A031368, A122111, A244321, A244322, A244989.
Looking at least instead of greatest prime index gives A026804.
The partitions with these Heinz numbers are counted by A027193.
The case where Omega is odd also is A340386.
A001222 counts prime factors.
A056239 adds up prime indices.
A300063 ranks partitions of odd numbers.
A061395 selects maximum prime index.
A066208 ranks partitions into odd parts.
A112798 lists the prime indices of each positive integer.
A340931 ranks odd-length partitions of odd numbers.
Cf. A000009, A058695, A072233, A160786, A300272, A340101, A340385, A340604.

Select[Range[100],OddQ[PrimePi[FactorInteger[#][[-1,1]]]]&] (* Gus Wiseman, Feb 08 2021 *)

For all n, A244989(a(n)) = n.

A340610 Numbers whose number of prime factors (A001222) divides their greatest prime index (A061395).

Original entry on oeis.org

2, 3, 5, 6, 7, 9, 11, 13, 14, 17, 19, 20, 21, 23, 26, 29, 30, 31, 35, 37, 38, 39, 41, 43, 45, 47, 49, 50, 52, 53, 56, 57, 58, 59, 61, 65, 67, 71, 73, 74, 75, 78, 79, 83, 84, 86, 87, 89, 91, 92, 95, 97, 101, 103, 106, 107, 109, 111, 113, 117, 122, 125, 126, 127

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}        29: {10}       56: {1,1,1,4}
     3: {2}        30: {1,2,3}    57: {2,8}
     5: {3}        31: {11}       58: {1,10}
     6: {1,2}      35: {3,4}      59: {17}
     7: {4}        37: {12}       61: {18}
     9: {2,2}      38: {1,8}      65: {3,6}
    11: {5}        39: {2,6}      67: {19}
    13: {6}        41: {13}       71: {20}
    14: {1,4}      43: {14}       73: {21}
    17: {7}        45: {2,2,3}    74: {1,12}
    19: {8}        47: {15}       75: {2,3,3}
    20: {1,1,3}    49: {4,4}      78: {1,2,6}
    21: {2,4}      50: {1,3,3}    79: {22}
    23: {9}        52: {1,1,6}    83: {23}
    26: {1,6}      53: {16}       84: {1,1,2,4}

Robert Israel, Table of n, a(n) for n = 1..10000

Note: Heinz numbers are given in parentheses below.
The case of equality is A047993 (A106529).
The case where all parts are multiples, not just the maximum part, is A143773 (A316428), with strict case A340830, while the case of factorizations is A340853.
These are the Heinz numbers of certain partitions counted by A168659.
The reciprocal version is A340609.
The squarefree case is A340828 (A340856).
A001222 counts prime factors.
A006141 counts partitions whose length equals their minimum (A324522).
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.
A200750 counts partitions with length coprime to maximum (A340608).
A325134 = A001222 + A061395.
A326845 = A056239 * A061395.
Cf. A244990/A244991, A326837, A326849 (A326848), A340653, A340691, A340693 (A340606), A340787/A340788.

filter:= proc(n) local F,m,g,t;
  F:= ifactors(n)[2];
  m:= add(t[2],t=F);
  g:= numtheory:-pi(max(seq(t[1],t=F)));
  g mod m = 0;
end proc:
select(filter, [$2..1000]); # Robert Israel, Feb 08 2021

Select[Range[2,100],Divisible[PrimePi[FactorInteger[#][[-1,1]]],PrimeOmega[#]]&]

A001222(a(n)) divides A061395(a(n)).

A340601 Number of integer partitions of n of even rank.

Original entry on oeis.org

1, 1, 0, 3, 1, 5, 3, 11, 8, 18, 16, 34, 33, 57, 59, 98, 105, 159, 179, 262, 297, 414, 478, 653, 761, 1008, 1184, 1544, 1818, 2327, 2750, 3480, 4113, 5137, 6078, 7527, 8899, 10917, 12897, 15715, 18538, 22431, 26430, 31805, 37403, 44766, 52556, 62620, 73379

Offset: 0

Gus Wiseman, Jan 21 2021

nonn

The Dyson rank of a nonempty partition is its maximum part minus its number of parts. For this sequence, the rank of an empty partition is 0.

