A162642 -id:A162642 - cp's OEIS frontend

A295662 Number of odd exponents larger than one in the canonical prime factorization of n.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Antti Karttunen, Nov 28 2017

			For n = 24 = 2^3 * 3^1 there are two odd exponents, but only the other is larger than 1, thus a(24) = 1.
For n = 216 = 2^3 * 3^3 there are two odd exponents larger than 1, thus a(216) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n.

Cf. A056169, A162642, A295659, A295663, A295664.

Cf. A295661 (positions of nonzero terms).

Array[Count[FactorInteger[#][[All, -1]], ?(And[OddQ@ #, # > 1] &)] &, 105] (* _Michael De Vlieger, Nov 28 2017 *)

a(n) = vecsum(apply(x -> x%2 - (x==1), factor(n)[, 2])); \\ Amiram Eldar, Sep 28 2023

Additive with a(p) = 0, a(p^e) = A000035(e) if e > 1.
a(1) = 0; and for n > 1, if A067029(n) = 1, a(n) = a(A028234(n)), otherwise A000035(A067029(n)) + a(A028234(n)).
a(n) = A162642(n) - A056169(n).
a(n) <= A295659(n).
a(n) = 0 iff A295663(n) = 0, and when A295663(n) > 0, a(n) <= A295663(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p^2*(p+1)) = 0.122017493776862257491... . - Amiram Eldar, Sep 28 2023

A055460 Number of primes with odd exponents in the prime power factorization of n!.

Original entry on oeis.org

0, 1, 2, 2, 3, 1, 2, 3, 3, 1, 2, 3, 4, 4, 4, 4, 5, 4, 5, 4, 6, 6, 7, 5, 5, 5, 6, 5, 6, 5, 6, 7, 9, 7, 7, 7, 8, 8, 8, 8, 9, 10, 11, 10, 9, 7, 8, 7, 7, 8, 10, 9, 10, 8, 10, 12, 14, 12, 13, 11, 12, 12, 11, 11, 13, 12, 13, 12, 12, 13, 14, 13, 14, 14, 15, 14, 14, 11, 12, 13, 13, 13, 14, 16, 16, 14

Offset: 1

Labos Elemer, Jun 26 2000

nonn

The products of the corresponding primes form A055204.
Also, the number of primes dividing the squarefree part of n! (=A055204(n)).
Also, the number of prime factors in the factorization of n! into distinct terms of A050376. See the references in A241289. - Vladimir Shevelev, Apr 16 2014

			For n = 100, the exponents of primes in the factorization of n! are {97,48,24,16,9,7,5,5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1}, and there are 17 odd values: {97,9,7,5,5,3,3,1,1,1,1,1,1,1,1,1,1}, so a(100) = 17.
The factorization of 6! into distinct terms of A050376 is 5*9*16 with only one prime, so a(6)=1. - _Vladimir Shevelev_, Apr 16 2014

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 (in Russian; MR 2000f: 11097, pp. 3912-3913).

Max Alekseyev, Table of n, a(n) for n = 1..100000
S. Litsyn and V. S. Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.

Cf. A000142, A000720, A007913, A008833, A162642, A348841.

Cf. A249016 (indices of records), A249017 (values of records)

Table[Count[FactorInteger[n!][[All, -1]], m_ /; OddQ@ m] - Boole[n == 1], {n, 100}] (* Michael De Vlieger, Feb 05 2017 *)

PARI
```
a(n) = omega(core(n!))
```

a(n) = A001221(A055204(n)). - Max Alekseyev, Oct 19 2014
From Wolfdieter Lang, Nov 06 2021: (Start)
a(n) = A162642(A000142(n)).
a(n) = A000720(n) - A348841(n), (End)

Edited by Max Alekseyev, Oct 19 2014

A352142 Numbers whose prime factorization has all odd indices and all odd exponents.

Original entry on oeis.org

1, 2, 5, 8, 10, 11, 17, 22, 23, 31, 32, 34, 40, 41, 46, 47, 55, 59, 62, 67, 73, 82, 83, 85, 88, 94, 97, 103, 109, 110, 115, 118, 125, 127, 128, 134, 136, 137, 146, 149, 155, 157, 160, 166, 167, 170, 179, 184, 187, 191, 194, 197, 205, 206, 211, 218, 227, 230

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 with all odd parts and all odd multiplicities, counted by A117958.

