A364462
Positive integers having a divisor of the form prime(a)*prime(b) such that prime(a+b) is also a divisor.
Original entry on oeis.org
12, 24, 30, 36, 48, 60, 63, 70, 72, 84, 90, 96, 108, 120, 126, 132, 140, 144, 150, 154, 156, 165, 168, 180, 189, 192, 204, 210, 216, 228, 240, 252, 264, 270, 273, 276, 280, 286, 288, 300, 308, 312, 315, 324, 325, 330, 336, 348, 350, 360, 372, 378, 384, 390
Offset: 1
The terms together with their prime indices begin:
12: {1,1,2}
24: {1,1,1,2}
30: {1,2,3}
36: {1,1,2,2}
48: {1,1,1,1,2}
60: {1,1,2,3}
63: {2,2,4}
70: {1,3,4}
72: {1,1,1,2,2}
84: {1,1,2,4}
90: {1,2,2,3}
96: {1,1,1,1,1,2}
108: {1,1,2,2,2}
120: {1,1,1,2,3}
126: {1,2,2,4}
132: {1,1,2,5}
140: {1,1,3,4}
144: {1,1,1,1,2,2}
Subsets not of this type are counted by
A085489, w/ re-usable parts
A007865.
Subsets of this type are counted by
A088809, with re-usable parts
A093971.
Partitions not of this type are counted by
A236912.
Partitions of this type are counted by
A237113.
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filter:= proc(n) local F, i,j,m;
F:= map(t -> `if`(t[2]>=2, numtheory:-pi(t[1])$2, numtheory:-pi(t[1])), ifactors(n)[2]);
for i from 1 to nops(F)-1 do for j from 1 to i-1 do
if member(F[i]+F[j],F) then return true fi
od od;
false
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 30 2023
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prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[prix[#], Total/@Subsets[prix[#],{2}]]!={}&]
A364910
Number of integer partitions of 2n whose distinct parts sum to n.
Original entry on oeis.org
1, 1, 1, 3, 3, 4, 12, 11, 19, 23, 54, 55, 103, 115, 178, 289, 389, 507, 757, 970, 1343, 2033, 2579, 3481, 4840, 6312, 8317, 10998, 15459, 19334, 26368, 33480, 44709, 56838, 74878, 93369, 128109, 157024, 206471, 258357, 338085, 417530, 544263, 669388, 859570, 1082758, 1367068
Offset: 0
The a(0) = 1 through a(7) = 11 partitions:
() (11) (22) (33) (44) (55) (66) (77)
(2211) (3311) (3322) (4422) (4433)
(21111) (311111) (4411) (5511) (5522)
(4111111) (33321) (6611)
(42222) (442211)
(322221) (4222211)
(332211) (4421111)
(3222111) (42221111)
(3321111) (422111111)
(32211111) (611111111)
(51111111) (4211111111)
(321111111)
The a(0) = 1 through a(7) = 11 linear combinations:
0 1*1 1*2 1*3 1*4 1*5 1*6 1*7
0*2+3*1 0*3+4*1 0*4+5*1 0*4+3*2 0*6+7*1
1*2+1*1 1*3+1*1 1*3+1*2 0*5+6*1 1*4+1*3
1*4+1*1 1*4+1*2 1*5+1*2
1*5+1*1 1*6+1*1
0*3+0*2+6*1 0*4+0*2+7*1
0*3+1*2+4*1 0*4+1*2+5*1
0*3+2*2+2*1 0*4+2*2+3*1
0*3+3*2+0*1 0*4+3*2+1*1
1*3+0*2+3*1 1*4+0*2+3*1
1*3+1*2+1*1 1*4+1*2+1*1
2*3+0*2+0*1
The case with no zero coefficients is
A000009.
A version based on Heinz numbers is
A364906.
Using all partitions (not just strict) we get
A364907.
Using strict partitions of any number from 1 to n gives
A365002.
These partitions have ranks
A365003.
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Table[Length[Select[IntegerPartitions[2n],Total[Union[#]]==n&]],{n,0,15}]
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a(n) = {my(res = 0); forpart(p = 2*n,s = Set(p); if(vecsum(s) == n, res++)); res} \\ David A. Corneth, Aug 20 2023
-
from sympy.utilities.iterables import partitions
def A364910(n): return sum(1 for d in partitions(n<<1,k=n) if sum(set(d))==n) # Chai Wah Wu, Sep 13 2023
A364348
Numbers with two possibly equal divisors prime(a) and prime(b) such that prime(a+b) is also a divisor.