			The a(1) = 1 through a(9) = 18 partitions (empty column indicated by dot):
  (1)  .  (3)    (22)  (5)      (42)    (7)        (44)      (9)
          (21)         (41)     (321)   (43)       (62)      (63)
          (111)        (311)    (2211)  (61)       (332)     (81)
                       (2111)           (322)      (521)     (333)
                       (11111)          (331)      (2222)    (522)
                                        (511)      (4211)    (531)
                                        (2221)     (32111)   (711)
                                        (4111)     (221111)  (4221)
                                        (31111)              (4311)
                                        (211111)             (6111)
                                        (1111111)            (32211)
                                                             (33111)
                                                             (51111)
                                                             (222111)
                                                             (411111)
                                                             (3111111)
                                                             (21111111)
                                                             (111111111)

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61.
FindStat, St000145: The Dyson rank of a partition

Note: Heinz numbers are given in parentheses below.
The positive case is A101708 (A340605).
The Heinz numbers of these partitions are A340602.
The odd version is A340692 (A340603).
- Rank -
A047993 counts partitions of rank 0 (A106529).
A072233 counts partitions by sum and length.
A101198 counts partitions of rank 1 (A325233).
A101707 counts partitions of odd positive rank (A340604).
A101708 counts partitions of even positive rank (A340605).
A257541 gives the rank of the partition with Heinz number n.
A340653 counts factorizations of rank 0.
- Even -
A024430 counts set partitions of even length.
A027187 counts partitions of even length (A028260).
A027187 (also) counts partitions of even maximum (A244990).
A034008 counts compositions of even length.
A035363 counts partitions into even parts (A066207).
A052841 counts ordered set partitions of even length.
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts even-length partitions of even numbers (A340784).
A339846 counts factorizations of even length.
Cf. A000041, A006141, A039900, A064174, A067538, A117409, A200750, A324516.

b:= proc(n, i, r) option remember; `if`(n=0, 1-max(0, r),
      `if`(i<1, 0, b(n, i-1, r) +b(n-i, min(n-i, i), 1-
      `if`(r<0, irem(i, 2), r))))
    end:
a:= n-> b(n$2, -1):
seq(a(n), n=0..55);  # Alois P. Heinz, Jan 22 2021

Table[If[n==0,1,Length[Select[IntegerPartitions[n],EvenQ[Max[#]-Length[#]]&]]],{n,0,30}]
(* Second program: *)
b[n_, i_, r_] := b[n, i, r] = If[n == 0, 1 - Max[0, r], If[i < 1, 0, b[n, i - 1, r] + b[n - i, Min[n - i, i], 1 - If[r < 0, Mod[i, 2], r]]]];
a[n_] := b[n, n, -1];
a /@ Range[0, 55] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

p_q(k) = {prod(j=1, k, 1-q^j); }
GB_q(N, M)= {if(N>=0 && M>=0,  p_q(N+M)/(p_q(M)*p_q(N)), 0 ); }
A_q(N) = {my(q='q+O('q^N), g=1+sum(i=1,N, sum(j=1,N/i, q^(i*j) * ( ((1/2)*(1+(-1)^(i+j))) + sum(k=1,N-(i*j), ((q^k)*GB_q(k,i-2)) * ((1/2)*(1+(-1)^(i+j+k)))))))); Vec(g)}
A_q(50) \\ John Tyler Rascoe, Apr 15 2024

G.f.: 1 + Sum_{i, j>0} q^(i*j) * ( (1+(-1)^(i+j))/2 + Sum_{k>0} q^k * q_binomial(k,i-2) * (1+(-1)^(i+j+k))/2 ). - John Tyler Rascoe, Apr 15 2024

a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)). - Vaclav Kotesovec, Apr 17 2024

A344296 Numbers with at least as many prime factors (counted with multiplicity) as half their sum of prime indices.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 48, 54, 56, 60, 64, 72, 80, 81, 84, 88, 90, 96, 100, 108, 112, 120, 128, 144, 160, 162, 168, 176, 180, 192, 200, 208, 216, 224, 240, 243, 252, 256, 264, 270, 280, 288, 300, 320, 324, 336, 352

Offset: 1

Gus Wiseman, May 16 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.

