A325259 -id:A325259 - cp's OEIS frontend

A325244 Number of integer partitions of n with one fewer distinct multiplicities than distinct parts.

0, 0, 0, 1, 1, 2, 3, 4, 7, 12, 16, 21, 33, 38, 50, 75, 87, 111, 150, 185, 244, 307, 373, 461, 585, 702, 856, 1043, 1255, 1498, 1822, 2143, 2565, 3064, 3607, 4251, 5064, 5920, 6953, 8174, 9503, 11064, 12927, 14921, 17320, 19986, 23067, 26485, 30499, 34894

Offset: 0

Gus Wiseman, Apr 15 2019

nonn

For example, (32211) has two distinct multiplicities (1, 2) and three distinct parts (1, 2, 3) so is counted under a(9).

The Heinz numbers of these partitions are given by A325259.

			The a(3) = 1 through a(10) = 16 partitions:
  (21)  (31)  (32)  (42)    (43)    (53)     (54)      (64)
              (41)  (51)    (52)    (62)     (63)      (73)
                    (2211)  (61)    (71)     (72)      (82)
                            (3211)  (3221)   (81)      (91)
                                    (3311)   (3321)    (3322)
                                    (4211)   (4221)    (4411)
                                    (32111)  (4311)    (5221)
                                             (5211)    (5311)
                                             (32211)   (6211)
                                             (42111)   (32221)
                                             (222111)  (33211)
                                             (321111)  (42211)
                                                       (43111)
                                                       (52111)
                                                       (421111)
                                                       (3211111)

Cf. A008284, A071625, A090858, A098859, A116608, A117571, A127002, A325242, A325259, A325270.

Table[Length[Select[IntegerPartitions[n],Length[Union[#]]==Length[Union[Length/@Split[#]]]+1&]],{n,0,30}]

A371781 Numbers with biquanimous prime signature.

Original entry on oeis.org

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 126, 129, 132, 133, 134, 140, 141, 142, 143, 145, 146, 150, 155, 156, 158, 159

Offset: 1

Gus Wiseman, Apr 09 2024

nonn

First differs from A320911 in lacking 900.
First differs from A325259 in having 1 and lacking 120.
A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 (aerated) and ranked by A357976.
Also numbers n with a unitary divisor d|n having exactly half as many prime factors as n, counting multiplicity.

			The prime signature of 120 is (3,1,1), which is not biquanimous, so 120 is not in the sequence.

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

A number's prime signature is given by A124010.
For prime indices we have A357976, counted by A002219 aerated.
The complement for prime indices is A371731, counted by A371795, A006827.
The complement is A371782, counted by A371840.
Partitions of this type are counted by A371839.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A237258 aerated counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A371783 counts k-quanimous partitions.
A371791 counts biquanimous sets, complement A371792.
Cf. A035470, A064914, A232466, A299702, A336137, A371737, A371796.
Subsequence of A028260.

biquanimous:= proc(L) local s,x,i,P; option remember;
  s:= convert(L,`+`); if s::odd then return false fi;
  P:= mul(1+x^i,i=L);
  coeff(P,x,s/2) > 0
end proc:
select(n -> biquanimous(ifactors(n)[2][..,2]), [$1..200]); # Robert Israel, Apr 22 2024

g[n_]:=Select[Divisors[n],GCD[#,n/#]==1&&PrimeOmega[#]==PrimeOmega[n/#]&];
Select[Range[100],g[#]!={}&]
(* second program: *)
q[n_] := Module[{e = FactorInteger[n][[;; , 2]], sum, x}, sum = Plus @@ e; EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, e}], x][[1 + sum/2]] > 0]; q[1] = True; Select[Range[200], q] (* Amiram Eldar, Jul 24 2024 *)

A325241 Numbers > 1 whose maximum prime exponent is one greater than their minimum.

Original entry on oeis.org

12, 18, 20, 28, 44, 45, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 108, 116, 117, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 171, 172, 175, 180, 188, 198, 200, 204, 207, 212, 220, 228, 234, 236, 242, 244, 245, 252, 260, 261, 268, 275, 276, 279

Offset: 1

Gus Wiseman, Apr 15 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose maximum multiplicity is one greater than their minimum (counted by A325279).

