A344414 -id:A344414 - cp's OEIS frontend

A344295 Heinz numbers of partitions of 2*n with at most n parts, none greater than 3, for some n.

1, 3, 9, 10, 25, 27, 30, 75, 81, 90, 100, 225, 243, 250, 270, 300, 625, 675, 729, 750, 810, 900, 1000, 1875, 2025, 2187, 2250, 2430, 2500, 2700, 3000, 5625, 6075, 6250, 6561, 6750, 7290, 7500, 8100, 9000, 10000, 15625, 16875, 18225, 18750, 19683, 20250, 21870

Offset: 1

Gus Wiseman, May 15 2021

nonn

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

			The sequence of terms together with their prime indices begins:
      1: {}
      3: {2}
      9: {2,2}
     10: {1,3}
     25: {3,3}
     27: {2,2,2}
     30: {1,2,3}
     75: {2,3,3}
     81: {2,2,2,2}
     90: {1,2,2,3}
    100: {1,1,3,3}
    225: {2,2,3,3}
    243: {2,2,2,2,2}
    250: {1,3,3,3}
    270: {1,2,2,2,3}
    300: {1,1,2,3,3}

These partitions are counted by A001399.
Allowing any number of parts and sum gives A051037.
Allowing parts > 3 and any length gives A300061.
Not requiring the sum of prime indices to be even gives A344293.
Allowing any number of parts (but still with even sum) gives A344297.
Allowing parts > 3 gives A344413.
A001358 lists semiprimes.
A025065 counts partitions of n with at least n/2 parts, ranked by A344296.
A035363 counts partitions of n of length n/2, ranked by A340387.
A056239 adds up prime indices, row sums of A112798.
A110618 counts partitions of n with at most n/2 parts, ranked by A344291.
A344414 counts partitions of n with all parts >= n/2, ranked by A344296.
Cf. A030229, A080193, A244990, A261144, A266755, A279622, A334433, A344294.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],EvenQ[Total[primeMS[#]]]&&PrimeOmega[#]<=Total[primeMS[#]]/2&&Max@@primeMS[#]<=3&]

Intersection of A300061 (even Heinz weight), A344291 (Omega > half Heinz weight), and A051037 (5-smooth).

A344292 Numbers m whose sum of prime indices A056239(m) is even and is at most twice the number of prime factors counted with multiplicity A001222(m).

Original entry on oeis.org

1, 3, 4, 9, 10, 12, 16, 27, 28, 30, 36, 40, 48, 64, 81, 84, 88, 90, 100, 108, 112, 120, 144, 160, 192, 208, 243, 252, 256, 264, 270, 280, 300, 324, 336, 352, 360, 400, 432, 448, 480, 544, 576, 624, 640, 729, 756, 768, 784, 792, 810, 832, 840, 880, 900, 972

Offset: 1

Gus Wiseman, May 22 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.

Also Heinz numbers of integer partitions of even numbers m with at least m/2 parts, counted by A000070 riffled with 0's, or A025065 with odd positions zeroed out.

			The sequence of terms together with their prime indices begins:
       1: {}                 84: {1,1,2,4}
       3: {2}                88: {1,1,1,5}
       4: {1,1}              90: {1,2,2,3}
       9: {2,2}             100: {1,1,3,3}
      10: {1,3}             108: {1,1,2,2,2}
      12: {1,1,2}           112: {1,1,1,1,4}
      16: {1,1,1,1}         120: {1,1,1,2,3}
      27: {2,2,2}           144: {1,1,1,1,2,2}
      28: {1,1,4}           160: {1,1,1,1,1,3}
      30: {1,2,3}           192: {1,1,1,1,1,1,2}
      36: {1,1,2,2}         208: {1,1,1,1,6}
      40: {1,1,1,3}         243: {2,2,2,2,2}
      48: {1,1,1,1,2}       252: {1,1,2,2,4}
      64: {1,1,1,1,1,1}     256: {1,1,1,1,1,1,1,1}
      81: {2,2,2,2}         264: {1,1,1,2,5}

These are the Heinz numbers of partitions counted by A000070 and A025065.
A subset of A300061 (sum of prime indices is even).
The conjugate opposite version is A320924, counted by A209816.
The conjugate opposite version allowing odds is A322109, counted by A110618.
The case of equality is A340387, counted by A000041.
The opposite version allowing odd weights is A344291, counted by A110618.
Allowing odd weights gives A344296, counted by A025065.
The opposite version is A344413, counted by A209816.
The conjugate version allowing odd weights is A344414, counted by A025065.
The case of equality in the conjugate case is A344415, counted by A035363.
The conjugate version is A344416, counted by A000070.
A001222 counts prime factors with multiplicity.
A027187 counts partitions of even length, ranked by A028260.
A056239 adds up prime indices, row sums of A112798.
A058696 counts partitions of even numbers, ranked by A300061.
A301987 lists numbers whose sum of prime indices equals their product.
A330950 counts partitions of n with Heinz number divisible by n.
A334201 adds up all prime indices except the greatest.
Cf. A001414, A067538, A316413, A316428, A325037, A325038, A325044, A338914, A344294, A344297.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],EvenQ[Total[primeMS[#]]]&&PrimeOmega[#]>=Total[primeMS[#]]/2&]

Members m of A300061 such that A056239(m) <= 2*A001222(m).

