A109297 -id:A109297 - cp's OEIS frontend

A352141 Numbers whose prime factorization has all even indices and all even exponents.

1, 9, 49, 81, 169, 361, 441, 729, 841, 1369, 1521, 1849, 2401, 2809, 3249, 3721, 3969, 5041, 6241, 6561, 7569, 7921, 8281, 10201, 11449, 12321, 12769, 13689, 16641, 17161, 17689, 19321, 21609, 22801, 25281, 26569, 28561, 29241, 29929, 32761, 33489, 35721

Offset: 1

Gus Wiseman, Mar 18 2022

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, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
These are the Heinz numbers of partitions with all even parts and all even multiplicities, counted by A035444.

			The terms together with their prime indices begin:
     1 = 1
     9 = prime(2)^2
    49 = prime(4)^2
    81 = prime(2)^4
   169 = prime(6)^2
   361 = prime(8)^2
   441 = prime(2)^2 prime(4)^2
   729 = prime(2)^6
   841 = prime(10)^2
  1369 = prime(12)^2
  1521 = prime(2)^2 prime(6)^2
  1849 = prime(14)^2
  2401 = prime(4)^4
  2809 = prime(16)^2
  3249 = prime(2)^2 prime(8)^2
  3721 = prime(18)^2
  3969 = prime(2)^4 prime(4)^2

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

The second condition alone (all even exponents) is A000290, counted by A035363.
The restriction to primes is A031215.
These partitions are counted by A035444.
The first condition alone is A066207, counted by A035363, squarefree A258117.
A056166 = exponents all prime, counted by A055923.
A066208 = prime indices all odd, counted by A000009.
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162641 counts even exponents, odd A162642.
A257991 counts odd indices, even A257992.
A325131 = disjoint indices from exponents, counted by A114639.
A346068 = indices and exponents all prime, counted by A351982.
A351979 = odd indices with even exponents, counted by A035457.
A352140 = even indices with odd exponents, counted by A055922 aerated.
A352142 = odd indices with odd exponents, counted by A117958.
Cf. A000720, A028260, A055396, A061395, A181819, A195017, A241638, A268335, A276078, A324524, A324525, A324588, A325698, A325700, A352143.

Select[Range[1000],#==1||And@@EvenQ/@PrimePi/@First/@FactorInteger[#]&&And@@EvenQ/@Last/@FactorInteger[#]&]

from itertools import count, islice
from sympy import factorint, primepi
def A352141_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda k:all(map(lambda x: not (x[1]%2 or primepi(x[0])%2), factorint(k).items())),count(max(startvalue,1)))
A352141_list = list(islice(A352141_gen(),30)) # Chai Wah Wu, Mar 18 2022

Intersection of A000290 and A066207.
A257991(a(n)) = A162642(a(n)) = 0.
A257992(a(n)) = A001222(a(n)).
A162641(a(n)) = A001221(a(n)).
Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2*k)^2) = 1.163719... . - Amiram Eldar, Sep 19 2022

A352492 Powerful numbers whose prime indices are all prime numbers.

Original entry on oeis.org

1, 9, 25, 27, 81, 121, 125, 225, 243, 289, 625, 675, 729, 961, 1089, 1125, 1331, 1681, 2025, 2187, 2601, 3025, 3125, 3267, 3375, 3481, 4489, 4913, 5625, 6075, 6561, 6889, 7225, 7803, 8649, 9801, 10125, 11881, 11979, 14641, 15125, 15129, 15625, 16129, 16875

Offset: 1

Gus Wiseman, Mar 24 2022

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 terms together with their prime indices (not prime factors) begin:
    1: {}
    9: {2,2}
   25: {3,3}
   27: {2,2,2}
   81: {2,2,2,2}
  121: {5,5}
  125: {3,3,3}
  225: {2,2,3,3}
  243: {2,2,2,2,2}
  289: {7,7}
  625: {3,3,3,3}
  675: {2,2,2,3,3}
  729: {2,2,2,2,2,2}
  961: {11,11}
For example, 675 = prime(2)^3 prime(3)^2 = 3^3 * 5^2.