			The terms together with their prime indices begin:
   1 = 1
   2 = prime(1)
   5 = prime(3)
   8 = prime(1)^3
  10 = prime(1) prime(3)
  11 = prime(5)
  17 = prime(7)
  22 = prime(1) prime(5)
  23 = prime(9)
  31 = prime(11)
  32 = prime(1)^5
  34 = prime(1) prime(7)
  40 = prime(1)^3 prime(3)

David A. Corneth, Table of n, a(n) for n = 1..10000

The restriction to primes is A031368.
The first condition alone is A066208, counted by A000009.
These partitions are counted by A117958.
The squarefree case is A258116, even A258117.
The second condition alone is A268335, counted by A055922.
The even-even version is A352141 counted by A035444.
A000290 = exponents all even, counted by A035363.
A056166 = exponents all prime, counted by A055923.
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.
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.
A352140 = even indices with odd exponents, counted by A055922 aerated.
A352143 = odd indices with odd conjugate indices, counted by A053253 aerated.
Cf. A000720, A028260, A055396, A061395, A106529, A181819, A195017, A241638, A276078, A324517, A324524, A324525, A325698, A325700.

Select[Range[100],#==1||And@@OddQ/@PrimePi/@First/@FactorInteger[#]&&And@@OddQ/@Last/@FactorInteger[#]&]

from itertools import count, islice
from sympy import primepi, factorint
def A352142_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda k:all(map(lambda x:x[1]%2 and primepi(x[0])%2, factorint(k).items())),count(max(startvalue,1)))
A352142_list = list(islice(A352142_gen(),30)) # Chai Wah Wu, Mar 18 2022

Intersection of A066208 and A268335.
A257991(a(n)) = A001222(a(n)).
A162642(a(n)) = A001221(a(n)).
A257992(a(n)) = A162641(a(n)) = 0.

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

Original entry on oeis.org

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

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)).

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

A366849 Odd numbers whose halved even prime indices are relatively prime.

Original entry on oeis.org

3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 91, 93, 99, 105, 111, 117, 123, 129, 135, 141, 147, 153, 159, 165, 171, 177, 183, 189, 195, 201, 203, 207, 213, 219, 225, 231, 237, 243, 247, 249, 255, 261, 267, 273, 279, 285, 291, 297, 301, 303, 309

Offset: 1

Gus Wiseman, Nov 01 2023

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 even prime indices of 91 are {4,6}, halved {2,3}, which are relatively prime, so 91 is in the sequence.
The prime indices of 665 are {3,4,8}, even {4,8}, halved {2,4}, which are not relatively prime, so 665 is not in the sequence.
The terms together with their prime indices begin:
   3: {2}
   9: {2,2}
  15: {2,3}
  21: {2,4}
  27: {2,2,2}
  33: {2,5}
  39: {2,6}
  45: {2,2,3}
  51: {2,7}
  57: {2,8}
  63: {2,2,4}
  69: {2,9}
  75: {2,3,3}
  81: {2,2,2,2}
  87: {2,10}
  91: {4,6}
  93: {2,11}
  99: {2,2,5}

For odd instead of halved even prime indices we have A366848.
A version for odd indices A366846, counted by A366850.
This is the odd restriction of A366847, counted by A366845.
A000041 counts integer partitions, strict A000009 (also into odds).
A035363 counts partitions into all even parts, ranks A066207.
A112798 lists prime indices, length A001222, sum A056239.
A162641 counts even prime exponents, odd A162642.
A257992 counts even prime indices, odd A257991.
A289509 lists numbers with relatively prime prime indices, ones of A289508, counted by A000837.
A366528 adds up odd prime indices, partition triangle A113685.
A366531 = 2*A366533 adds up even prime indices, triangle A113686/A174713.
Cf. A000720, A055396, A061395, A066208, A302697, A325698, A366842, A366843.