A366318 Heinz numbers of integer partitions that are of length 2 or begin with n/2, where n is the sum of all parts.

Original entry on oeis.org

4, 6, 9, 10, 12, 14, 15, 21, 22, 25, 26, 30, 33, 34, 35, 38, 39, 40, 46, 49, 51, 55, 57, 58, 62, 63, 65, 69, 70, 74, 77, 82, 84, 85, 86, 87, 91, 93, 94, 95, 106, 111, 112, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 154, 155, 158, 159

Offset: 1

Gus Wiseman, Oct 08 2023

nonn

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 terms together with their prime indices begin:
     4: {1,1}      38: {1,8}         77: {4,5}
     6: {1,2}      39: {2,6}         82: {1,13}
     9: {2,2}      40: {1,1,1,3}     84: {1,1,2,4}
    10: {1,3}      46: {1,9}         85: {3,7}
    12: {1,1,2}    49: {4,4}         86: {1,14}
    14: {1,4}      51: {2,7}         87: {2,10}
    15: {2,3}      55: {3,5}         91: {4,6}
    21: {2,4}      57: {2,8}         93: {2,11}
    22: {1,5}      58: {1,10}        94: {1,15}
    25: {3,3}      62: {1,11}        95: {3,8}
    26: {1,6}      63: {2,2,4}      106: {1,16}
    30: {1,2,3}    65: {3,6}        111: {2,12}
    33: {2,5}      69: {2,9}        112: {1,1,1,1,4}
    34: {1,7}      70: {1,3,4}      115: {3,9}
    35: {3,4}      74: {1,12}       118: {1,17}

The first condition alone is A001358, counted by A004526.
The complement of the first condition is A100959, counted by A058984.
The partitions with these Heinz numbers are counted by A238628.
The second condition alone is A344415, counted by A035363.
The complement of the second condition is A366319, counted by A086543.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
A322109 ranks partitions of n with no part > n/2, counted by A110618.
A334201 adds up all prime indices except the greatest.
A344296 solves for k in A001222(k) >= A056239(k)/2, counted by A025065.
Cf. A001414, A316413, A320924, A325044, A340387, A344414, A344416.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Length[prix[#]]==2||MemberQ[prix[#],Total[prix[#]]/2]&]

Union of A001358 and A344415.

A344417 Number of palindromic factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 7, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, May 22 2021

nonn

A palindrome is a sequence that is the same whether it is read forward or in reverse. A palindromic factorization of n is a finite multiset of positive integers > 1 with product n that can be permuted into a palindrome.

			The palindromic factorizations for n = 2, 4, 16, 36, 64, 144:
  (2)  (4)    (16)       (36)       (64)           (144)
       (2*2)  (4*4)      (6*6)      (8*8)          (12*12)
              (2*2*4)    (2*2*9)    (4*4*4)        (4*4*9)
              (2*2*2*2)  (3*3*4)    (2*2*16)       (4*6*6)
                         (2*2*3*3)  (2*2*4*4)      (2*2*36)
                                    (2*2*2*2*4)    (3*3*16)
                                    (2*2*2*2*2*2)  (2*2*6*6)
                                                   (3*3*4*4)
                                                   (2*2*2*2*9)
                                                   (2*2*3*3*4)
                                                   (2*2*2*2*3*3)

Positions of 1's are A005117.
The case of palindromic compositions is A016116.
The additive version (palindromic partitions) is A025065.
The case of palindromic prime signature is A242414.
The case of palindromic plane trees is A319436.
A001055 counts factorizations.
A229153 ranks non-palindromic partitions.
A265640 ranks palindromic partitions.
Cf. A000041, A000070, A000188, A004526, A030229, A056503, A057567, A082293, A344414.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
palQ[y_]:=Select[Permutations[y],#==Reverse[#]&]!={};
Table[Length[Select[facs[n],palQ]],{n,50}]

a(2^n) = A025065(n).

a(n) = A057567(A000188(n)). - Andrew Howroyd, May 22 2021

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A344295 Heinz numbers of partitions of 2*n with at most n parts, none greater than 3, for some n.

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A344292 Numbers m whose sum of prime indices A056239(m) is even and is at most twice the number of prime factors counted with multiplicity A001222(m).

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A366318 Heinz numbers of integer partitions that are of length 2 or begin with n/2, where n is the sum of all parts.

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A344417 Number of palindromic factorizations of n.

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