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

Powerful numbers are A001694, counted by A007690.
The version for prime exponents instead of indices is A056166, counted by A055923.
This is the powerful case of A076610 (products of A006450), counted by A000607.
The partitions with these Heinz numbers are counted by A339218.
A000040 lists primes.
A031368 lists primes of odd index, products A066208.
A101436 counts exponents in prime factorization that are themselves prime.
A112798 lists prime indices, reverse A296150, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A053810 lists all numbers p^q with p and q prime, counted by A230595.
A257994 counts prime indices that are themselves prime, complement A330944.
Cf. A000720, A000961, A001597, A007821, A007916, A052485, A109297, A140319, A164336, A181819, A325131, A330945, A346068.

Select[Range[1000],#==1||And@@PrimeQ/@PrimePi/@First/@FactorInteger[#]&&Min@@Last/@FactorInteger[#]>1&]

Intersection of A001694 and A076610.

Sum_{n>=1} 1/a(n) = Product_{p in A006450} (1 + 1/(p*(p-1))) = 1.24410463... - Amiram Eldar, May 04 2022

A352140 Numbers whose prime factorization has all even prime indices and all odd exponents.

Original entry on oeis.org

1, 3, 7, 13, 19, 21, 27, 29, 37, 39, 43, 53, 57, 61, 71, 79, 87, 89, 91, 101, 107, 111, 113, 129, 131, 133, 139, 151, 159, 163, 173, 181, 183, 189, 193, 199, 203, 213, 223, 229, 237, 239, 243, 247, 251, 259, 263, 267, 271, 273, 281, 293, 301, 303, 311, 317

Offset: 1

Gus Wiseman, Mar 11 2022

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, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
Also Heinz numbers of integer partitions with all even parts and all odd multiplicities, counted by A055922 aerated.
All terms are odd. - Michael S. Branicky, Mar 12 2022

			The terms together with their prime indices begin:
      1 = 1
      3 = prime(2)^1
      7 = prime(4)^1
     13 = prime(6)^1
     19 = prime(8)^1
     21 = prime(4)^1 prime(2)^1
     27 = prime(2)^3
     29 = prime(10)^1
     37 = prime(12)^1
     39 = prime(6)^1 prime(2)^1
     43 = prime(14)^1
     53 = prime(16)^1
     57 = prime(8)^1 prime(2)^1
     61 = prime(18)^1
     71 = prime(20)^1

The restriction to primes is A031215.
These partitions are counted by A055922 (aerated).
The first condition alone is A066207, counted by A035363.
The squarefree case is A258117.
The second condition alone is A268335, counted by A055922.
A056166 = exponents all prime, counted by A055923.
A066208 = prime indices all odd, counted by A000009.
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162641 counts even prime exponents, odd A162642.
A257991 counts odd prime indices, even A257992.
A325131 = disjoint indices from exponents, counted by A114639.
A346068 = indices and exponents all prime, counted by A351982.
A351979 = odd indices with even exponents, counted by A035457.
A352141 = even indices with even exponents, counted by A035444.
A352142 = odd indices with odd exponents, counted by A117958.
Cf. A000720, A028260, A055396, A061395, A181819, A195017, A241638, A276078, A324517, A324524, A324525, A325698.

Select[Range[100],And@@EvenQ/@PrimePi/@First/@FactorInteger[#]&&And@@OddQ/@Last/@FactorInteger[#]&]

from sympy import factorint, primepi
def ok(n):
    if n%2 == 0: return False
    return all(primepi(p)%2==0 and e%2==1 for p, e in factorint(n).items())
print([k for k in range(318) if ok(k)]) # Michael S. Branicky, Mar 12 2022

Intersection of A066207 and A268335.
A257991(a(n)) = A162641(a(n)) = 0.
A162642(a(n)) = A001221(a(n)).
A257992(a(n)) = A001222(a(n)).

A325708 Numbers n whose prime indices cover an initial interval of positive integers and include all prime exponents of n.

Original entry on oeis.org

1, 2, 6, 12, 18, 30, 36, 60, 90, 120, 150, 180, 210, 270, 300, 360, 420, 450, 540, 600, 630, 750, 840, 900, 1050, 1080, 1260, 1350, 1470, 1500, 1680, 1800, 1890, 2100, 2250, 2310, 2520, 2700, 2940, 3000, 3150, 3780, 4200, 4410, 4500, 4620, 5040, 5250, 5400

Offset: 1

Gus Wiseman, May 18 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 covering an initial interval of positive integers and containing all of their distinct multiplicities. The enumeration of these partitions by sum is given by A325707.

