A325225 -id:A325225 - cp's OEIS frontend

A325231 Numbers of the form 2 * p or 3 * 2^k, p prime, k > 1.

6, 10, 12, 14, 22, 24, 26, 34, 38, 46, 48, 58, 62, 74, 82, 86, 94, 96, 106, 118, 122, 134, 142, 146, 158, 166, 178, 192, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 384, 386, 394, 398, 422, 446, 454, 458, 466

Offset: 1

Gus Wiseman, Apr 13 2019

nonn

Also numbers n such that the sum of prime indices of n minus the greater of the number of prime factors of n counted with multiplicity and the largest prime index of n is 1. 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, and their sum is A056239.

			The sequence of terms together with their prime indices begins:
    6: {1,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   34: {1,7}
   38: {1,8}
   46: {1,9}
   48: {1,1,1,1,2}
   58: {1,10}
   62: {1,11}
   74: {1,12}
   82: {1,13}
   86: {1,14}
   94: {1,15}
   96: {1,1,1,1,1,2}
  106: {1,16}
  118: {1,17}

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Positions of 1's in A325223.

Cf. A001222, A056239, A060687, A061395, A093641, A112798, A174090, A257541, A265283, A325224, A325225, A325227, A325232.

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

from sympy import isprime
A325231_list = [n for n in range(6,10**6) if ((not n % 2) and isprime(n//2)) or (bin(n)[2:4] == '11' and bin(n).count('1') == 2)] # Chai Wah Wu, Apr 16 2019

A325223 Sum of the prime indices of n minus the greater of the number of prime factors of n counted with multiplicity and the largest prime index of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 1, 0, 1, 2, 0, 0, 2, 0, 2, 2, 1, 0, 1, 3, 1, 3, 2, 0, 3, 0, 0, 2, 1, 3, 2, 0, 1, 2, 2, 0, 3, 0, 2, 4, 1, 0, 1, 4, 4, 2, 2, 0, 3, 3, 3, 2, 1, 0, 3, 0, 1, 4, 0, 3, 3, 0, 2, 2, 4, 0, 2, 0, 1, 5, 2, 4, 3, 0, 2, 4, 1, 0, 4, 3, 1, 2

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

A prime index of n is a number m such that prime(m) divides n.

Also the number of squares in the Young diagram of the integer partition with Heinz number n after the first row or the first column, whichever is larger, is removed. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			88 has 4 prime indices {1,1,1,5} with sum 8 and maximum 5, so a(88) = 8 - max(4,5) = 3.

FindStat, St000784: The maximum of the length and the largest part of the integer partition

Positions of 0's are A174090. Positions of 1's are A325231.

Cf. A001222, A051924, A052126, A056239, A061395, A064989, A065770, A112798, A257990, A263297, A325134, A325169, A325224, A325225, A325227.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[primeMS[n]]-Max[Length[primeMS[n]],Max[primeMS[n]]],{n,100}]

a(n) = A056239(n) - max(A001222(n), A061395(n)) = A056239(n) - A263297(n).

A325234 Heinz numbers of integer partitions with Dyson rank -1.

Original entry on oeis.org

4, 12, 18, 27, 40, 60, 90, 100, 112, 135, 150, 168, 225, 250, 252, 280, 352, 375, 378, 392, 420, 528, 567, 588, 625, 630, 700, 792, 832, 880, 882, 945, 980, 1050, 1188, 1232, 1248, 1320, 1323, 1372, 1470, 1575, 1750, 1782, 1848, 1872, 1936, 1980, 2058, 2080

Offset: 1

Gus Wiseman, Apr 13 2019

nonn

Numbers whose maximum prime index is one fewer than their number of prime indices counted with multiplicity.

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:
     4: {1,1}
    12: {1,1,2}
    18: {1,2,2}
    27: {2,2,2}
    40: {1,1,1,3}
    60: {1,1,2,3}
    90: {1,2,2,3}
   100: {1,1,3,3}
   112: {1,1,1,1,4}
   135: {2,2,2,3}
   150: {1,2,3,3}
   168: {1,1,1,2,4}
   225: {2,2,3,3}
   250: {1,3,3,3}
   252: {1,1,2,2,4}
   280: {1,1,1,3,4}
   352: {1,1,1,1,1,5}
   375: {2,3,3,3}
   378: {1,2,2,2,4}
   392: {1,1,1,4,4}

FindStat, St000145: The Dyson rank of a partition

Positions of -1's in A257541.