Select[Range[100], OddQ[#]&&GCD@@Select[PrimePi/@First/@FactorInteger[#], EvenQ]==2&]

A375033 The maximum even exponent in the prime factorization of n, or 0 if no such exponent exists.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 0, 0

Offset: 1

Amiram Eldar, Jul 28 2024

First differs from A350386 at n = 36.
The asymptotic density of the occurrences of 0's is d(0) = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442... (A065463; the asymptotic density of the exponentially odd numbers, A268335).
The asymptotic density of the occurrences of 2*k, for k = 1, 2, ..., is d(k) = Product_{p prime} (1 - 1/(p^(2*k+1)*(p+1))) - Product_{p prime} (1 - 1/(p^(2*k-1)*(p+1))).
For example, the asymptotic density of the occurrences of 2's is d(1) = Product_{p prime} (1 - 1/(p^3*(p+1))) - Product_{p prime}(1 - 1/(p*(p+1))) = 0.243291... (the asymptotic density of A375031).

Cf. A051903, A065463, A162642, A268335, A350386, A350388, A375031, A375032.

a[n_] := Max[0, Max[Select[FactorInteger[n][[;; , 2]], EvenQ]]]; a[1] = 0; Array[a, 100]

a(n) = {my(e = select(x -> !(x % 2), factor(n)[,2])); if(#e == 0, 0, vecmax(e));}

max(a(n), A375032(n)) = A051903(n).
a(n) = 0 if and only if n is an exponentially odd number (A268335).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} (2*k) * d(k) = 0.72584606502990528747..., where d(k) is defined in the Comments section above.
a(n) = A051903(A350388(n)). - Amiram Eldar, Aug 17 2024

A386430 Odd numbers k such that there are no prime factors p of sigma(k) such that p does not divide A003961(k) and the valuation(k, p) is different from valuation(sigma(k), p), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 3, 7, 15, 21, 27, 31, 33, 57, 69, 87, 91, 93, 105, 127, 141, 177, 189, 195, 217, 231, 237, 273, 285, 301, 381, 399, 447, 465, 483, 495, 513, 567, 573, 597, 609, 627, 651, 717, 775, 819, 837, 861, 889, 903, 987, 1023, 1029, 1149, 1185, 1239, 1311, 1365, 1419, 1431, 1437, 1455, 1497, 1561, 1653, 1659, 1687, 1743

Offset: 1

Antti Karttunen, Aug 22 2025

Conjecture: After the initial 1, and apart from any hypothetical odd perfect numbers, all other terms are in A248150, i.e., sigma(k) == 0 (mod 4). This would imply (with the same caveat), that this sequence has no common terms with A228058 and no squares larger than one. This is true at least for the first 709203 terms (terms in range [1..2^34]).

Terms k such that A162642(k) = 1 are rare: 3, 7, 27, 31, 127, 567, 775, 8191, 27783, 131071, 524287, 2147483647, ... (odd terms of A387160).

			a(386548) = 5919068925 = 3^4 * 5^2 * 7^2 * 11^2 * 17 * 29. sigma(5919068925) = 15355618740 = 2^2 * 3^4 * 5 * 7 * 11^2 * 19^2 * 31. The "don't care primes" is given by A003961(A007947(5919068925))) = 2947945 = 5*7*11*13*19*31, thus only odd prime factor that matters here is 3, which in case has the same exponent (4) in both n = 5919068925 and sigma(n). In a way, this number is very close to satisfying Euler's criterion for odd perfect numbers (A228058), except that it has two unitary prime factors of the form 4k+1, instead of just one, apart from the square factor. Both n/17 and n/29 are in A228058.

Odd terms of A351554.
Cf. A000203, A003961, A056169, A162642, A228058, A248150, A351555, A386424, A387160.
Cf. A349169 (subsequence).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351555(n) = { my(s=sigma(n),f=factor(s),u=A003961(n)); sum(k=1,#f~,if((f[k,1]%2) && 0!=(u%f[k,1]), (valuation(n,f[k,1])!=f[k,2]), 0)); };
isA386430(n) = ((n%2) && (0==A351555(n)));

{k | k odd, A351555(k) = 0}.

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.

cp's OEIS Frontend

A295662 Number of odd exponents larger than one in the canonical prime factorization of n.

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A055460 Number of primes with odd exponents in the prime power factorization of n!.

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A352142 Numbers whose prime factorization has all odd indices and all odd exponents.

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

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

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

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A366849 Odd numbers whose halved even prime indices are relatively prime.

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