Original entry on oeis.org
6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 65, 66, 70, 72, 78, 84, 90, 96, 102, 105, 108, 114, 120, 126, 130, 132, 133, 138, 140, 144, 147, 150, 154, 156, 162, 165, 168, 174, 180, 186, 189, 192, 195, 198, 204, 210, 216, 222, 228, 231, 234, 240, 246, 252
Offset: 1
We have 6 because prime(1) and prime(1) are both divisors of 6, and prime(1+1) is also.
The terms together with their prime indices begin:
6: {1,2}
12: {1,1,2}
18: {1,2,2}
21: {2,4}
24: {1,1,1,2}
30: {1,2,3}
36: {1,1,2,2}
42: {1,2,4}
48: {1,1,1,1,2}
54: {1,2,2,2}
60: {1,1,2,3}
63: {2,2,4}
65: {3,6}
66: {1,2,5}
70: {1,3,4}
72: {1,1,1,2,2}
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prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[prix[#],Total/@Tuples[prix[#],2]]!={}&]
A364533
Number of strict integer partitions of n containing the sum of no pair of distinct parts. A variation of sum-free strict partitions.
Original entry on oeis.org
1, 1, 1, 2, 2, 3, 3, 5, 5, 8, 7, 11, 11, 15, 15, 21, 22, 28, 32, 38, 40, 51, 55, 65, 74, 83, 94, 111, 119, 136, 160, 174, 196, 222, 252, 273, 315, 341, 391, 425, 477, 518, 602, 636, 719, 782, 886, 944, 1073, 1140, 1302, 1380, 1553, 1651, 1888, 1995, 2224, 2370
Offset: 0
The a(1) = 1 through a(12) = 11 partitions (A..C = 10..12):
1 2 3 4 5 6 7 8 9 A B C
21 31 32 42 43 53 54 64 65 75
41 51 52 62 63 73 74 84
61 71 72 82 83 93
421 521 81 91 92 A2
432 631 A1 B1
531 721 542 543
621 632 732
641 741
731 831
821 921
Allowing re-used parts gives
A364346.
The linear combination-free version is
A364350.
The complement in strict partitions is
A364670, w/ re-used parts
A363226.
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Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Intersection[#, Total/@Subsets[#,{2}]] == {}&]],{n,0,30}]
A350844
Number of strict integer partitions of n with no difference -2.
Original entry on oeis.org
1, 1, 1, 2, 1, 3, 3, 4, 4, 7, 7, 8, 11, 12, 15, 18, 21, 23, 31, 32, 40, 45, 54, 59, 73, 78, 94, 106, 122, 136, 161, 177, 203, 231, 259, 293, 334, 372, 417, 476, 525, 592, 663, 742, 821, 931, 1020, 1147, 1271, 1416, 1558, 1752, 1916, 2137, 2357, 2613, 2867
Offset: 0
The a(1) = 1 through a(12) = 11 partitions (A..C = 10..12):
1 2 3 4 5 6 7 8 9 A B C
21 32 51 43 62 54 73 65 84
41 321 52 71 63 82 74 93
61 521 72 91 83 A2
81 541 92 B1
432 721 A1 543
621 4321 632 651
821 732
741
921
6321
The version for no difference 0 is
A000009.
The version for subsets of prescribed maximum is
A005314.
A027187 counts partitions of even length.
Cf.
A000929,
A003000,
A018819,
A040039,
A045690,
A045691,
A154402,
A303362,
A323094,
A342095,
A342097.
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Table[Length[Select[IntegerPartitions[n],FreeQ[Differences[#],0|-2]&]],{n,0,30}]
A364531
Positive integers with no prime index equal to the sum of prime indices of any nonprime divisor.
Original entry on oeis.org
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 77
Offset: 1
A299701 counts distinct subset-sums of prime indices.
A363260 counts partitions disjoint from differences, complement
A364467.
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prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[prix[#],Total/@Subsets[prix[#],{2,Length[prix[#]]}]]=={}&]
A364670
Number of strict integer partitions of n with a part equal to the sum of two distinct others. A variation of sum-full strict partitions.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3, 1, 4, 3, 7, 6, 10, 10, 14, 16, 24, 25, 34, 39, 48, 59, 71, 81, 103, 120, 136, 166, 194, 226, 260, 312, 353, 419, 473, 557, 636, 742, 824, 974, 1097, 1266, 1418, 1646, 1837, 2124, 2356, 2717, 3029, 3469, 3830, 4383, 4884, 5547
Offset: 0
The a(6) = 1 through a(16) = 10 strict partitions (A = 10):
321 . 431 . 532 5321 642 5431 743 6432 853
541 651 6421 752 6531 862
4321 5421 7321 761 7431 871
6321 5432 7521 6532
6431 9321 6541
6521 54321 7432
8321 7621
8431
A321
64321
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Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Intersection[#, Total/@Subsets[#,{2}]]!={}&]],{n,0,30}]
A364467
Number of integer partitions of n where some part is the difference of two consecutive parts.