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     6: {1,2}
    12: {1,1,2}
    18: {1,2,2}
    30: {1,2,3}
    36: {1,1,2,2}
    60: {1,1,2,3}
    90: {1,2,2,3}
   120: {1,1,1,2,3}
   150: {1,2,3,3}
   180: {1,1,2,2,3}
   210: {1,2,3,4}
   270: {1,2,2,2,3}
   300: {1,1,2,3,3}
   360: {1,1,1,2,2,3}
   420: {1,1,2,3,4}
   450: {1,2,2,3,3}
   540: {1,1,2,2,2,3}
   600: {1,1,1,2,3,3}

Cf. A055932, A109297, A114639, A114640, A181819, A290689, A290822, A325705, A325706, A325707.

Select[Range[1000],#==1||Range[PrimeNu[#]]==PrimePi/@First/@FactorInteger[#]&&SubsetQ[PrimePi/@First/@FactorInteger[#],Last/@FactorInteger[#]]&]

A351979 Numbers whose prime factorization has all odd prime indices and all even prime exponents.

Original entry on oeis.org

1, 4, 16, 25, 64, 100, 121, 256, 289, 400, 484, 529, 625, 961, 1024, 1156, 1600, 1681, 1936, 2116, 2209, 2500, 3025, 3481, 3844, 4096, 4489, 4624, 5329, 6400, 6724, 6889, 7225, 7744, 8464, 8836, 9409, 10000, 10609, 11881, 12100, 13225, 13924, 14641, 15376

Offset: 1

Gus Wiseman, Mar 11 2022

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, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
Also Heinz numbers of integer partitions with all odd parts and all even multiplicities, counted by A035457 (see Emeric Deutsch's comment there).

			The terms together with their prime indices begin:
     1: 1
     4: prime(1)^2
    16: prime(1)^4
    25: prime(3)^2
    64: prime(1)^6
   100: prime(1)^2 prime(3)^2
   121: prime(5)^2
   256: prime(1)^8
   289: prime(7)^2
   400: prime(1)^4 prime(3)^2
   484: prime(1)^2 prime(5)^2
   529: prime(9)^2
   625: prime(3)^4
   961: prime(11)^2
  1024: prime(1)^10
  1156: prime(1)^2 prime(7)^2
  1600: prime(1)^6 prime(3)^2
  1681: prime(13)^2
  1936: prime(1)^4 prime(5)^2

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

The second condition alone (exponents all even) is A000290, counted by A035363.
The distinct prime factors of terms all come from A031368.
These partitions are counted by A035457 or A000009 aerated.
The first condition alone (indices all odd) is A066208, counted by A000009.
The squarefree square roots are A258116, even A258117.
A056166 = exponents all prime, counted by A055923.
A066207 = indices all even, counted by complement of A086543.
A076610 = indices all prime, counted by A000607.
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162641 counts even exponents, odd A162642.
A257991 counts odd indices, even A257992.
A268335 = exponents all odd, counted by A055922.
A325131 = disjoint indices from exponents, counted by A114639.
A346068 = indices and exponents all prime, counted by A351982.
A352140 = even indices with odd exponents, counted by A055922 (aerated).
A352141 = even indices with even exponents, counted by A035444.
A352142 = odd indices and odd multiplicities, counted by A117958.
Cf. A000720, A028260, A045931, A055396, A061395, A106529, A181819, A195017, A276078, A324588, A325698, A325700.

Select[Range[1000],#==1||And@@OddQ/@PrimePi/@First/@FactorInteger[#]&&And@@EvenQ/@Last/@FactorInteger[#]&]

from sympy import factorint, primepi
def ok(n):
    return all(primepi(p)%2==1 and e%2==0 for p, e in factorint(n).items())
print([k for k in range(15500) if ok(k)]) # Michael S. Branicky, Mar 12 2022

Squares of elements of A066208.
Intersection of A066208 and A000290.
A257991(a(n)) = A001222(a(n)).
A162641(a(n)) = A001221(a(n)).
A162642(a(n)) = A257992(a(n)) = 0.
Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2*k-1)^2) = 1.4135142... . - Amiram Eldar, Sep 19 2022