The asymptotic density of this sequence is 1/zeta(3) - 1/zeta(2) = A088453 - A059956 = 0.22398... . - Amiram Eldar, Jan 30 2023

			The sequence of terms together with their prime indices begins:
  12: {1,1,2}
  18: {1,2,2}
  20: {1,1,3}
  28: {1,1,4}
  44: {1,1,5}
  45: {2,2,3}
  50: {1,3,3}
  52: {1,1,6}
  60: {1,1,2,3}
  63: {2,2,4}
  68: {1,1,7}
  72: {1,1,1,2,2}
  75: {2,3,3}
  76: {1,1,8}
  84: {1,1,2,4}
  90: {1,2,2,3}
  92: {1,1,9}
  98: {1,4,4}
  99: {2,2,5}

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Positions of 1's in A062977. Supersequence of A054753, A096156.
Cf. A001221, A001222, A001694, A051903, A051904, A052485, A056239, A112798, A118914, A325240, A325259, A325270, A325279.
Cf. A059956, A088453.

Select[Range[100],Max@@FactorInteger[#][[All,2]]-Min@@FactorInteger[#][[All,2]]==1&]
Select[Range[300],  Min[e = FactorInteger[#][[;; , 2]]] +1 == Max[e] &] (* Amiram Eldar, Jan 30 2023 *)

is(n)={my(e=factor(n)[,2]); n>1 && vecmin(e) + 1 == vecmax(e); } \\ Amiram Eldar, Jan 30 2023

from sympy import factorint
def ok(n):
    e = sorted(factorint(n).values())
    return n > 1 and max(e) == 1 + min(e)
print([k for k in range(280) if ok(k)]) # Michael S. Branicky, Dec 18 2021

A051903(a(n)) - A051904(a(n)) = 1.

A325270 Numbers with 1 fewer distinct prime exponents than (not necessarily distinct) prime factors.

Original entry on oeis.org

4, 6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 25, 26, 28, 33, 34, 35, 38, 39, 44, 45, 46, 49, 50, 51, 52, 55, 57, 58, 62, 63, 65, 68, 69, 74, 75, 76, 77, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 106, 111, 115, 116, 117, 118, 119, 121, 122, 123, 124, 129, 133

Offset: 1

Gus Wiseman, Apr 18 2019

nonn

Also Heinz numbers of integer partitions with 1 fewer distinct multiplicities than parts, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The enumeration of these partitions by sum is given by A117571.

Also numbers whose sorted prime signature is (1,1), (2), or (1,2). - Gus Wiseman, Jul 03 2019

			The sequence of terms together with their prime indices begins:
    4: {1,1}
    6: {1,2}
    9: {2,2}
   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}
   25: {3,3}
   26: {1,6}
   28: {1,1,4}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   38: {1,8}
   39: {2,6}
   44: {1,1,5}

Cf. A001221, A001222, A000961, A005117, A060687, A062770, A071625, A072774, A090858, A117571, A118914, A130091, A325244, A325259.

Select[Range[100],PrimeOmega[#]==Length[Union[Last/@FactorInteger[#]]]+1&]

A325281 Numbers of the form ab, aab, or aabc where a, b, and c are distinct primes. Numbers with sorted prime signature (1,1), (1,2), or (1,1,2).

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 26, 28, 33, 34, 35, 38, 39, 44, 45, 46, 50, 51, 52, 55, 57, 58, 60, 62, 63, 65, 68, 69, 74, 75, 76, 77, 82, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 98, 99, 106, 111, 115, 116, 117, 118, 119, 122, 123, 124, 126, 129, 132

Offset: 1

Gus Wiseman, Apr 18 2019

nonn

Also numbers whose adjusted frequency depth is one plus their number of prime factors counted with multiplicity. The adjusted frequency depth of a positive integer n is 0 if n = 1, and otherwise it is one plus the number of times one must apply A181819 to reach a prime number, where A181819(k = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of k. For example, 180 has adjusted frequency depth 5 because we have: 180 -> 18 -> 6 -> 4 -> 3.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose adjusted frequency depth is equal to their length plus 1. The enumeration of these partitions by sum is given by A127002.