Cf. A001222, A047993, A056239, A061395, A063995, A101198, A106529, A112798, A257990, A263297, A325225, A325233, A325235.

Select[Range[1000],PrimePi[FactorInteger[#][[-1,1]]]-PrimeOmega[#]==-1&]

A325235 Heinz numbers of integer partitions with Dyson rank 1 or -1.

Original entry on oeis.org

3, 4, 10, 12, 15, 18, 25, 27, 28, 40, 42, 60, 63, 70, 88, 90, 98, 100, 105, 112, 132, 135, 147, 150, 168, 175, 198, 208, 220, 225, 245, 250, 252, 280, 297, 308, 312, 330, 343, 352, 375, 378, 392, 420, 462, 468, 484, 495, 520, 528, 544, 550, 567, 588, 625, 630

Offset: 1

Gus Wiseman, Apr 13 2019

nonn

Numbers whose maximum prime index and number of prime indices counted with multiplicity differ by 1.

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:
    3: {2}
    4: {1,1}
   10: {1,3}
   12: {1,1,2}
   15: {2,3}
   18: {1,2,2}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}
   40: {1,1,1,3}
   42: {1,2,4}
   60: {1,1,2,3}
   63: {2,2,4}
   70: {1,3,4}
   88: {1,1,1,5}
   90: {1,2,2,3}
   98: {1,4,4}
  100: {1,1,3,3}
  105: {2,3,4}
  112: {1,1,1,1,4}

FindStat, St000145: The Dyson rank of a partition

Positions of 1's and -1's in A257541.

Cf. A001222, A047993, A056239, A061395, A063995, A101198, A106529, A112798, A257990, A263297, A325225, A325233, A325234.

Select[Range[1000],Abs[PrimePi[FactorInteger[#][[-1,1]]]-PrimeOmega[#]]==1&]

A325228 Number of integer partitions of n such that the lesser of the maximum part and the number of parts is 3.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 6, 9, 13, 16, 20, 24, 28, 32, 38, 42, 48, 54, 60, 66, 74, 80, 88, 96, 104, 112, 122, 130, 140, 150, 160, 170, 182, 192, 204, 216, 228, 240, 254, 266, 280, 294, 308, 322, 338, 352, 368, 384, 400, 416, 434, 450, 468, 486, 504, 522, 542, 560

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

			The a(5) = 1 through a(10) = 16 partitions:
  (311)  (321)   (322)    (332)     (333)      (433)
         (411)   (331)    (422)     (432)      (442)
         (3111)  (421)    (431)     (441)      (532)
                 (511)    (521)     (522)      (541)
                 (3211)   (611)     (531)      (622)
                 (31111)  (3221)    (621)      (631)
                          (3311)    (711)      (721)
                          (32111)   (3222)     (811)
                          (311111)  (3321)     (3322)
                                    (32211)    (3331)
                                    (33111)    (32221)
                                    (321111)   (33211)
                                    (3111111)  (322111)
                                               (331111)
                                               (3211111)
                                               (31111111)

Column k = 3 of A325227.

Cf. A051924, A096771, A115720, A265283, A325224, A325225, A325229, A325231, A325232.

Table[Length[Select[IntegerPartitions[n],Min[Length[#],Max[#]]==3&]],{n,30}]

cp's OEIS Frontend

A325231 Numbers of the form 2 * p or 3 * 2^k, p prime, k > 1.

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A325223 Sum of the prime indices of n minus the greater of the number of prime factors of n counted with multiplicity and the largest prime index of n.

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A325234 Heinz numbers of integer partitions with Dyson rank -1.

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A325235 Heinz numbers of integer partitions with Dyson rank 1 or -1.

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A325228 Number of integer partitions of n such that the lesser of the maximum part and the number of parts is 3.

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