A342081
Numbers without an inferior odd divisor > 1.
Original entry on oeis.org
1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 26, 28, 29, 31, 32, 34, 37, 38, 41, 43, 44, 46, 47, 52, 53, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 79, 82, 83, 86, 88, 89, 92, 94, 97, 101, 103, 104, 106, 107, 109, 113, 116, 118, 122, 124
Offset: 1
The divisors > 1 of 72 are {2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72}, of which {3, 9} are odd and {2, 3, 4, 6, 8} are inferior, with intersection {3}, so 72 is not in the sequence.
The strictly inferior version is the same with
A001248 added.
A006530 selects the greatest prime factor.
A020639 selects the smallest prime factor.
A038548 counts superior (or inferior) divisors, with strict case
A056924.
- Odd -
A026424 lists numbers with odd Omega.
A027193 counts odd-length partitions.
A058695 counts partitions of odd numbers.
A340101 counts factorizations into odd factors;
A340102 also has odd length.
A341594 counts strictly superior odd divisors
A341675 counts superior odd divisors.
Cf.
A000005,
A000203,
A001055,
A001221,
A001222,
A001414,
A207375,
A244991,
A300272,
A340832,
A340931.
-
Select[Range[100],Function[n,Select[Divisors[n]//Rest,OddQ[#]&&#<=n/#&]=={}]]
-
is(n) = #select(x -> x > 2 && x^2 <= n, factor(n)[, 1]) == 0; \\ Amiram Eldar, Nov 01 2024
-
from sympy import primefactors
A342081_list = [n for n in range(1,10**3) if len([p for p in primefactors(n) if p > 2 and p*p <= n]) == 0] # Chai Wah Wu, Mar 08 2021
A342082
Numbers with an inferior odd divisor > 1.
Original entry on oeis.org
9, 12, 15, 18, 21, 24, 25, 27, 30, 33, 35, 36, 39, 40, 42, 45, 48, 49, 50, 51, 54, 55, 56, 57, 60, 63, 65, 66, 69, 70, 72, 75, 77, 78, 80, 81, 84, 85, 87, 90, 91, 93, 95, 96, 98, 99, 100, 102, 105, 108, 110, 111, 112, 114, 115, 117, 119, 120, 121, 123, 125
Offset: 1
The divisors > 1 of 72 are {2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72}, of which {3, 9} are odd and {2, 3, 4, 6, 8} are inferior, with intersection {3}, so 72 is in the sequence.
The strictly inferior version is the same with
A001248 removed.
A006530 selects the greatest prime factor.
A020639 selects the smallest prime factor.
A038548 counts superior (or inferior) divisors, with strict case
A056924.
- Odd -
A026424 lists numbers with odd Omega.
A027193 counts odd-length partitions.
A058695 counts partitions of odd numbers.
A340101 counts factorizations into odd factors;
A340102 also has odd length.
A341594 counts strictly superior odd divisors
A341675 counts superior odd divisors.
Cf.
A000005,
A000203,
A001055,
A001221,
A001222,
A001414,
A207375,
A244991,
A300272,
A340832,
A340931.
-
Select[Range[100],Function[n,Select[Divisors[n]//Rest,OddQ[#]&&#<=n/#&]!={}]]
-
is(n) = #select(x -> x > 2 && x^2 <= n, factor(n)[, 1]) > 0; \\ Amiram Eldar, Nov 01 2024
-
from sympy import primefactors
A342082_list = [n for n in range(1,10**3) if len([p for p in primefactors(n) if p > 2 and p*p <= n]) > 0] # Chai Wah Wu, Mar 09 2021
A346634
Number of strict odd-length integer partitions of 2n + 1.
Original entry on oeis.org
1, 1, 1, 2, 4, 6, 9, 14, 19, 27, 38, 52, 71, 96, 128, 170, 224, 293, 380, 491, 630, 805, 1024, 1295, 1632, 2048, 2560, 3189, 3958, 4896, 6038, 7424, 9100, 11125, 13565, 16496, 20013, 24223, 29250, 35244, 42378, 50849, 60896, 72789, 86841, 103424, 122960, 145937
Offset: 0
The a(0) = 1 through a(7) = 14 partitions:
(1) (3) (5) (7) (9) (11) (13) (15)
(4,2,1) (4,3,2) (5,4,2) (6,4,3) (6,5,4)
(5,3,1) (6,3,2) (6,5,2) (7,5,3)
(6,2,1) (6,4,1) (7,4,2) (7,6,2)
(7,3,1) (7,5,1) (8,4,3)
(8,2,1) (8,3,2) (8,5,2)
(8,4,1) (8,6,1)
(9,3,1) (9,4,2)
(10,2,1) (9,5,1)
(10,3,2)
(10,4,1)
(11,3,1)
(12,2,1)
(5,4,3,2,1)
The even version is the even bisection of
A067661.
The case of all odd parts is counted by
A069911 (non-strict:
A078408).
A340385 counts partitions with odd length and maximum, ranked by
A340386.
Other cases of odd length:
-
b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
`if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
end:
a:= n-> b(2*n+1$2, 0):
seq(a(n), n=0..80); # Alois P. Heinz, Aug 05 2021
-
Table[Length[Select[IntegerPartitions[2n+1],UnsameQ@@#&&OddQ[Length[#]]&]],{n,0,15}]
A341448
Heinz numbers of integer partitions of type OO.
Original entry on oeis.org
6, 14, 15, 24, 26, 33, 35, 38, 51, 54, 56, 58, 60, 65, 69, 74, 77, 86, 93, 95, 96, 104, 106, 119, 122, 123, 126, 132, 135, 140, 141, 142, 143, 145, 150, 152, 158, 161, 177, 178, 185, 201, 202, 204, 209, 214, 215, 216, 217, 219, 221, 224, 226, 232, 234, 240
Offset: 1
The sequence of partitions together with their Heinz numbers begins:
6: (2,1) 74: (12,1) 141: (15,2)
14: (4,1) 77: (5,4) 142: (20,1)
15: (3,2) 86: (14,1) 143: (6,5)
24: (2,1,1,1) 93: (11,2) 145: (10,3)
26: (6,1) 95: (8,3) 150: (3,3,2,1)
33: (5,2) 96: (2,1,1,1,1,1) 152: (8,1,1,1)
35: (4,3) 104: (6,1,1,1) 158: (22,1)
38: (8,1) 106: (16,1) 161: (9,4)
51: (7,2) 119: (7,4) 177: (17,2)
54: (2,2,2,1) 122: (18,1) 178: (24,1)
56: (4,1,1,1) 123: (13,2) 185: (12,3)
58: (10,1) 126: (4,2,2,1) 201: (19,2)
60: (3,2,1,1) 132: (5,2,1,1) 202: (26,1)
65: (6,3) 135: (3,2,2,2) 204: (7,2,1,1)
69: (9,2) 140: (4,3,1,1) 209: (8,5)
Note: A-numbers of ranking sequences are in parentheses below.
The case of odd parts, length, and sum is counted by
A078408 (
A300272).
These partitions (for odd n) are counted by
A236914.
A340101 counts factorizations into odd factors.
Cf.
A000700,
A024429,
A027187,
A106529,
A117409,
A174725,
A257541,
A325134,
A339890,
A340102,
A340604.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],OddQ[Count[primeMS[#],?EvenQ]]&&OddQ[Count[primeMS[#],?OddQ]]&]
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