These are the Heinz numbers of certain partitions counted by A025065, but different from palindromic partitions, which have Heinz numbers A265640.

			The sequence of terms together with their prime indices begins:
      1: {}            30: {1,2,3}
      2: {1}           32: {1,1,1,1,1}
      3: {2}           36: {1,1,2,2}
      4: {1,1}         40: {1,1,1,3}
      6: {1,2}         48: {1,1,1,1,2}
      8: {1,1,1}       54: {1,2,2,2}
      9: {2,2}         56: {1,1,1,4}
     10: {1,3}         60: {1,1,2,3}
     12: {1,1,2}       64: {1,1,1,1,1,1}
     16: {1,1,1,1}     72: {1,1,1,2,2}
     18: {1,2,2}       80: {1,1,1,1,3}
     20: {1,1,3}       81: {2,2,2,2}
     24: {1,1,1,2}     84: {1,1,2,4}
     27: {2,2,2}       88: {1,1,1,5}
     28: {1,1,4}       90: {1,2,2,3}

The case with difference at least 1 is A322136.
The case of equality is A340387, counted by A000041 or A035363.
The opposite version is A344291, counted by A110618.
The conjugate version is A344414, with even-weight case A344416.
A025065 counts palindromic partitions, ranked by A265640.
A056239 adds up prime indices, row sums of A112798.
A300061 lists numbers whose sum of prime indices is even.
Cf. A001399, A002865, A025147, A027336, A036036, A067712, A244990, A261144, A325691, A344293, A344295.

Select[Range[100],PrimeOmega[#]>=Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/2&]

A056239(a(n)) <= 2*A001222(a(n)).

a(n) = A322136(n)/4.

A340609 Numbers whose number of prime factors (A001222) is divisible by their greatest prime index (A061395).

Original entry on oeis.org

2, 4, 6, 8, 9, 16, 20, 24, 30, 32, 36, 45, 50, 54, 56, 64, 75, 81, 84, 96, 125, 126, 128, 140, 144, 160, 176, 189, 196, 210, 216, 240, 256, 264, 294, 315, 324, 350, 360, 384, 396, 400, 416, 440, 441, 486, 490, 512, 525, 540, 576, 594, 600, 616, 624, 660, 686

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.

If n is a term, then so is n^k for k > 1. - Robert Israel, Feb 08 2021

			The sequence of terms together with their prime indices begins:
      2: {1}             64: {1,1,1,1,1,1}      216: {1,1,1,2,2,2}
      4: {1,1}           75: {2,3,3}            240: {1,1,1,1,2,3}
      6: {1,2}           81: {2,2,2,2}          256: {1,1,1,1,1,1,1,1}
      8: {1,1,1}         84: {1,1,2,4}          264: {1,1,1,2,5}
      9: {2,2}           96: {1,1,1,1,1,2}      294: {1,2,4,4}
     16: {1,1,1,1}      125: {3,3,3}            315: {2,2,3,4}
     20: {1,1,3}        126: {1,2,2,4}          324: {1,1,2,2,2,2}
     24: {1,1,1,2}      128: {1,1,1,1,1,1,1}    350: {1,3,3,4}
     30: {1,2,3}        140: {1,1,3,4}          360: {1,1,1,2,2,3}
     32: {1,1,1,1,1}    144: {1,1,1,1,2,2}      384: {1,1,1,1,1,1,1,2}
     36: {1,1,2,2}      160: {1,1,1,1,1,3}      396: {1,1,2,2,5}
     45: {2,2,3}        176: {1,1,1,1,5}        400: {1,1,1,1,3,3}
     50: {1,3,3}        189: {2,2,2,4}          416: {1,1,1,1,1,6}
     54: {1,2,2,2}      196: {1,1,4,4}          440: {1,1,1,3,5}
     56: {1,1,1,4}      210: {1,2,3,4}          441: {2,2,4,4}

Robert Israel, Table of n, a(n) for n = 1..10000

Note: Heinz numbers are given in parentheses below.
The case of equality is A047993 (A106529).
These are the Heinz numbers of certain partitions counted by A168659.
The reciprocal version is A340610, with strict case A340828 (A340856).
If all parts (not just the greatest) are divisors we get A340693 (A340606).
A001222 counts prime factors.
A006141 counts partitions whose length equals their minimum (A324522).
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.
A200750 counts partitions with length coprime to maximum (A340608).
A325134 = A001222 + A061395.
A326845 = A056239 * A061395.
Cf. A039900, A064174, A143773, A244990/A244991, A326837, A326849 (A326848), A340653, A340787/A340788.

