A324571 -id:A324571 - cp's OEIS frontend

A324588 Heinz numbers of integer partitions of n into perfect squares (A001156).

1, 2, 4, 7, 8, 14, 16, 23, 28, 32, 46, 49, 53, 56, 64, 92, 97, 98, 106, 112, 128, 151, 161, 184, 194, 196, 212, 224, 227, 256, 302, 311, 322, 343, 368, 371, 388, 392, 419, 424, 448, 454, 512, 529, 541, 604, 622, 644, 661, 679, 686, 736, 742, 776, 784, 827, 838

Offset: 1

Gus Wiseman, Mar 08 2019

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Also products of elements of A011757.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   4: {1,1}
   7: {4}
   8: {1,1,1}
  14: {1,4}
  16: {1,1,1,1}
  23: {9}
  28: {1,1,4}
  32: {1,1,1,1,1}
  46: {1,9}
  49: {4,4}
  53: {16}
  56: {1,1,1,4}
  64: {1,1,1,1,1,1}
  92: {1,1,9}
  97: {25}
  98: {1,4,4}

Cf. A001156, A011757, A033461, A052335, A056239, A062457, A078135, A112798, A117144, A118914, A276078.

Cf. A109298, A324524, A324525, A324571, A324572, A324587.

Select[Range[100],And@@Cases[FactorInteger[#],{p_,_}:>IntegerQ[Sqrt[PrimePi[p]]]]&]

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

A325127 Numbers in whose prime factorization the exponent of prime(k) is greater than k for all prime indices k.

Original entry on oeis.org

1, 4, 8, 16, 27, 32, 64, 81, 108, 128, 216, 243, 256, 324, 432, 512, 625, 648, 729, 864, 972, 1024, 1296, 1728, 1944, 2048, 2187, 2500, 2592, 2916, 3125, 3456, 3888, 4096, 5000, 5184, 5832, 6561, 6912, 7776, 8192, 8748, 10000, 10368, 11664, 12500, 13824, 15552

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 more than k times. Such partitions are counted by A115584.

			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}
  625: {3,3,3,3}
  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}

Amiram Eldar, Table of n, a(n) for n = 1..2847 (terms up to 10^12)

Cf. A000720, A056239, A062457, A109298, A112798, A115584, A118914, A276078.

Cf. A324524, A324525, A324571, A325128, A325130.

Select[Range[1000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k>PrimePi[p]]&]
With[{k = 4}, m = Prime[k]^(k + 1); s = {}; Do[p = Prime[i]; AppendTo[s, Join[{1}, p^Range[i + 1, Floor[Log[p, m]]]]], {i, 1, k}]; Union @ Select[Times @@@ Tuples[s], # <= m &]] (* Amiram Eldar, Oct 24 2020 *)

Sum_{n>=1} 1/a(n) = Product_{k>=1} 1 + 1/(prime(k)^k * (prime(k)-1)) = 1.58661114052385082598.... - Amiram Eldar, Oct 24 2020

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]]&]

A324587 Heinz numbers of integer partitions of n into distinct perfect squares (A033461).

Original entry on oeis.org

1, 2, 7, 14, 23, 46, 53, 97, 106, 151, 161, 194, 227, 302, 311, 322, 371, 419, 454, 541, 622, 661, 679, 742, 827, 838, 1009, 1057, 1082, 1193, 1219, 1322, 1358, 1427, 1589, 1619, 1654, 1879, 2018, 2114, 2143, 2177, 2231, 2386, 2437, 2438, 2741, 2854, 2933

Offset: 1

Gus Wiseman, Mar 08 2019

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Also products of distinct elements of A011757.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    7: {4}
   14: {1,4}
   23: {9}
   46: {1,9}
   53: {16}
   97: {25}
  106: {1,16}
  151: {36}
  161: {4,9}
  194: {1,25}
  227: {49}
  302: {1,36}
  311: {64}
  322: {1,4,9}
  371: {4,16}
  419: {81}
  454: {1,49}
  541: {100}

Cf. A001156, A005117, A011757, A033461, A052335, A056239, A062457, A078135, A112798, A117144, A276078.

Cf. A109298, A324524, A324525, A324571, A324572, A324588.

Select[Range[1000],And@@Cases[FactorInteger[#],{p_,k_}:>k==1&&IntegerQ[Sqrt[PrimePi[p]]]]&]

A325129 Heinz numbers of integer partitions into nonsquares (A087153).

Original entry on oeis.org

1, 3, 5, 9, 11, 13, 15, 17, 19, 25, 27, 29, 31, 33, 37, 39, 41, 43, 45, 47, 51, 55, 57, 59, 61, 65, 67, 71, 73, 75, 79, 81, 83, 85, 87, 89, 93, 95, 99, 101, 103, 107, 109, 111, 113, 117, 121, 123, 125, 127, 129, 131, 135, 137, 139, 141, 143, 145, 149, 153, 155

Offset: 1

Gus Wiseman, Apr 01 2019

nonn

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:
   1: {}
   3: {2}
   5: {3}
   9: {2,2}
  11: {5}
  13: {6}
  15: {2,3}
  17: {7}
  19: {8}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  33: {2,5}
  37: {12}
  39: {2,6}
  41: {13}
  43: {14}
  45: {2,2,3}

Cf. A001156, A011757, A033461, A056239, A062457, A066328, A087153, A112798.

Cf. A324571, A324587, A324588, A325128, A325130.

Select[Range[100],!MemberQ[If[#==1,{},FactorInteger[#]],{p_,_}/;IntegerQ[Sqrt[PrimePi[p]]]]&]

cp's OEIS Frontend

A324588 Heinz numbers of integer partitions of n into perfect squares (A001156).

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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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A325127 Numbers in whose prime factorization the exponent of prime(k) is greater 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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A324587 Heinz numbers of integer partitions of n into distinct perfect squares (A033461).

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Mathematica

A325129 Heinz numbers of integer partitions into nonsquares (A087153).

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Mathematica