A263297 -id:A263297 - cp's OEIS frontend

A325229 Heinz numbers of integer partitions such that lesser of the maximum part and the number of parts is 2.

6, 9, 10, 12, 14, 15, 18, 21, 22, 24, 25, 26, 27, 33, 34, 35, 36, 38, 39, 46, 48, 49, 51, 54, 55, 57, 58, 62, 65, 69, 72, 74, 77, 81, 82, 85, 86, 87, 91, 93, 94, 95, 96, 106, 108, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 144, 145, 146

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

The enumeration of these partitions by sum is given by A265283.

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:
    6: {1,2}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   25: {3,3}
   26: {1,6}
   27: {2,2,2}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   39: {2,6}
   46: {1,9}

Cf. A056239, A061395, A093641, A112798, A252464, A257541, A263297, A265283, A325224, A325225, A325227, A325230, A325232.

Select[Range[300],Min[PrimeOmega[#],PrimePi[FactorInteger[#][[-1,1]]]]==2&]

A325194 Regular triangle read by rows where T(n,k) is the number of integer partitions of n with co-rank n - k, where co-rank is the greater of the length and the largest part.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 7, 1, 0, 0, 0

Offset: 0

Gus Wiseman, Apr 13 2019

			Triangle begins:
  1
  0  0
  0  1  0
  0  0  0  0
  0  0  2  0  0
  0  0  0  1  0  0
  0  0  0  2  1  0  0
  0  0  0  0  2  0  0  0
  0  0  0  0  2  3  0  0  0
  0  0  0  0  0  2  3  0  0  0
  0  0  0  0  0  2  4  2  0  0  0
  0  0  0  0  0  0  2  5  1  0  0  0
  0  0  0  0  0  0  2  4  7  1  0  0  0
  0  0  0  0  0  0  0  2  6  6  0  0  0  0
  0  0  0  0  0  0  0  2  4  9  7  0  0  0  0
  0  0  0  0  0  0  0  0  2  6 11  5  0  0  0  0
  0  0  0  0  0  0  0  0  2  4 10 14  5  0  0  0  0
  0  0  0  0  0  0  0  0  0  2  6 13 15  3  0  0  0  0
  0  0  0  0  0  0  0  0  0  2  4 10 19 17  2  0  0  0  0
  0  0  0  0  0  0  0  0  0  0  2  6 14 22 17  1  0  0  0  0
  0  0  0  0  0  0  0  0  0  0  2  4 10 21 29 17  1  0  0  0  0
Row n = 16 counts the following partitions:
  (8)         (72)       (64)      (533)    (444)
  (11111111)  (711)      (622)     (542)    (3333)
              (2211111)  (631)     (551)    (4332)
              (3111111)  (6211)    (5222)   (4422)
                         (61111)   (5321)   (4431)
                         (222211)  (5411)
                         (322111)  (32222)
                         (331111)  (33221)
                         (421111)  (33311)
                         (511111)  (42221)
                                   (43211)
                                   (44111)
                                   (52211)
                                   (53111)

Column sums are A000041. Row sums are A325193.

Cf. A051924, A071724, A096771, A115720, A263297, A325178, A325192.

Table[Length[Select[IntegerPartitions[k],Max[Length[#],Max[#]]==n-k&]],{n,0,10},{k,0,n}]

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&]

A263323 The greater of maximal exponent and maximal prime index in the prime factorization of n.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 4, 3, 2, 3, 5, 2, 6, 4, 3, 4, 7, 2, 8, 3, 4, 5, 9, 3, 3, 6, 3, 4, 10, 3, 11, 5, 5, 7, 4, 2, 12, 8, 6, 3, 13, 4, 14, 5, 3, 9, 15, 4, 4, 3, 7, 6, 16, 3, 5, 4, 8, 10, 17, 3, 18, 11, 4, 6, 6, 5, 19, 7, 9, 4, 20, 3, 21, 12, 3, 8, 5, 6, 22, 4

