A325127 -id:A325127 - cp's OEIS frontend

A109297 Primal codes of finite permutations on positive integers.

1, 2, 9, 12, 18, 40, 112, 125, 250, 352, 360, 540, 600, 675, 832, 1008, 1125, 1350, 1500, 2176, 2250, 2268, 2352, 2401, 3168, 3969, 4802, 4864, 7488, 7938, 10692, 11616, 11776, 14000, 19584, 21609, 27440, 28812, 29403, 29696, 32448, 35000, 37908, 43218, 43776

Offset: 1

Jon Awbrey, Jul 08 2005

nonn

A finite permutation is a bijective mapping from a finite set to itself, counting the empty mapping as a permutation of the empty set.

Also Heinz numbers of integer partitions where the set of distinct parts is equal to the set of distinct multiplicities. These partitions are counted by A114640. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Apr 02 2019

			Writing (prime(i))^j as i:j, we have the following table:
Primal Codes of Finite Permutations on Positive Integers
` ` ` 1 = { }
` ` ` 2 = 1:1
` ` ` 9 = 2:2
` ` `12 = 1:2 2:1
` ` `18 = 1:1 2:2
` ` `40 = 1:3 3:1
` ` 112 = 1:4 4:1
` ` 125 = 3:3
` ` 250 = 1:1 3:3
` ` 352 = 1:5 5:1
` ` 360 = 1:3 2:2 3:1
` ` 540 = 1:2 2:3 3:1
` ` 600 = 1:3 2:1 3:2
` ` 675 = 2:3 3:2
` ` 832 = 1:6 6:1
` `1008 = 1:4 2:2 4:1
` `1125 = 2:2 3:3
` `1350 = 1:1 2:3 3:2
` `1500 = 1:2 2:1 3:3
` `2176 = 1:7 7:1
` `2250 = 1:1 2:2 3:3

Amiram Eldar, Table of n, a(n) for n = 1..300 (terms 1..100 from Alois P. Heinz)
Jon Awbrey, Riffs and Rotes.

Subsequence of A130091.
Cf. A076954, A106177, A108352, A108371, A109298, A109301, A056239, A112798, A114640, A118914.
Cf. A324524, A324525, A324571, A325127, A325128, A325130, A325131.

a:= proc(n) option remember; local k; for k from 1+`if`(n=1, 0,
      a(n-1)) while (l-> sort(map(i-> i[2], l)) <> sort(map(
      i-> numtheory[pi](i[1]), l)))(ifactors(k)[2]) do od; k
    end:
seq(a(n), n=1..45);  # Alois P. Heinz, Mar 08 2019

Select[Range[1000],#==1||Union[PrimePi/@First/@FactorInteger[#]]==Union[Last/@FactorInteger[#]]&] (* Gus Wiseman, Apr 02 2019 *)

is(n) = {my(f = factor(n), p = f[,1], e = vecsort(f[,2])); for(i=1, #p, if(primepi(p[i]) != e[i], return(0))); 1}; \\ Amiram Eldar, Jul 30 2022

More terms from Franklin T. Adams-Watters, Dec 19 2005

Offset set to 1 by Alois P. Heinz, Mar 08 2019

A325131 Heinz numbers of integer partitions where the set of distinct parts is disjoint from the set of distinct multiplicities.

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 64, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 100, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 127

Offset: 1

Gus Wiseman, Apr 01 2019

nonn

The enumeration of these partitions by sum is given by A114639.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers where the prime indices are disjoint from the prime exponents.

			The sequence of terms together with their prime indices begins:
   1: {}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
  11: {5}
  13: {6}
  15: {2,3}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  32: {1,1,1,1,1}
  33: {2,5}

Cf. A000720, A001222, A056239, A109298, A112798, A114639, A118914.

Cf. A324571, A325127, A325128, A325129, A325130.

