A028260 -id:A028260 - cp's OEIS frontend

A320892 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into distinct semiprimes.

16, 64, 81, 96, 144, 160, 224, 256, 324, 352, 384, 400, 416, 486, 544, 576, 608, 625, 640, 729, 736, 784, 864, 896, 928, 960, 992, 1024, 1184, 1215, 1296, 1312, 1344, 1376, 1408, 1440, 1504, 1536, 1600, 1664, 1696, 1701, 1888, 1936, 1944, 1952, 2016, 2025

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

A semiprime (A001358) is a product of any two not necessarily distinct primes.
If A025487(k) is in the sequence then so is every number with the same prime signature. - David A. Corneth, Oct 23 2018
Numbers for which A001222(n) is even and A322353(n) is zero. - Antti Karttunen, Dec 06 2018

			A complete list of all factorizations of 1296 into semiprimes is:
  1296 = (4*4*9*9)
  1296 = (4*6*6*9)
  1296 = (6*6*6*6)
None of these is strict, so 1296 belongs to the sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001055, A001358, A005117, A006881, A007717, A025487, A028260, A045778, A318871, A318953, A320462, A320655, A320656, A320891, A320893, A320894, A322353.

strsemfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strsemfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],PrimeOmega[#]==2&]}]];
Select[Range[1000],And[EvenQ[PrimeOmega[#]],strsemfacs[#]=={}]&]

A322353(n, m=n, facs=List([])) = if(1==n, my(u=apply(bigomega,Vec(facs))); (0==length(u)||(2==vecmin(u)&&2==vecmax(u))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A322353(n/d, d-1, newfacs))); (s));
isA300892(n) = if(bigomega(n)%2,0,(0==A322353(n))); \\ Antti Karttunen, Dec 06 2018

A046337 Odd numbers with an even number of prime factors (counted with multiplicity).

Original entry on oeis.org

1, 9, 15, 21, 25, 33, 35, 39, 49, 51, 55, 57, 65, 69, 77, 81, 85, 87, 91, 93, 95, 111, 115, 119, 121, 123, 129, 133, 135, 141, 143, 145, 155, 159, 161, 169, 177, 183, 185, 187, 189, 201, 203, 205, 209, 213, 215, 217, 219, 221, 225, 235, 237, 247, 249, 253, 259

Offset: 1

Patrick De Geest, Jun 15 1998

nonn

John Cerkan, Table of n, a(n) for n = 1..10000

Intersection of A005408 and A028260.
Setwise difference A005408 \ A067019.
Setwise difference A028260 \ A063745.
Union of A359161 and A359163.
Union of A327862 and A360110.
Subsequence of A345452, of A356312 and of A359371.
Positions of positive terms in A166698, positions of even terms in A327858 and A356299.
Subsequences: A002557, A046315 (odd semiprimes), A056913, A359596, A359607, A359608 (without its term 2).
Cf. A000035, A008836, A046338, A046470, A353557 (characteristic function), A358777.
Cf. also A036349, A297845.

Select[Range[1,301,2],EvenQ[PrimeOmega[#]]&] (* Harvey P. Dale, Jul 25 2011 *)

lista(nn) = {forstep(n=1, nn, 2, if (bigomega(n) % 2 == 0, print1(n, ", ")));} \\ Michel Marcus, Jul 04 2015

{k | A000035(k) > 0 and A008836(k) > 0}. - Antti Karttunen, Jan 13 2023

A235992 Numbers with an even arithmetic derivative, cf. A003415.

Original entry on oeis.org

0, 1, 4, 8, 9, 12, 15, 16, 20, 21, 24, 25, 28, 32, 33, 35, 36, 39, 40, 44, 48, 49, 51, 52, 55, 56, 57, 60, 64, 65, 68, 69, 72, 76, 77, 80, 81, 84, 85, 87, 88, 91, 92, 93, 95, 96, 100, 104, 108, 111, 112, 115, 116, 119, 120, 121, 123, 124, 128, 129, 132, 133

Offset: 1

Reinhard Zumkeller, Mar 11 2014

nonn

A165560(a(n)) = 0; A003415(a(n)) mod 2 = 0.
For n > 1: A007814(a(n)) <> 1, A006519(a(n)) <> 2.
Union of multiples of 4 and odd numbers with an even number of prime factors with multiplicity. - Charlie Neder, Feb 25 2019
After two initial terms (0 and 1), numbers n such that A086134(n) = 2. - Antti Karttunen, Sep 30 2019
A multiplicative semigroup; if m and n are in the sequence then so is m*n. (See also comments in A359780.) - Antti Karttunen, Jan 17 2023

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A235991 (complement).
Union of A327862 and A327864.
Union of A359829 (primitive elements) and A359831 (nonprimitive elements).
Cf. A003415, A086134, A327863, A327865, A327933, A327935, A358680 (characteristic function).
Positions of multiples of 4 in A358669 (and in A358765).
Cf. also A028260, A036349, A046337, A332820 (other multiplicative semigroups), and comments in A359780.

a235992 n = a235992_list !! (n-1)
a235992_list = filter (even . a003415) [0..]