A325707 Number of integer partitions of n covering an initial interval of positive integers and containing all of their distinct multiplicities.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 2, 1, 2, 2, 4, 4, 5, 6, 7, 8, 10, 11, 13, 16, 18, 23, 26, 32, 36, 43, 48, 57, 64, 74, 84, 98, 110, 127, 145, 165, 189, 215, 244, 277, 316, 356, 403, 455, 513, 577, 650, 727, 817, 913, 1024, 1143, 1279, 1425, 1592, 1773, 1977, 2198, 2448, 2717

Offset: 0

Gus Wiseman, May 18 2019

nonn

The Heinz numbers of these partitions are given by A325708.

			The initial terms count the following partitions:
   1: (1)
   3: (21)
   4: (211)
   5: (221)
   6: (321)
   6: (2211)
   7: (3211)
   8: (3221)
   8: (32111)
   9: (3321)
   9: (32211)
  10: (4321)
  10: (33211)
  10: (32221)
  10: (322111)
  11: (43211)
  11: (33221)
  11: (332111)
  11: (322211)
  12: (43221)
  12: (432111)
  12: (33321)
  12: (332211)
  12: (3222111)

Cf. A000009 (partitions covering an initial interval), A055932, A109297, A114639, A114640, A290689, A324753, A325702, A325706, A325708.

Table[Length[Select[IntegerPartitions[n],Range[Length[Union[#]]]==Union[#]&&SubsetQ[Sort[#],Sort[Length/@Split[#]]]&]],{n,0,30}]

A328830 The second prime shadow of n: a(1) = 1; for n > 1, a(n) = a(A003557(n)) * prime(A056169(n)) when A056169(n) > 0, otherwise a(n) = a(A003557(n)).

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 4, 2, 3, 3, 2, 2, 4, 2, 4, 3, 3, 2, 4, 2, 3, 2, 4, 2, 5, 2, 2, 3, 3, 3, 3, 2, 3, 3, 4, 2, 5, 2, 4, 4, 3, 2, 4, 2, 4, 3, 4, 2, 4, 3, 4, 3, 3, 2, 6, 2, 3, 4, 2, 3, 5, 2, 4, 3, 5, 2, 4, 2, 3, 4, 4, 3, 5, 2, 4, 2, 3, 2, 6, 3, 3, 3, 4, 2, 6, 3, 4, 3, 3, 3, 4, 2, 4, 4, 3, 2, 5, 2, 4, 5

Offset: 1

Antti Karttunen, Oct 29 2019

nonn

a(n) depends only on prime signature of n (cf. A025487).

			For n = 30 = 2 * 3 * 5, there are three unitary prime factors, while A003557(30) = 1, which terminates the recursion, thus a(30) = prime(3) = 5.
For n = 60060 = 2^2 * 3 * 5 * 7 * 11 * 13, there are 5 unitary prime factors, while in A003557(60060) = 2 there is only one, thus a(60060) = prime(5) * prime(1) = 11 * 2 = 22.
The number 1260 = 2^2*3^2*5*7 has prime exponents (2,2,1,1) so its prime shadow is prime(2)*prime(2)*prime(1)*prime(1) = 36.  Next, 36 = 2^2*3^2 has prime exponents (2,2) so its prime shadow is prime(2)*prime(2) = 9. In fact, the term a(1260) = 9 is the first appearance of 9 in the sequence. - _Gus Wiseman_, Apr 28 2022

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Column 2 of A353510.
Cf. A001221, A001222, A003557, A008578, A056169, A071625, A323022.
Differs from A182860 for the first time at a(30) = 5, while A182860(30) = 4.
Cf. A182863 for the first appearances.
A005361 gives product of prime exponents.
A112798 gives prime indices, sum A056239.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A325131 lists numbers relatively prime to their prime shadow.
A325755 lists numbers divisible by their prime shadow.
Cf. A109297, A182850, A353393, A353507.