Select[Range[100],Intersection[PrimePi/@First/@FactorInteger[#],Last/@FactorInteger[#]]=={}&]

A115584 Number of partitions of n in which each part k occurs more than k times.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 4, 3, 6, 4, 7, 7, 8, 8, 12, 9, 15, 14, 17, 18, 24, 21, 29, 29, 35, 35, 46, 42, 56, 54, 65, 67, 81, 77, 98, 95, 115, 114, 139, 135, 164, 165, 190, 195, 230, 225, 272, 271, 313, 321, 370, 374, 433, 441, 501, 514, 589, 592, 681, 698, 778, 809, 907

Offset: 0

Vladeta Jovovic, Mar 09 2006

The Heinz numbers of these partitions are given by A325127. - Gus Wiseman, Apr 02 2019

			a(2) = 1 because we have [1,1]; a(10) = 4 because we have [2,2,2,2,2], [2,2,2,2,1,1], [2,2,2,1,1,1,1] and [1^10].
From _Gus Wiseman_, Apr 02 2019: (Start)
The initial terms count the following integer partitions:
   0: ()
   2: (11)
   3: (111)
   4: (1111)
   5: (11111)
   6: (222)
   6: (111111)
   7: (1111111)
   8: (2222)
   8: (22211)
   8: (11111111)
   9: (222111)
   9: (111111111)
  10: (22222)
  10: (222211)
  10: (2221111)
  10: (1111111111)
  11: (2222111)
  11: (22211111)
  11: (11111111111)
  12: (3333)
  12: (222222)
  12: (2222211)
  12: (22221111)
  12: (222111111)
  12: (111111111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A052335, A087153, A114639, A117144, A276429, A324572, A325127.

g:=product((1-x^k+x^(k*(k+1)))/(1-x^k),k=1..30): gser:=series(g,x=0,75): seq(coeff(gser,x,n),n=0..70); # Emeric Deutsch, Mar 12 2006
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +add(b(n-i*j, i-1), j=i+1..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, Feb 09 2017

CoefficientList[ Series[ Product[(1 - x^k + x^(k(k + 1)))/(1 - x^k), {k, 14}], {x, 0, 66}], x] (* Robert G. Wilson v, Mar 12 2006 *)
Table[Length[Select[IntegerPartitions[n],And@@Table[Count[#,i]>i,{i,Union[#]}]&]],{n,0,30}] (* Gus Wiseman, Apr 02 2019 *)

G.f.: Product_{k>=1} (1-x^k+x^(k*(k+1)))/(1-x^k).

More terms from Robert G. Wilson v and Emeric Deutsch, Mar 12 2006

A054744 p-full numbers: numbers such that if any prime p divides it, then so does p^p.

Original entry on oeis.org

1, 4, 8, 16, 27, 32, 64, 81, 108, 128, 216, 243, 256, 324, 432, 512, 648, 729, 864, 972, 1024, 1296, 1728, 1944, 2048, 2187, 2592, 2916, 3125, 3456, 3888, 4096, 5184, 5832, 6561, 6912, 7776, 8192, 8748, 10368, 11664, 12500, 13824, 15552, 15625, 16384

Offset: 1

James Sellers, Apr 22 2000

A027748(a(n),k) <= A124010(a(n),k), 1<=k<=A001221(a(n)). [Reinhard Zumkeller, Apr 28 2012]

Heinz numbers of integer partitions where the multiplicity of each part k is at least prime(k). These partitions are counted by A325132. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Apr 02 2019

			8 is an element because 8 = 2^3 and 2<=3, while 25 is not an element because 25 = 5^2 and 5>2.
From _Gus Wiseman_, Apr 02 2019: (Start)
The sequence of terms together with their prime indices begins:
    1: {}
    4: {1,1}
    8: {1,1,1}
   16: {1,1,1,1}
   27: {2,2,2}
   32: {1,1,1,1,1}
   64: {1,1,1,1,1,1}
   81: {2,2,2,2}
  108: {1,1,2,2,2}
  128: {1,1,1,1,1,1,1}
  216: {1,1,1,2,2,2}
  243: {2,2,2,2,2}
  256: {1,1,1,1,1,1,1,1}
  324: {1,1,2,2,2,2}
  432: {1,1,1,1,2,2,2}
  512: {1,1,1,1,1,1,1,1,1}
  648: {1,1,1,2,2,2,2}
  729: {2,2,2,2,2,2}
  864: {1,1,1,1,1,2,2,2}
  972: {1,1,2,2,2,2,2}
(End)