			The sequence of terms together with their prime indices and their omega-sequences (see A323023) begins:
   6:     {1,2} (2,2,1)
  10:     {1,3} (2,2,1)
  12:   {1,1,2} (3,2,2,1)
  14:     {1,4} (2,2,1)
  15:     {2,3} (2,2,1)
  18:   {1,2,2} (3,2,2,1)
  20:   {1,1,3} (3,2,2,1)
  21:     {2,4} (2,2,1)
  22:     {1,5} (2,2,1)
  26:     {1,6} (2,2,1)
  28:   {1,1,4} (3,2,2,1)
  33:     {2,5} (2,2,1)
  34:     {1,7} (2,2,1)
  35:     {3,4} (2,2,1)
  38:     {1,8} (2,2,1)
  39:     {2,6} (2,2,1)
  44:   {1,1,5} (3,2,2,1)
  45:   {2,2,3} (3,2,2,1)
  46:     {1,9} (2,2,1)
  50:   {1,3,3} (3,2,2,1)
  51:     {2,7} (2,2,1)
  52:   {1,1,6} (3,2,2,1)
  55:     {3,5} (2,2,1)
  57:     {2,8} (2,2,1)
  58:    {1,10} (2,2,1)
  60: {1,1,2,3} (4,3,2,2,1)

Cf. A056239, A112798, A118914, A127002, A181819, A323023, A325246, A325259, A325266, A325270, A325277, A325282.

Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).

fdadj[n_Integer]:=If[n==1,0,Length[NestWhileList[Times@@Prime/@Last/@FactorInteger[#]&,n,!PrimeQ[#]&]]];
Select[Range[100],fdadj[#]==PrimeOmega[#]+1&]

A325284 Numbers whose prime indices form an initial interval with a single hole: (1, 2, ..., x, x + 2, ..., m - 1, m), where x can be 0 but must be less than m - 1.

Original entry on oeis.org

3, 9, 10, 15, 20, 27, 40, 42, 45, 50, 70, 75, 80, 81, 84, 100, 105, 126, 135, 140, 160, 168, 200, 225, 243, 250, 252, 280, 294, 315, 320, 330, 336, 350, 375, 378, 400, 405, 462, 490, 500, 504, 525, 560, 588, 640, 660, 672, 675, 700, 729, 735, 756, 770, 800

Offset: 1

Gus Wiseman, Apr 19 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.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose distinct parts form an initial interval with a single hole. The enumeration of these partitions by sum is given by A090858.

			The sequence of terms together with their prime indices begins:
    3: {2}
    9: {2,2}
   10: {1,3}
   15: {2,3}
   20: {1,1,3}
   27: {2,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   45: {2,2,3}
   50: {1,3,3}
   70: {1,3,4}
   75: {2,3,3}
   80: {1,1,1,1,3}
   81: {2,2,2,2}
   84: {1,1,2,4}
  100: {1,1,3,3}
  105: {2,3,4}
  126: {1,2,2,4}
  135: {2,2,2,3}
  140: {1,1,3,4}

Cf. A055932, A056239, A061395, A090858, A112798, A124010, A127002, A130091, A325241, A325251, A325259, A325270.

Select[Range[100],Length[Complement[Range[PrimePi[FactorInteger[#][[-1,1]]]],PrimePi/@First/@FactorInteger[#]]]==1&]

cp's OEIS Frontend

A325244 Number of integer partitions of n with one fewer distinct multiplicities than distinct parts.

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A371781 Numbers with biquanimous prime signature.

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A325241 Numbers > 1 whose maximum prime exponent is one greater than their minimum.

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A325270 Numbers with 1 fewer distinct prime exponents than (not necessarily distinct) prime factors.

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A325281 Numbers of the form a*b, a*a*b, or a*a*b*c where a, b, and c are distinct primes. Numbers with sorted prime signature (1,1), (1,2), or (1,1,2).

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A325284 Numbers whose prime indices form an initial interval with a single hole: (1, 2, ..., x, x + 2, ..., m - 1, m), where x can be 0 but must be less than m - 1.

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Mathematica

A325281 Numbers of the form ab, aab, or aabc where a, b, and c are distinct primes. Numbers with sorted prime signature (1,1), (1,2), or (1,1,2).