Select[Range[0, 133], EvenQ@ If[Abs@ # < 2, 0, # Total[#2/#1 & @@@ FactorInteger[Abs@ #]]] &] (* Michael De Vlieger, Sep 30 2019 *)

from itertools import count, islice
from sympy import factorint
def A235992_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n: not n&3 or (n&1 and not sum(factorint(n).values())&1), count(max(startvalue,0)))
A235992_list = list(islice(A235992_gen(),40)) # Chai Wah Wu, Nov 04 2022

A340387 Numbers whose sum of prime indices is twice their number, counted with multiplicity in both cases.

Original entry on oeis.org

1, 3, 9, 10, 27, 28, 30, 81, 84, 88, 90, 100, 208, 243, 252, 264, 270, 280, 300, 544, 624, 729, 756, 784, 792, 810, 840, 880, 900, 1000, 1216, 1632, 1872, 2080, 2187, 2268, 2352, 2376, 2430, 2464, 2520, 2640, 2700, 2800, 2944, 3000, 3648, 4896, 5440, 5616

Offset: 1

Gus Wiseman, Jan 09 2021

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Also Heinz numbers of integer partitions whose sum is twice their length, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). Like partitions in general (A000041), these are also counted by A000041.

			The sequence of terms together with their prime indices begins:
      1: {}
      3: {2}
      9: {2,2}
     10: {1,3}
     27: {2,2,2}
     28: {1,1,4}
     30: {1,2,3}
     81: {2,2,2,2}
     84: {1,1,2,4}
     88: {1,1,1,5}
     90: {1,2,2,3}
    100: {1,1,3,3}
    208: {1,1,1,1,6}
    243: {2,2,2,2,2}
    252: {1,1,2,2,4}

Partitions of 2n into n parts are counted by A000041.
The number of prime indices alone is A001222.
The sum of prime indices alone is A056239.
Allowing sum to be any multiple of length gives A067538, ranked by A316413.
A000569 counts graphical partitions, ranked by A320922.
A027187 counts partitions of even length, ranked by A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A301987 lists numbers whose sum of prime indices equals their product, with nonprime case A301988.
Cf. A000720, A001221, A001414, A006125, A006129, A112798, A316428, A320911, A325037, A325044, A330950, A331385, A331416.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Total[primeMS[#]]==2*PrimeOmega[#]&]

All terms satisfy A056239(a(n)) = 2*A001222(a(n)).

A119899 Integers i such that bigomega(i) (A001222) and tau(i) (A000005) are both even.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 24, 26, 33, 34, 35, 38, 39, 40, 46, 51, 54, 55, 56, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 104, 106, 111, 115, 118, 119, 122, 123, 126, 129, 132, 133, 134, 135, 136, 140, 141, 142, 143, 145, 146, 150

Offset: 1

Antti Karttunen, Jun 04 2006

nonn

Also numbers whose alternating sum of prime indices is < 0. Equivalently, numbers with even bigomega whose conjugate prime indices are not all even. This is the intersection of A028260 and A000037. - Gus Wiseman, Jun 20 2021

			From _Gus Wiseman_, Jun 20 2021: (Start)
The sequence of terms together with their prime indices begins:
       6: {1,2}          51: {2,7}          86: {1,14}
      10: {1,3}          54: {1,2,2,2}      87: {2,10}
      14: {1,4}          55: {3,5}          88: {1,1,1,5}
      15: {2,3}          56: {1,1,1,4}      90: {1,2,2,3}
      21: {2,4}          57: {2,8}          91: {4,6}
      22: {1,5}          58: {1,10}         93: {2,11}
      24: {1,1,1,2}      60: {1,1,2,3}      94: {1,15}
      26: {1,6}          62: {1,11}         95: {3,8}
      33: {2,5}          65: {3,6}          96: {1,1,1,1,1,2}
      34: {1,7}          69: {2,9}         104: {1,1,1,6}
      35: {3,4}          74: {1,12}        106: {1,16}
      38: {1,8}          77: {4,5}         111: {2,12}
      39: {2,6}          82: {1,13}        115: {3,9}
      40: {1,1,1,3}      84: {1,1,2,4}     118: {1,17}
      46: {1,9}          85: {3,7}         119: {4,7}
(End)

Superset: A119847. Subset: A006881. The intersection of A028260 and A000037.
Positions of negative terms in A316524.
The partitions with these Heinz numbers are counted by A344608.
Complement of A344609.
Cf. A000041, A000070, A000097, A027187, A103919, A116406, A239829, A239830, A343941, A344607, A344651.

