A325197
Heinz numbers of integer partitions such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 2.
Original entry on oeis.org
5, 8, 14, 21, 24, 25, 27, 28, 35, 36, 40, 54, 56, 66, 98, 99, 110, 120, 125, 132, 135, 147, 154, 165, 168, 175, 180, 189, 196, 198, 200, 220, 225, 231, 245, 250, 252, 264, 270, 275, 280, 297, 300, 308, 375, 378, 385, 390, 392, 396, 440, 450, 500, 546, 585, 594
Offset: 1
The sequence of terms together with their prime indices begins:
5: {3}
8: {1,1,1}
14: {1,4}
21: {2,4}
24: {1,1,1,2}
25: {3,3}
27: {2,2,2}
28: {1,1,4}
35: {3,4}
36: {1,1,2,2}
40: {1,1,1,3}
54: {1,2,2,2}
56: {1,1,1,4}
66: {1,2,5}
98: {1,4,4}
99: {2,2,5}
110: {1,3,5}
120: {1,1,1,2,3}
125: {3,3,3}
132: {1,1,2,5}
Cf.
A195086,
A065770,
A325166,
A325168,
A325169,
A325170,
A325180,
A325182,
A325188,
A325189,
A325195,
A325196,
A325198,
A325199,
A325200.
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primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Select[Range[1000],otbmax[primeptn[#]]-otb[primeptn[#]]==2&]
A325198
Positive numbers whose maximum prime index minus minimum prime index is 2.
Original entry on oeis.org
10, 20, 21, 30, 40, 50, 55, 60, 63, 80, 90, 91, 100, 105, 120, 147, 150, 160, 180, 187, 189, 200, 240, 247, 250, 270, 275, 300, 315, 320, 360, 385, 391, 400, 441, 450, 480, 500, 525, 540, 551, 567, 600, 605, 637, 640, 713, 720, 735, 750, 800, 810, 900, 945
Offset: 1
The sequence of terms together with their prime indices begins:
10: {1,3}
20: {1,1,3}
21: {2,4}
30: {1,2,3}
40: {1,1,1,3}
50: {1,3,3}
55: {3,5}
60: {1,1,2,3}
63: {2,2,4}
80: {1,1,1,1,3}
90: {1,2,2,3}
91: {4,6}
100: {1,1,3,3}
105: {2,3,4}
120: {1,1,1,2,3}
147: {2,4,4}
150: {1,2,3,3}
160: {1,1,1,1,1,3}
180: {1,1,2,2,3}
187: {5,7}
Cf.
A000961,
A008805,
A046660,
A056239,
A093641,
A112798,
A118914,
A174090,
A195086,
A256617,
A325180,
A325197.
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N:= 1000: # for terms <= N
q:= 2: r:= 3:
Res:= NULL:
do
p:= q; q:= r; r:= nextprime(r);
if p*r > N then break fi;
for i from 1 do
pi:= p^i;
if pi*r > N then break fi;
for j from 0 do
piqj:= pi*q^j;
if piqj*r > N then break fi;
Res:= Res, seq(piqj*r^k,k=1 .. floor(log[r](N/piqj)))
od
od
od:
sort([Res]); # Robert Israel, Apr 12 2019
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Select[Range[100],PrimePi[FactorInteger[#][[-1,1]]]-PrimePi[FactorInteger[#][[1,1]]]==2&]
A307682
Products of four primes, two of which are distinct.
Original entry on oeis.org
24, 36, 40, 54, 56, 88, 100, 104, 135, 136, 152, 184, 189, 196, 225, 232, 248, 250, 296, 297, 328, 344, 351, 375, 376, 424, 441, 459, 472, 484, 488, 513, 536, 568, 584, 621, 632, 664, 676, 686, 712, 776, 783, 808, 824, 837, 856, 872, 875, 904, 999, 1016, 1029
Offset: 1
-
Select[Range@ 1050, And[PrimeNu@ # == 2, PrimeOmega@ # == 4] &] (* Michael De Vlieger, Apr 21 2019 *)
-
isok(n) = (bigomega(n) == 4) && (omega(n) == 2); \\ Michel Marcus, Apr 22 2019
-
import sympy
def bigomega(n): return sympy.primeomega(n)
def omega(n): return len(sympy.primefactors(n))
print([n for n in range(1, 1000) if bigomega(n) == 4 and omega(n) == 2])
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