filter:= proc(n) local F,m,g,t;
  F:= ifactors(n)[2];
  m:= add(t[2],t=F);
  g:= numtheory:-pi(max(seq(t[1],t=F)));
  m mod g = 0;
end proc:
seelect(filter, [$2..1000]); # Robert Israel, Feb 08 2021

Select[Range[2,100],Divisible[PrimeOmega[#],PrimePi[FactorInteger[#][[-1,1]]]]&]

A061395(a(n)) divides A001222(a(n)).

A347044 Greatest divisor of n with half (rounded up) as many prime factors (counting multiplicity) as n.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 4, 3, 5, 11, 6, 13, 7, 5, 4, 17, 9, 19, 10, 7, 11, 23, 6, 5, 13, 9, 14, 29, 15, 31, 8, 11, 17, 7, 9, 37, 19, 13, 10, 41, 21, 43, 22, 15, 23, 47, 12, 7, 25, 17, 26, 53, 9, 11, 14, 19, 29, 59, 15, 61, 31, 21, 8, 13, 33, 67, 34, 23, 35, 71

Offset: 1

Gus Wiseman, Aug 16 2021

nonn

Appears to contain each positive integer at least once, but only a finite number of times.

			The divisors of 123456 with half bigomega are: 16, 24, 5144, 7716, so a(123456) = 7716.

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

The greatest divisor without the condition is A006530 (smallest: A020639).
Divisors of this type are counted by A096825 (exact: A345957).
The case of powers of 2 is A163403.
The smallest divisor of this type is given by A347043 (exact: A347045).
The exact version is A347046.
A000005 counts divisors.
A001221 counts distinct prime factors.
A001222 counts all prime factors (also called bigomega).
A038548 counts inferior (or superior) divisors (strict: A056924).
A056239 adds up prime indices, row sums of A112798.
A207375 lists central divisors (min: A033676, max: A033677).
A340387 lists numbers whose sum of prime indices is twice bigomega.
A340609 lists numbers whose maximum prime index divides bigomega.
A340610 lists numbers whose maximum prime index is divisible by bigomega.
A347042 counts divisors d|n such that bigomega(d) divides bigomega(n).
Cf. A000720, A027746, A060775, A106529, A217581, A244990, A335433, A335448.

Table[Max[Select[Divisors[n],PrimeOmega[#]==Ceiling[PrimeOmega[n]/2]&]],{n,100}]
a[n_] := Module[{p = Flatten[Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]], np}, np = Length[p]; Times @@ p[[Floor[np/2] + 1;; np]]]; Array[a, 100] (* Amiram Eldar, Nov 02 2024 *)

from sympy import divisors, factorint
def a(n):
    npf = len(factorint(n, multiple=True))
    for d in divisors(n)[::-1]:
        if len(factorint(d, multiple=True)) == (npf+1)//2: return d
    return 1
print([a(n) for n in range(1, 72)]) # Michael S. Branicky, Aug 18 2021

from math import prod
from sympy import factorint
def A347044(n):
    fs = factorint(n,multiple=True)
    l = len(fs)
    return prod(fs[l//2:]) # Chai Wah Wu, Aug 20 2021

a(n) = Product_{k=floor(A001222(n)/2)+1..A001222(n)} A027746(n,k). - Amiram Eldar, Nov 02 2024

A340602 Heinz numbers of integer partitions of even rank.

