A319613
a(n) = prime(n) * prime(2n).
Original entry on oeis.org
6, 21, 65, 133, 319, 481, 731, 1007, 1403, 2059, 2449, 3293, 4141, 4601, 5311, 6943, 8201, 9211, 10921, 12283, 13213, 15247, 16517, 19847, 22213, 24139, 25853, 28141, 29539, 31753, 37211, 40741, 43429, 46843, 52001, 54209, 58561, 62429, 66299, 70757, 75359
Offset: 1
Cf.
A000040,
A001358,
A018819,
A031215,
A031368,
A056239,
A087897,
A101417,
A112798,
A120641,
A320340,
A323092,
A323093,
A323094.
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a:= n-> (p-> p(n)*p(2*n))(ithprime):
seq(a(n), n=1..50); # Alois P. Heinz, Jan 08 2019
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Table[Prime[n]*Prime[2*n],{n,50}]
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a(n) = prime(n)*prime(2*n) \\ Felix Fröhlich, Jan 09 2019
A350845
Heinz numbers of integer partitions with at least two adjacent parts of quotient 2.
Original entry on oeis.org
6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 65, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 130, 132, 133, 138, 144, 147, 150, 156, 162, 168, 174, 180, 186, 189, 192, 195, 198, 204, 210, 216, 222, 228, 231, 234, 240, 246, 252, 258, 260, 264, 266, 270
Offset: 1
The terms and corresponding partitions begin:
6: (2,1)
12: (2,1,1)
18: (2,2,1)
21: (4,2)
24: (2,1,1,1)
30: (3,2,1)
36: (2,2,1,1)
42: (4,2,1)
48: (2,1,1,1,1)
54: (2,2,2,1)
60: (3,2,1,1)
63: (4,2,2)
65: (6,3)
66: (5,2,1)
72: (2,2,1,1,1)
78: (6,2,1)
84: (4,2,1,1)
90: (3,2,2,1)
96: (2,1,1,1,1,1)
The strict complement is counted by
A350840.
These partitions are counted by
A350846.
A000045 = sets containing n with all differences > 2.
A325160 ranks strict partitions with no successions, counted by
A003114.
Cf.
A000929,
A001105,
A018819,
A045690,
A045691,
A094537,
A154402,
A319613,
A323093,
A337135,
A342094,
A342095,
A342098,
A342191.
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primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],MemberQ[Divide@@@Partition[primeptn[#],2,1],2]&]
A350846
Number of integer partitions of n with at least two adjacent parts of quotient 2.
Original entry on oeis.org
0, 0, 0, 1, 1, 2, 4, 5, 8, 12, 18, 25, 36, 48, 65, 89, 119, 157, 207, 269, 350, 448, 574, 729, 927, 1166, 1465, 1830, 2282, 2827, 3501, 4309, 5300, 6483, 7923, 9641, 11718, 14187, 17155, 20674, 24885, 29860, 35787, 42772, 51054, 60791, 72289, 85772, 101641
Offset: 0
The a(3) = 1 through a(9) = 12 partitions:
(21) (211) (221) (42) (421) (422) (63)
(2111) (321) (2221) (521) (621)
(2211) (3211) (3221) (3321)
(21111) (22111) (4211) (4221)
(211111) (22211) (5211)
(32111) (22221)
(221111) (32211)
(2111111) (42111)
(222111)
(321111)
(2211111)
(21111111)
Cf.
A000929,
A003000,
A003114,
A018819,
A045690,
A045691,
A116931,
A120641,
A154402,
A323093,
A342094,
A342095,
A342096,
A342098.
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Table[Length[Select[IntegerPartitions[n], MemberQ[Divide@@@Partition[#,2,1],2]&]],{n,0,30}]
A323053
Number of integer partitions of n with no 1's such that no part is a power of any other (unequal) part.
Original entry on oeis.org
1, 0, 1, 1, 2, 2, 3, 4, 6, 7, 9, 12, 15, 19, 25, 30, 38, 47, 58, 71, 87, 106, 131, 156, 190, 228, 275, 328, 394, 468, 556, 661, 784, 923, 1089, 1283, 1507, 1766, 2068, 2416, 2821, 3284, 3822, 4438, 5148, 5961, 6898, 7968, 9195, 10593, 12198, 14019, 16102, 18472
Offset: 0
The a(2) = 1 through a(11) = 12 integer partitions (A = 10, B = 11):
(2) (3) (4) (5) (6) (7) (8) (9) (A) (B)
(22) (32) (33) (43) (44) (54) (55) (65)
(222) (52) (53) (63) (64) (74)
(322) (62) (72) (73) (83)
(332) (333) (433) (92)
(2222) (522) (532) (443)
(3222) (622) (533)
(3322) (632)
(22222) (722)
(3332)
(5222)
(32222)
Cf.
A001597,
A002865,
A007916,
A052410,
A101417,
A102430,
A108917,
A305148,
A305630,
A305631,
A321346,
A323093.
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stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],And[FreeQ[#,1],stableQ[#,IntegerQ[Log[#1,#2]]&]]&]],{n,30}]
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