Offset: 1

Alexei Kourbatov, Oct 14 2015

nonn

Also: minimal m such that n divides (prime(m)#)^m. Here prime(m)# denotes the primorial A002110(m), i.e., the product of all primes from 2 to prime(m). - Charles R Greathouse IV, Oct 15 2015
Also: minimal m such that n is the product of at most m distinct primes not exceeding prime(m), with multiplicity at most m.
By convention, a(1)=0, as 1 is the empty product.
Those n with a(n) <= k fill a k-hypercube whose 1-sides span from 0 to k.
A263297 is a similar construction, with a k-simplex instead of a hypercube.
Each nonnegative integer occurs finitely often; in particular:
- Terms a(n) <= k occur A000169(k+1) = (k+1)^k times.
- The term a(n) = 0 occurs exactly once.
- The term a(n) = k > 0 occurs exactly A178922(k) = (k+1)^k - k^(k-1) times.

			a(36)=2 because 36 is the product of 2 distinct primes (2*2*3*3), each not exceeding prime(2)=3, with multiplicity not exceeding 2.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000169, A001221, A002110, A051903, A061395, A178922, A263297.

f[n_] := Max[ PrimePi[ Max @@ First /@ FactorInteger@n], Max @@ Last /@ FactorInteger@n]; Array[f, 80]

a(n) = if (n==1, 0, my(f = factor(n)); max(vecmax(f[,2]), primepi(f[#f~,1]))); \\ Michel Marcus, Oct 15 2015

a(n) = max(A051903(n), A061395(n)).

a(n) <= pi(n), with equality if n=1 or prime.

A383174 Permutation of the natural numbers formed by ordering by max(gpfi,bigomega), then bigomega, then numerically, where gpfi(k) = A061395(k) and bigomega(k) = A001222(k).

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 5, 10, 15, 25, 8, 12, 18, 20, 27, 30, 45, 50, 75, 125, 7, 14, 21, 35, 49, 28, 42, 63, 70, 98, 105, 147, 175, 245, 343, 16, 24, 36, 40, 54, 56, 60, 81, 84, 90, 100, 126, 135, 140, 150, 189, 196, 210, 225, 250, 294, 315, 350, 375, 441, 490, 525

Offset: 1

Bassam Abdul-Baki, Apr 18 2025

nonn

The sequence can be constructed starting with term 1 and then:
At step number m >= 1, append terms k with gpfi(k) <= m and bigomega(k) <= m, and which have not already appeared, and ordered first by bigomega and then numerically.
Terms which have not appeared are exactly those with gpfi(k) = m or bigomega(k) = m, and in particular they start with prime(m) and end with prime(m)^m.
First differs from A344844 at n=30, with its ordering by prime exponents differing from here ordering numerically. - Michael S. Branicky, Apr 20 2025

			At step m=2, the new terms added are 3, 4, 6, 9, being those with gpfi(k) = 2, or with bigomega(k) = 2.

Michael S. Branicky, Table of n, a(n) for n = 1..12870 (all terms involving only primes <= 19, so ending with 19^8)
Index entries for sequences that are permutations of the natural numbers

Cf. A061395, A001222, A263297, A344844.

from math import prod
from sympy import nextprime
from itertools import count, islice, combinations_with_replacement as cwr
def agen(): # generator of terms
    aset, plst = set(), [1, 2]
    for n in count(1):
        row = []
        for mc in cwr(plst, n):
            p = prod(mc)
            if p not in aset:
                row.append((n-mc.count(1), p))
                aset.add(p)
        plst.append(nextprime(plst[-1]))
        yield from (p for m, p in sorted(row))
print(list(islice(agen(), 80))) # Michael S. Branicky, Apr 18 2025

a(33) and on corrected by Michael S. Branicky, Apr 19 2025

cp's OEIS Frontend

A325229 Heinz numbers of integer partitions such that lesser of the maximum part and the number of parts is 2.

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A325194 Regular triangle read by rows where T(n,k) is the number of integer partitions of n with co-rank n - k, where co-rank is the greater of the length and the largest part.

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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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A263323 The greater of maximal exponent and maximal prime index in the prime factorization of n.

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A383174 Permutation of the natural numbers formed by ordering by max(gpfi,bigomega), then bigomega, then numerically, where gpfi(k) = A061395(k) and bigomega(k) = A001222(k).

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