Original entry on oeis.org

1, 2, 5, 6, 8, 9, 11, 14, 17, 20, 21, 23, 24, 26, 30, 31, 32, 35, 36, 38, 39, 41, 44, 45, 47, 49, 50, 54, 56, 57, 58, 59, 65, 66, 67, 68, 73, 74, 75, 80, 81, 83, 84, 86, 87, 91, 92, 95, 96, 97, 99, 102, 103, 104, 106, 109, 110, 111, 120, 122, 124, 125, 126, 127

Offset: 1

Gus Wiseman, Jan 21 2021

nonn

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is 0.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence of partitions with their Heinz numbers begins:
     1: ()           31: (11)           58: (10,1)
     2: (1)          32: (1,1,1,1,1)    59: (17)
     5: (3)          35: (4,3)          65: (6,3)
     6: (2,1)        36: (2,2,1,1)      66: (5,2,1)
     8: (1,1,1)      38: (8,1)          67: (19)
     9: (2,2)        39: (6,2)          68: (7,1,1)
    11: (5)          41: (13)           73: (21)
    14: (4,1)        44: (5,1,1)        74: (12,1)
    17: (7)          45: (3,2,2)        75: (3,3,2)
    20: (3,1,1)      47: (15)           80: (3,1,1,1,1)
    21: (4,2)        49: (4,4)          81: (2,2,2,2)
    23: (9)          50: (3,3,1)        83: (23)
    24: (2,1,1,1)    54: (2,2,2,1)      84: (4,2,1,1)
    26: (6,1)        56: (4,1,1,1)      86: (14,1)
    30: (3,2,1)      57: (8,2)          87: (10,2)

FindStat, St000145: The Dyson rank of a partition

Taking only length gives A001222.
Taking only maximum part gives A061395.
These partitions are counted by A340601.
The complement is A340603.
The case of positive rank is A340605.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A101198 counts partitions of rank 1 (A325233).
A101707 counts partitions of odd positive rank (A340604).
A101708 counts partitions of even positive rank (A340605).
A257541 gives the rank of the partition with Heinz number n.
A324516 counts partitions with rank = maximum minus minimum part (A324515).
A340653 counts factorizations of rank 0.
A340692 counts partitions of odd rank (A340603).
- Even -
A024430 counts set partitions of even length.
A027187 counts partitions of even length (A028260).
A027187 (also) counts partitions of even maximum (A244990).
A034008 counts compositions of even length.
A035363 counts partitions into even parts (A066207).
A052841 counts ordered set partitions of even length.
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts even-length partitions of even numbers (A340784).
A339846 counts factorizations of even length.
Cf. A000041, A006141, A056239, A072233, A112798, A168659, A325134, A326836, A326845, A340386, A340387.

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

Either n = 1 or A061395(n) - A001222(n) is even.

A344291 Numbers whose sum of prime indices is at least twice their number of prime indices (counted with multiplicity).

Original entry on oeis.org

1, 3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85

Offset: 1

Gus Wiseman, May 15 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: {}       25: {3,3}      43: {14}       62: {1,11}
     3: {2}      26: {1,6}      44: {1,1,5}    63: {2,2,4}
     5: {3}      27: {2,2,2}    45: {2,2,3}    65: {3,6}
     7: {4}      28: {1,1,4}    46: {1,9}      66: {1,2,5}
     9: {2,2}    29: {10}       47: {15}       67: {19}
    10: {1,3}    30: {1,2,3}    49: {4,4}      68: {1,1,7}
    11: {5}      31: {11}       50: {1,3,3}    69: {2,9}
    13: {6}      33: {2,5}      51: {2,7}      70: {1,3,4}
    14: {1,4}    34: {1,7}      52: {1,1,6}    71: {20}
    15: {2,3}    35: {3,4}      53: {16}       73: {21}
    17: {7}      37: {12}       55: {3,5}      74: {1,12}
    19: {8}      38: {1,8}      57: {2,8}      75: {2,3,3}
    21: {2,4}    39: {2,6}      58: {1,10}     76: {1,1,8}
    22: {1,5}    41: {13}       59: {17}       77: {4,5}
    23: {9}      42: {1,2,4}    61: {18}       78: {1,2,6}
For example, the prime indices of 45 are {2,2,3} with sum 7 >= 2*3, so 45 is in the sequence.

The partitions with these Heinz numbers are counted by A110618.
The conjugate version is A322109.
The case of equality is A340387, counted by A035363.
The 5-smooth case is A344293, with non-3-smooth case A344294.
The opposite version is A344296.
The conjugate opposite version is A344414.
The conjugate case of equality is A344415.
A001221 counts distinct prime indices.
A001222 counts prime indices with multiplicity.
A056239 adds up prime indices, row sums of A112798.
Cf. A001358, A025065, A084127, A244990, A344292.