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A056169(n) = { my(f=factor(n)[, 2]); sum(i=1, #f, f[i]==1); }; \\ From A056169
A328830(n) = if(1==n,n, my(u=A056169(n)); if(0==u,1,prime(u)) * A328830(A003557(n)));

a(1) = 1; for n > 1, a(n) = A008578(1+A056169(n)) * a(A003557(n)).
A001221(a(n)) = A323022(n).
A001222(a(n)) = A071625(n).
a(n) = A181819(A181819(n)). - Gus Wiseman, Apr 27 2022

Added Gus Wiseman's new name to the front of the definition. - Antti Karttunen, Apr 27 2022

A352143 Numbers whose prime indices and conjugate prime indices are all odd.

Original entry on oeis.org

1, 2, 5, 8, 11, 17, 20, 23, 31, 32, 41, 44, 47, 59, 67, 68, 73, 80, 83, 92, 97, 103, 109, 124, 125, 127, 128, 137, 149, 157, 164, 167, 176, 179, 188, 191, 197, 211, 227, 233, 236, 241, 257, 268, 269, 272, 275, 277, 283, 292, 307, 313, 320, 331, 332, 347, 353

Offset: 1

Gus Wiseman, Mar 18 2022

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, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
These are the Heinz numbers of integer partitions whose parts and conjugate parts are all odd. They are counted by A053253.

			The terms together with their prime indices begin:
   1: {}
   2: {1}
   5: {3}
   8: {1,1,1}
  11: {5}
  17: {7}
  20: {1,1,3}
  23: {9}
  31: {11}
  32: {1,1,1,1,1}
  41: {13}
  44: {1,1,5}
  47: {15}
  59: {17}
  67: {19}
  68: {1,1,7}
  73: {21}
  80: {1,1,1,1,3}

The restriction to primes is A031368.
These partitions appear to be counted by A053253.
The even version is A066207^2.
For even instead of odd conjugate parts we get A066208^2.
The first condition alone (all odd indices) is A066208, counted by A000009.
The second condition alone is A346635, counted by A000009.
A055922 counts partitions with odd multiplicities, ranked by A268335.
A066207 = indices all even, counted by A035363 (complement A086543).
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162642 counts odd prime exponents, even A162641.
A238745 gives the Heinz number of the conjugate prime signature.
A257991 counts odd indices, even A257992.
A258116 ranks strict partitions with all odd parts, even A258117.
A351979 = odd indices and even multiplicities, counted by A035457.
A352140 = even indices and odd multiplicities, counted by A055922 aerated.
A352141 = even indices and even multiplicities, counted by A035444.
A352142 = odd indices and odd multiplicities, counted by A117958.
Cf. A000290, A000701, A000720, A028260, A045931, A046682, A055396, A061395, A195017, A241638, A325698, A325700.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],And@@OddQ/@primeMS[#]&&And@@OddQ/@conj[primeMS[#]]&]

Intersection of A066208 and A346635.

A352518 Numbers > 1 that are not a prime power and whose prime indices and exponents are all themselves prime numbers.

Original entry on oeis.org

225, 675, 1089, 1125, 2601, 3025, 3267, 3375, 6075, 7225, 7803, 8649, 11979, 15125, 15129, 24025, 25947, 27225, 28125, 29403, 30375, 31329, 33275, 34969, 35937, 36125, 40401, 42025, 44217, 45387, 54675, 62001, 65025, 70227, 81675, 84375, 87025, 93987

Offset: 1

Gus Wiseman, Mar 24 2022

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 terms together with their prime indices (not factors) begin:
     225: {2,2,3,3}
     675: {2,2,2,3,3}
    1089: {2,2,5,5}
    1125: {2,2,3,3,3}
    2601: {2,2,7,7}
    3025: {3,3,5,5}
    3267: {2,2,2,5,5}
    3375: {2,2,2,3,3,3}
    6075: {2,2,2,2,2,3,3}
    7225: {3,3,7,7}
    7803: {2,2,2,7,7}
    8649: {2,2,11,11}
   11979: {2,2,5,5,5}
   15125: {3,3,3,5,5}
   15129: {2,2,13,13}
   24025: {3,3,11,11}
   25947: {2,2,2,11,11}
   27225: {2,2,3,3,5,5}
   28125: {2,2,3,3,3,3,3}
For example, 7803 = prime(1)^3 prime(4)^2.