Original entry on oeis.org
0, 0, 0, 1, 1, 2, 4, 5, 9, 13, 21, 28, 42, 55, 78, 106, 144, 187, 255, 325, 429, 554, 717, 906, 1165, 1460, 1853, 2308, 2899, 3582, 4468, 5489, 6779, 8291, 10173, 12363, 15079, 18247, 22124, 26645, 32147, 38555, 46285, 55310, 66093, 78684, 93674, 111104
Offset: 0
The a(3) = 1 through a(9) = 13 partitions:
(21) (211) (221) (42) (421) (422) (63)
(2111) (321) (2221) (431) (621)
(2211) (3211) (521) (3321)
(21111) (22111) (3221) (4221)
(211111) (4211) (4311)
(22211) (5211)
(32111) (22221)
(221111) (32211)
(2111111) (42111)
(222111)
(321111)
(2211111)
(21111111)
For all differences of pairs parts we have
A363225, complement
A364345.
The complement is counted by
A363260.
These partitions have ranks
A364537.
A325325 counts partitions with distinct first differences.
Cf.
A002865,
A025065,
A093971,
A108917,
A196723,
A229816,
A236912,
A237113,
A237667,
A320347,
A326083.
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Table[Length[Select[IntegerPartitions[n],Intersection[#,-Differences[#]]!={}&]],{n,0,30}]
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from collections import Counter
from sympy.utilities.iterables import partitions
def A364467(n): return sum(1 for s,p in map(lambda x: (x[0],tuple(sorted(Counter(x[1]).elements()))), partitions(n,size=True)) if not set(p).isdisjoint({p[i+1]-p[i] for i in range(s-1)})) # Chai Wah Wu, Sep 26 2023
A350837
Number of integer partitions of n with no adjacent parts of quotient 2.
Original entry on oeis.org
1, 1, 2, 2, 4, 5, 7, 10, 14, 18, 24, 31, 41, 53, 70, 87, 112, 140, 178, 221, 277, 344, 428, 526, 648, 792, 971, 1180, 1436, 1738, 2103, 2533, 3049, 3660, 4387, 5242, 6259, 7450, 8860, 10511, 12453, 14723, 17387, 20489, 24121, 28343, 33269, 38982, 45632, 53327
Offset: 0
The a(1) = 1 through a(7) = 10 partitions:
(1) (2) (3) (4) (5) (6) (7)
(11) (111) (22) (32) (33) (43)
(31) (41) (51) (52)
(1111) (311) (222) (61)
(11111) (411) (322)
(3111) (331)
(111111) (511)
(4111)
(31111)
(1111111)
The sets version (subsets of prescribed maximum) is
A045691.
These partitions are ranked by
A350838.
Cf.
A000070,
A003000,
A003114, `
A003242,
A051424, `
A101417,
A120641,
A154402,
A305148,
A323093,
A323094,
A342095,
A350839.
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Table[Length[Select[IntegerPartitions[n], FreeQ[Divide@@@Partition[#,2,1],2]&]],{n,0,15}]
A350838
Heinz numbers of partitions with no adjacent parts of quotient 2.
Original entry on oeis.org
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83
Offset: 1
The terms and their prime indices begin:
1: {} 19: {8} 38: {1,8}
2: {1} 20: {1,1,3} 39: {2,6}
3: {2} 22: {1,5} 40: {1,1,1,3}
4: {1,1} 23: {9} 41: {13}
5: {3} 25: {3,3} 43: {14}
7: {4} 26: {1,6} 44: {1,1,5}
8: {1,1,1} 27: {2,2,2} 45: {2,2,3}
9: {2,2} 28: {1,1,4} 46: {1,9}
10: {1,3} 29: {10} 47: {15}
11: {5} 31: {11} 49: {4,4}
13: {6} 32: {1,1,1,1,1} 50: {1,3,3}
14: {1,4} 33: {2,5} 51: {2,7}
15: {2,3} 34: {1,7} 52: {1,1,6}
16: {1,1,1,1} 35: {3,4} 53: {16}
17: {7} 37: {12} 55: {3,5}
The sets version (subsets of prescribed maximum) is counted by
A045691.
These partitions are counted by
A350837.
The strict case is counted by
A350840.
A000045 = sets containing n with all differences > 2.
Cf.
A000302,
A001105,
A003000,
A018819,
A094537,
A120641,
A154402,
A319613,
A323093,
A337135,
A342097,
A342095.
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primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],And@@Table[FreeQ[Divide@@@Partition[primeptn[#],2,1],2],{i,2,PrimeOmega[#]}]&]
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