Select[Range[100],PrimeOmega[#]<=Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/2&]

A056239(a(n)) >= 2*A001222(a(n)).

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&]

A340784 Heinz numbers of even-length integer partitions of even numbers.

Original entry on oeis.org

1, 4, 9, 10, 16, 21, 22, 25, 34, 36, 39, 40, 46, 49, 55, 57, 62, 64, 81, 82, 84, 85, 87, 88, 90, 91, 94, 100, 111, 115, 118, 121, 129, 133, 134, 136, 144, 146, 155, 156, 159, 160, 166, 169, 183, 184, 187, 189, 194, 196, 198, 203, 205, 206, 210, 213, 218, 220

Offset: 1

Gus Wiseman, Jan 30 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are positive integers whose number of prime indices and sum of prime indices are both even, counting multiplicity in both cases.

A multiplicative semigroup: if m and n are in the sequence, then so is m*n. - Antti Karttunen, Jul 28 2024

			The sequence of partitions together with their Heinz numbers begins:
      1: ()            57: (8,2)            118: (17,1)
      4: (1,1)         62: (11,1)           121: (5,5)
      9: (2,2)         64: (1,1,1,1,1,1)    129: (14,2)
     10: (3,1)         81: (2,2,2,2)        133: (8,4)
     16: (1,1,1,1)     82: (13,1)           134: (19,1)
     21: (4,2)         84: (4,2,1,1)        136: (7,1,1,1)
     22: (5,1)         85: (7,3)            144: (2,2,1,1,1,1)
     25: (3,3)         87: (10,2)           146: (21,1)
     34: (7,1)         88: (5,1,1,1)        155: (11,3)
     36: (2,2,1,1)     90: (3,2,2,1)        156: (6,2,1,1)
     39: (6,2)         91: (6,4)            159: (16,2)
     40: (3,1,1,1)     94: (15,1)           160: (3,1,1,1,1,1)
     46: (9,1)        100: (3,3,1,1)        166: (23,1)
     49: (4,4)        111: (12,2)           169: (6,6)
     55: (5,3)        115: (9,3)            183: (18,2)

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

Note: A-numbers of Heinz-number sequences are in parentheses below.
The case of prime powers is A056798.
These partitions are counted by A236913.
The odd version is A160786 (A340931).
A000009 counts partitions into odd parts (A066208).
A001222 counts prime factors.
A047993 counts balanced partitions (A106529).
A056239 adds up prime indices.
A058695 counts partitions of odd numbers (A300063).
A061395 selects the maximum prime index.
A072233 counts partitions by sum and length.
A112798 lists the prime indices of each positive integer.
- Even -
A027187 counts partitions of even length/maximum (A028260/A244990).
A034008 counts compositions of even length.
A035363 counts partitions into even parts (A066207).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A339846 counts factorizations of even length.
A340601 counts partitions of even rank (A340602).
A340785 counts factorizations into even factors.
A340786 counts even-length factorizations into even factors.
Cf. A026424, A257541, A300272, A326837, A326845, A340385 (A340386), A340604, A353331 (characteristic function), A353332, A353333, A353334.
Squares (A000290) is a subsequence.
Not a subsequence of A329609 (30 is the first term of A329609 not occurring here, and 210 is the first term here not present in A329609).
Positions of even terms in A373381.

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

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
A353331(n) = ((!(bigomega(n)%2)) && (!(A056239(n)%2)));
isA340784(n) = A353331(n); \\ Antti Karttunen, Apr 14 2022

Intersection of A028260 and A300061.

cp's OEIS Frontend

A244991 Numbers whose greatest prime factor is a prime with an odd index; n such that A006530(n) is in A031368.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A340610 Numbers whose number of prime factors (A001222) divides their greatest prime index (A061395).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A340601 Number of integer partitions of n of even rank.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A344296 Numbers with at least as many prime factors (counted with multiplicity) as half their sum of prime indices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A340609 Numbers whose number of prime factors (A001222) is divisible by their greatest prime index (A061395).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A347044 Greatest divisor of n with half (rounded up) as many prime factors (counting multiplicity) as n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Python

Formula

A340602 Heinz numbers of integer partitions of even rank.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs