A325400
Heinz numbers of reversed integer partitions whose k-th differences are weakly increasing for all k >= 0.
Original entry on oeis.org
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74
Offset: 1
Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:
18: {1,2,2}
36: {1,1,2,2}
50: {1,3,3}
54: {1,2,2,2}
60: {1,1,2,3}
70: {1,3,4}
72: {1,1,1,2,2}
75: {2,3,3}
90: {1,2,2,3}
98: {1,4,4}
100: {1,1,3,3}
108: {1,1,2,2,2}
120: {1,1,1,2,3}
126: {1,2,2,4}
140: {1,1,3,4}
144: {1,1,1,1,2,2}
147: {2,4,4}
150: {1,2,3,3}
154: {1,4,5}
162: {1,2,2,2,2}
Cf.
A007294,
A056239,
A112798,
A240026,
A325354,
A325360,
A325362,
A325394,
A325397,
A325398,
A325399,
A325405,
A325467.
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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[Greater@@Differences[primeptn[#],k],{k,0,PrimeOmega[#]}]&]
A109428
Numbers n such that sigma(n)/omega(n) is not an integer [sigma(n) =sum of divisors of n; omega(n)=number of distinct prime factors of n].
Original entry on oeis.org
18, 36, 50, 72, 84, 98, 100, 144, 156, 162, 196, 200, 225, 228, 242, 252, 273, 288, 300, 324, 336, 338, 364, 372, 392, 399, 400, 441, 444, 468, 484, 516, 525, 532, 576, 578, 624, 648, 651, 676, 684, 700, 722, 732, 741, 756, 777, 784, 800, 804, 819, 868, 876
Offset: 1
The number 36 is in the sequence because sigma(36)=91 (1+2+3+4+6+9+12+18+36) and omega(36)=2 (2,3) and so sigma(36)/omega(36)=91/2.
The number 12 is not in the sequence because sigma(12)=28 (1+2+3+4+6+12) and omega(12)=2 (2,3) and so sigma(12)/omega(12)=14.
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with(numtheory): b:=proc(n) if type(sigma(n)/nops(factorset(n)),integer)=false then n else fi end: seq(b(n),n=2..1000);
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Select[Range[2,1000],!IntegerQ[DivisorSigma[1,#]/PrimeNu[#]]&] (* Harvey P. Dale, Aug 09 2012 *)
A175785
Numbers n such that the number of distinct prime divisors of n does not divide phi(n).
Original entry on oeis.org
30, 60, 66, 102, 110, 120, 132, 138, 150, 165, 170, 174, 204, 220, 230, 240, 246, 255, 264, 276, 282, 290, 300, 318, 340, 345, 348, 354, 374, 408, 410, 426, 435, 440, 460, 470, 480, 492, 498, 506, 528, 530, 534, 550, 552, 561, 564, 580, 590, 600, 606, 615
Offset: 1
30 is in this sequence because omega(30)=3 does not divide phi(30)=8.
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Select[Range[2,700],Mod[EulerPhi[#],PrimeNu[#]]!=0&] (* Harvey P. Dale, Dec 29 2019 *)
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isok(n) = (eulerphi(n) % omega(n) != 0) \\ Michel Marcus, Jun 12 2013
Showing 1-3 of 3 results.
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