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

These partitions are counted by A352493.
This is the restriction of A346068 to numbers that are not a prime power.
The prime-power version is A352519, counted by A230595.
A000040 lists the primes.
A000961 lists prime powers.
A001694 lists powerful numbers, counted by A007690.
A038499 counts partitions of prime length.
A053810 lists all numbers p^q for p and q prime, counted by A001221.
A056166 = prime exponents are all prime, counted by A055923.
A076610 = prime indices are all prime, counted by A000607, powerful A339218.
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, sum A056239.
A124010 gives prime signature, sorted A118914, sum A001222.
A257994 counts prime indices that are themselves prime, nonprime A330944.
A325131 = disjoint indices from exponents, counted by A114639.
Cf. A000720, A001597, A002035, A007821, A007916, A101436, A117958, A164336, A268335, A330945.

Select[Range[10000],!PrimePowerQ[#]&& And@@PrimeQ/@PrimePi/@First/@FactorInteger[#]&& And@@PrimeQ/@Last/@FactorInteger[#]&]

Sum_{n>=1} 1/a(n) = (Product_{p prime-indexed prime} (1 + Sum_{q prime} 1/p^q)) - (Sum_{p prime-indexed prime} Sum_{q prime} 1/p^q) - 1 = 0.0106862606... . - Amiram Eldar, Aug 04 2024

A352519 Numbers of the form prime(p)^q where p and q are primes. Prime powers whose prime index and exponent are both prime.

Original entry on oeis.org

9, 25, 27, 121, 125, 243, 289, 961, 1331, 1681, 2187, 3125, 3481, 4489, 4913, 6889, 11881, 16129, 24649, 29791, 32041, 36481, 44521, 58081, 68921, 76729, 78125, 80089, 109561, 124609, 134689, 160801, 161051, 177147, 185761, 205379, 212521, 259081, 299209

Offset: 1

Gus Wiseman, Mar 26 2022

nonn

Alternatively, numbers of the form prime(prime(i))^prime(j) for some positive integers i, j.

			The terms together with their prime indices begin:
      9: {2,2}
     25: {3,3}
     27: {2,2,2}
    121: {5,5}
    125: {3,3,3}
    243: {2,2,2,2,2}
    289: {7,7}
    961: {11,11}
   1331: {5,5,5}
   1681: {13,13}
   2187: {2,2,2,2,2,2,2}
   3125: {3,3,3,3,3}
   3481: {17,17}
   4489: {19,19}
   4913: {7,7,7}
   6889: {23,23}
  11881: {29,29}
  16129: {31,31}
  24649: {37,37}
  29791: {11,11,11}

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

Numbers of the form p^q for p and q prime are A053810, counted by A001221.
These partitions are counted by A230595.
This is the prime power case of A346068.
For numbers that are not a prime power we have A352518, counted by A352493.
A000040 lists the primes.
A000961 lists prime powers.
A001597 lists perfect powers.
A001694 lists powerful numbers, counted by A007690.
A056166 = prime exponents are all prime, counted by A055923.
A076610 = prime indices are all prime, counted by A000607, powerful A339218.
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, sum A056239.
A124010 gives prime signature, sorted A118914, sum A001222.
A164336 lists all possible power-towers of prime numbers.
A257994 counts prime indices that are themselves prime, nonprime A330944.
A325131 = disjoint indices from exponents, counted by A114639.
Cf. A000720, A002035, A005117, A007821, A038499, A101436, A181819, A181821.

N:= 10^7: # for terms <= N
M:=numtheory:-pi(numtheory:-pi(isqrt(N))):
PP:= {seq(ithprime(ithprime(i)),i=1..M)}:
R:= NULL:
for p in PP do
  q:= 1:
  do
    q:= nextprime(q);
    t:= p^q;
    if t > N then break fi;
    R:= R, t;
  od;
od:
sort([R]); # Robert Israel, Dec 08 2022

Select[Range[10000],PrimePowerQ[#]&&MatchQ[FactorInteger[#],{{?(PrimeQ[PrimePi[#]]&),k?PrimeQ}}]&]

from sympy import primepi, integer_nthroot, primerange
def A352519(n):
    def f(x): return int(n+x-sum(primepi(primepi(integer_nthroot(x,p)[0])) for p in primerange(x.bit_length())))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 12 2024

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A325707 Number of integer partitions of n covering an initial interval of positive integers and containing all of their distinct multiplicities.

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A352143 Numbers whose prime indices and conjugate prime indices are all odd.

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