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A048103, A001694, A036966.
Cf. A100716.
Cf. A056239, A109298, A112798, A115584, A117144, A124010, A276078, A324525, A324571, A325127, A325128, A325130, A325131, A325132.

a054744 n = a054744_list !! (n-1)
a054744_list = filter (\x -> and $
   zipWith (<=) (a027748_row x) (map toInteger $ a124010_row x)) [1..]
-- Reinhard Zumkeller, Apr 28 2012

Select[Range[1000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k>=p]&] (* Gus Wiseman, Apr 02 2019 *)

If n = Product p_i^e_i then p_i<=e_i for all i.

Sum_{n>=1} 1/a(n) = Product_{p prime} 1 + 1/(p^(p-1)*(p-1)) = 1.58396891058853238595.... - Amiram Eldar, Oct 24 2020

A325128 Numbers in whose prime factorization the exponent of prime(k) is less than k for all prime indices k.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 127, 129, 131, 133, 137, 139, 141

Offset: 1

Gus Wiseman, Apr 01 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 where each part k appears fewer than k times. Such partitions are counted by A087153.
The asymptotic density of this sequence is Product_{k>=1} (1 - 1/prime(k)^k) = 0.44070243286030291209... - Amiram Eldar, Feb 02 2021

			The sequence of terms together with their prime indices begins:
   1: {}
   3: {2}
   5: {3}
   7: {4}
  11: {5}
  13: {6}
  15: {2,3}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  29: {10}
  31: {11}
  33: {2,5}
  35: {3,4}
  37: {12}
  39: {2,6}
  41: {13}
  43: {14}
  47: {15}
  49: {4,4}

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

Cf. A056239, A062457, A087153, A109298, A112798, A115584, A118914, A276078.

Cf. A324524, A324525, A324571, A325127, A325130.

Select[Range[100],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k

A325130 Numbers in whose prime factorization the exponent of prime(k) is not equal to k for any prime index k.

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 37, 39, 40, 41, 43, 44, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 64, 65, 67, 68, 69, 71, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 88, 89, 91, 92, 93, 95, 96

Offset: 1

Gus Wiseman, Apr 01 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 the integer partitions counted by A276429.
The asymptotic density of this sequence is Product_{k>=1} (1 - 1/prime(k)^k + 1/prime(k)^(k+1)) = 0.68974964705635552968... - Amiram Eldar, Jan 09 2021

			The sequence of terms together with their prime indices begins:
   1: {}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
  11: {5}
  12: {1,1,2}
  13: {6}
  15: {2,3}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  23: {9}
  24: {1,1,1,2}
  25: {3,3}
  27: {2,2,2}
  28: {1,1,4}

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A056239, A087153, A112798, A124010, A276078, A276429.
Cf. A324525, A324571, A325127, A325128, A325130, A325131.
Complement of A276936.

q:= n-> andmap(i-> numtheory[pi](i[1])<>i[2], ifactors(n)[2]):
a:= proc(n) option remember; local k; for k from 1+
     `if`(n=1, 0, a(n-1)) while not q(k) do od; k
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Oct 28 2019

Select[Range[100],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k!=PrimePi[p]]&]

A380085 The largest unitary divisor of n that is a term in A276078.

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 1, 9, 10, 11, 3, 13, 14, 15, 1, 17, 18, 19, 5, 21, 22, 23, 3, 25, 26, 1, 7, 29, 30, 31, 1, 33, 34, 35, 9, 37, 38, 39, 5, 41, 42, 43, 11, 45, 46, 47, 3, 49, 50, 51, 13, 53, 2, 55, 7, 57, 58, 59, 15, 61, 62, 63, 1, 65, 66, 67, 17, 69, 70, 71

Offset: 1

Amiram Eldar, Jan 11 2025

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

Cf. A000720, A077610, A276078, A325127, A377515, A380086, A380087, A380088.

f[p_, e_] := p^If[e <= PrimePi[p], e, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^if(f[i,2] <= primepi(f[i,1]), f[i,2], 0));}

Multiplicative with a(p^e) = p^e if e <= pi(p) = A000720(p), and 1 otherwise.
a(n) = 1 if and only if n is in A325127.
a(n) = n if and only if n is in A276078.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{k>=1} (p(k)^(2*(k+1)) + p(k)^(2*k+1) - p(k)^(k+1) - p(k)^k + 1)/(p(k)^(2*k+1) * (p(k)+1)) = 0.76189494803691349595..., where p(k) = prime(k).

