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Numbers whose prime factorization has all odd indices and all odd exponents.
+10
8
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
COMMENTS
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.
EXAMPLE
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)
MATHEMATICA
Select[Range[100], #==1||And@@OddQ/@PrimePi/@First/@FactorInteger[#]&&And@@OddQ/@Last/@FactorInteger[#]&]
PROG
(Python)
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)))
CROSSREFS
The restriction to primes is A031368. These partitions are counted by A117958. 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.
Numbers whose prime factorization has all even indices and all even exponents.
+10
7
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
COMMENTS
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.
FORMULA
Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2*k)^2) = 1.163719... . - Amiram Eldar, Sep 19 2022
EXAMPLE
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
MATHEMATICA
Select[Range[1000], #==1||And@@EvenQ/@PrimePi/@First/@FactorInteger[#]&&And@@EvenQ/@Last/@FactorInteger[#]&]
PROG
(Python)
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)))
CROSSREFS
The second condition alone (all even exponents) is A000290, counted by A035363. The restriction to primes is A031215. These partitions are counted by A035444. A352140 = even indices with odd exponents, counted by A055922 aerated.
Cf. A000720, A028260, A055396, A061395, A181819, A195017, A241638, A268335, A276078, A324524, A324525, A324588, A325698, A325700, A352143.
Numbers whose prime factorization has all even prime indices and all odd exponents.
+10
6
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
COMMENTS
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.
EXAMPLE
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
MATHEMATICA
Select[Range[100], And@@EvenQ/@PrimePi/@First/@FactorInteger[#]&&And@@OddQ/@Last/@FactorInteger[#]&]
PROG
(Python)
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())
CROSSREFS
The restriction to primes is A031215. These partitions are counted by A055922 (aerated). Cf. A000720, A028260, A055396, A061395, A181819, A195017, A241638, A276078, A324517, A324524, A324525, A325698.
Numbers whose prime factorization has all odd prime indices and all even prime exponents.
+10
5
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
COMMENTS
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).
FORMULA
Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2*k-1)^2) = 1.4135142... . - Amiram Eldar, Sep 19 2022
EXAMPLE
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
MATHEMATICA
Select[Range[1000], #==1||And@@OddQ/@PrimePi/@First/@FactorInteger[#]&&And@@EvenQ/@Last/@FactorInteger[#]&]
PROG
(Python)
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())
CROSSREFS
The second condition alone (exponents all even) is A000290, counted by A035363. The distinct prime factors of terms all come from A031368. The first condition alone (indices all odd) is A066208, counted by A000009. A352140 = even indices with odd exponents, counted by A055922 (aerated).
Cf. A000720, A028260, A045931, A055396, A061395, A106529, A181819, A195017, A276078, A324588, A325698, A325700.
Numbers with all prime indices and exponents > 2.
+10
5
1, 125, 343, 625, 1331, 2197, 2401, 3125, 4913, 6859, 12167, 14641, 15625, 16807, 24389, 28561, 29791, 42875, 50653, 68921, 78125, 79507, 83521, 103823, 117649, 130321, 148877, 161051, 166375, 205379, 214375, 226981, 274625, 279841, 300125, 300763, 357911
COMMENTS
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.
FORMULA
Sum_{n>=1} 1/a(n) = Product_{p prime > 3} (1 + 1/(p^2*(p-1))) = (72/95)* A065483 = 1.0154153584... . - Amiram Eldar, May 28 2022
EXAMPLE
The initial terms together with their prime indices:
1: {}
125: {3,3,3}
343: {4,4,4}
625: {3,3,3,3}
1331: {5,5,5}
2197: {6,6,6}
2401: {4,4,4,4}
3125: {3,3,3,3,3}
4913: {7,7,7}
6859: {8,8,8}
12167: {9,9,9}
14641: {5,5,5,5}
15625: {3,3,3,3,3,3}
16807: {4,4,4,4,4}
24389: {10,10,10}
28561: {6,6,6,6}
29791: {11,11,11}
42875: {3,3,3,4,4,4}
MATHEMATICA
Select[Range[10000], #==1||!MemberQ[FactorInteger[#], {_?(#<5&), _}|{_, _?(#<3&)}]&]
CROSSREFS
The version for <= 2 instead of > 2 is A018256, # of compositions A137200. The version for indices and exponents prime (instead of > 2) is:
The partitions with these Heinz numbers are counted by A353501. A000726 counts partitions with multiplicities <= 2, compositions A128695.