A380086 The number of unitary divisors of n that are terms in A276078.

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 1, 2, 4, 2, 2, 2, 4, 4, 1, 2, 4, 2, 2, 4, 4, 2, 2, 2, 4, 1, 2, 2, 8, 2, 1, 4, 4, 4, 2, 2, 4, 4, 2, 2, 8, 2, 2, 4, 4, 2, 2, 2, 4, 4, 2, 2, 2, 4, 2, 4, 4, 2, 4, 2, 4, 4, 1, 4, 8, 2, 2, 4, 8, 2, 2, 2, 4, 4, 2, 4, 8, 2, 2, 1, 4, 2, 4, 4, 4, 4

Offset: 1

Amiram Eldar, Jan 11 2025

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

Cf. A000720, A005117, A034444, A077610, A276078, A325127, A377516, A380085, A380087, A380089.

f[p_, e_] := If[e <= PrimePi[p], 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~,if(f[i,2] <= primepi(f[i,1]), 2, 1));}

a(n) = A034444(A380085(n)).
Multiplicative with a(p^e) = 2 if e <= pi(p) = A000720(p), and 1 otherwise.
a(n) = 1 if and only if n is in A325127.
a(n) < A034444(n) if and only if n is in A276079.
a(n) = A034444(n) if and only if n is in A276078.
a(n) = A377516(n) if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s - 1/p^((pi(p)+1)*s)).
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A034444(k) = Product_{p prime} (1 - 1/(2*p^(pi(p)+1))) = 0.85808348184674088116... .

A380087 The sum of the unitary divisors of n that are terms in A276078.

Original entry on oeis.org

1, 3, 4, 1, 6, 12, 8, 1, 10, 18, 12, 4, 14, 24, 24, 1, 18, 30, 20, 6, 32, 36, 24, 4, 26, 42, 1, 8, 30, 72, 32, 1, 48, 54, 48, 10, 38, 60, 56, 6, 42, 96, 44, 12, 60, 72, 48, 4, 50, 78, 72, 14, 54, 3, 72, 8, 80, 90, 60, 24, 62, 96, 80, 1, 84, 144, 68, 18, 96, 144

Offset: 1

Amiram Eldar, Jan 11 2025

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

Cf. A000720, A005117, A034448, A077610, A276078, A377517, A380085, A380086, A380090.

f[p_, e_] := If[e <= PrimePi[p], p^e, 0] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] <= primepi(f[i,1]), f[i,1]^f[i,2], 0) + 1);}

a(n) = A034448(A380085(n)).
Multiplicative with a(p^e) = p^e + 1 if e <= pi(p) = A000720(p), and 1 otherwise.
a(n) = 1 if and only if n is in A325127.
a(n) < A034448(n) if and only if n is in A276079.
a(n) = A034448(n) if and only if n is in A276078.
a(n) = A377517(n) if and only if n is squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{k>=1} (p(k)^(k+2) + p(k)^(k+1) + p(k)^k - p(k) - 1)/(p(k)^(k+1) * (p(k)+1)) = 1.03676..., where p(k) = prime(k).

cp's OEIS Frontend

A109297 Primal codes of finite permutations on positive integers.

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A325131 Heinz numbers of integer partitions where the set of distinct parts is disjoint from the set of distinct multiplicities.

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A115584 Number of partitions of n in which each part k occurs more than k times.

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A054744 p-full numbers: numbers such that if any prime p divides it, then so does p^p.

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A325128 Numbers in whose prime factorization the exponent of prime(k) is less than k for all prime indices k.

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A325130 Numbers in whose prime factorization the exponent of prime(k) is not equal to k for any prime index k.

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A380085 The largest unitary divisor of n that is a term in A276078.

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A380086 The number of unitary divisors of n that are terms in A276078.