A295341 counts partitions with some multiplicity > 2, compositions A335464.
Number of integer compositions of n with all prime parts and all prime run-lengths.
+10
4
1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 4, 0, 2, 2, 5, 4, 9, 1, 5, 12, 20, 11, 19, 18, 31, 43, 54, 37, 63, 95, 121, 124, 154, 178, 261, 353, 393, 417, 565, 770, 952, 1138, 1326, 1647, 2186, 2824, 3261, 3917, 4941, 6423, 7935, 9719, 11554, 14557, 18536, 23380, 27985
EXAMPLE
The a(13) = 2 through a(16) = 9 compositions:
(22333) (77) (555) (3355)
(33322) (2255) (33333) (5533)
(5522) (222333) (22255)
(223322) (333222) (55222)
(2222222) (332233)
(2222233)
(2223322)
(2233222)
(3322222)
MAPLE
b:= proc(n, h) option remember; `if`(n=0, 1, add(`if`(i<>h and isprime(i),
add(`if`(isprime(j), b(n-i*j, i), 0), j=2..n/i), 0), i=2..n/2))
end:
a:= n-> b(n, 0):
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], And@@PrimeQ/@#&&And@@PrimeQ/@Length/@Split[#]&]], {n, 0, 15}]
CROSSREFS
A106356 counts compositions by number of adjacent equal parts.
A114901 counts compositions with no runs of length 1, ranked by A353427.
Number of non-constant integer partitions of n into prime parts with prime multiplicities.
+10
3
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 3, 0, 1, 4, 5, 3, 1, 3, 5, 7, 3, 5, 6, 8, 8, 11, 7, 6, 8, 15, 14, 14, 10, 15, 17, 21, 18, 23, 20, 28, 25, 31, 27, 35, 32, 33, 37, 46, 41, 50, 45, 58, 56, 63, 59, 78, 69, 76, 81, 85, 80, 103, 107, 111, 114, 127
EXAMPLE
The a(n) partitions for selected n (B = 11):
n = 10 16 19 20 25 28
---------------------------------------------------------------
3322 5533 55333 7733 77722 BB33
55222 55522 77222 5533333 BB222
3322222 3333322 553322 5553322 775522
33322222 5522222 55333222 55533322
332222222 55522222 772222222
333333322 3322222222222
3333322222
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], !SameQ@@#&&And@@PrimeQ/@#&& And@@PrimeQ/@Length/@Split[#]&]], {n, 0, 30}]
CROSSREFS
These partitions are ranked by A352518. A038499 counts partitions of prime length.
A055923 counts partitions with prime multiplicities, ranked by A056166.
A257994 counts prime indices that are themselves prime.
A339218 counts powerful partitions into prime parts, ranked by A352492.
Cf. A000005, A007690, A031368, A035444, A052485, A056239, A066208, A089723, A114639, A320628, A330945.
Number of integer partitions of n with all parts and all multiplicities > 2.
+10
2
1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 1, 0, 2, 0, 2, 3, 0, 0, 6, 2, 0, 6, 3, 2, 9, 2, 5, 11, 3, 5, 18, 6, 4, 20, 13, 8, 26, 10, 17, 37, 14, 16, 51, 23, 24, 58, 38, 32, 75, 44, 52, 100, 52, 59, 143, 75, 77, 159, 114, 112, 203, 132, 154, 266, 175
EXAMPLE
The a(n) partitions for selected n (A = 10):
n=9: n=12: n=21: n=24: n=30:
------------------------------------------------------
(333) (444) (777) (888) (AAA)
(3333) (444333) (6666) (66666)
(3333333) (444444) (555555)
(555333) (666444)
(4443333) (777333)
(33333333) (6663333)
(55533333)
(444333333)
(3333333333)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Min@@#>2&&Min@@Length/@Split[#]>2&]], {n, 0, 30}]
CROSSREFS
The version for only parts > 2 is A008483. The version for only multiplicities > 2 is A100405. The version for parts and multiplicities > 1 is A339222, ranked by A062739. For prime parts and multiplicities we have A351982, compositions A353429. These partitions are ranked by A353502. A137200 counts compositions with all parts and run-lengths <= 2.
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