Select[Range[200],And@@EvenQ[{PrimeOmega[#],DivisorSigma[0,#]}]&] (* Harvey P. Dale, Jan 24 2013 *)

A320912 Numbers with an even number of prime factors (counted with multiplicity) that can be factored into distinct semiprimes.

Original entry on oeis.org

1, 4, 6, 9, 10, 14, 15, 21, 22, 24, 25, 26, 33, 34, 35, 36, 38, 39, 40, 46, 49, 51, 54, 55, 56, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 100, 104, 106, 111, 115, 118, 119, 121, 122, 123, 126, 129, 132, 133, 134, 135, 136, 140

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

A semiprime (A001358) is a product of any two not necessarily distinct primes.

			9000 is in the sequence and can be factored in either of two ways: (4*6*15*25) or (4*9*10*25).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001055, A001222, A001358, A007717, A028260, A320655, A320892, A320893, A320894, A320911.

strsemfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strsemfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],PrimeOmega[#]==2&]}]];
Select[Range[100],And[EvenQ[PrimeOmega[#]],strsemfacs[#]!={}]&]

A344609 Numbers whose alternating sum of prime indices is >= 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 25, 27, 28, 29, 30, 31, 32, 36, 37, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 59, 61, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 89, 92, 97, 98, 99, 100, 101, 102, 103, 105, 107

Offset: 1

Gus Wiseman, May 30 2021

nonn

Also Heinz numbers of partitions whose reverse-alternating sum is >= 0. These are partitions whose conjugate parts are all even or whose length is odd.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The sequence of terms together with their prime indices begins:
      1: {}            20: {1,1,3}         45: {2,2,3}
      2: {1}           23: {9}             47: {15}
      3: {2}           25: {3,3}           48: {1,1,1,1,2}
      4: {1,1}         27: {2,2,2}         49: {4,4}
      5: {3}           28: {1,1,4}         50: {1,3,3}
      7: {4}           29: {10}            52: {1,1,6}
      8: {1,1,1}       30: {1,2,3}         53: {16}
      9: {2,2}         31: {11}            59: {17}
     11: {5}           32: {1,1,1,1,1}     61: {18}
     12: {1,1,2}       36: {1,1,2,2}       63: {2,2,4}
     13: {6}           37: {12}            64: {1,1,1,1,1,1}
     16: {1,1,1,1}     41: {13}            66: {1,2,5}
     17: {7}           42: {1,2,4}         67: {19}
     18: {1,2,2}       43: {14}            68: {1,1,7}
     19: {8}           44: {1,1,5}         70: {1,3,4}
For example, the prime indices of 70 are {1,3,4} with alternating sum 1 - 3 + 4 = 2, so 70 is in the sequence. On the other hand, the prime indices of 24 are {1,1,1,2} with alternating sum 1 - 1 + 1 - 2 = -1, so 24 is not in the sequence.

The opposite (nonpositive) version is A028260, counted by A027187.
The strict case (n > 0) is counted by A067659, odd bisection A344650.
Permutations of prime indices of these terms are counted by A116406.
Complement of A119899, Heinz numbers of the partitions counted by A344608.
Positions of nonnegative terms in A316524 or A344617.
Heinz numbers of the partitions counted by A344607.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A000070 counts partitions with alternating sum 1.
A000097 counts partitions with alternating sum 2.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum.
A120452 counts partitions with reverse-alternating sum 2.
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A335433/A335448 rank separable/inseparable partitions.
A344604 counts wiggly compositions with twins.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344612 counts partitions by sum and reverse-alternating sum.
A344618 gives reverse-alternating sums of standard compositions.
Cf. A001222, A001250, A003242, A005649, A026424, A071321/A071322, A124754, A239829, A343938, A344611, A344651, A344653/A344742, A344739.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[100],ats[primeMS[#]]>=0&]

A325700 Numbers with as many distinct even as distinct odd prime indices.

Original entry on oeis.org

1, 6, 12, 14, 15, 18, 24, 26, 28, 33, 35, 36, 38, 45, 48, 51, 52, 54, 56, 58, 65, 69, 72, 74, 75, 76, 77, 86, 93, 95, 96, 98, 99, 104, 106, 108, 112, 116, 119, 122, 123, 135, 141, 142, 143, 144, 145, 148, 152, 153, 158, 161, 162, 172, 175, 177, 178, 185, 192

Offset: 1

Gus Wiseman, May 17 2019

nonn

These are the Heinz numbers of the integer partitions counted by A241638.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
    1: {}
    6: {1,2}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   33: {2,5}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   45: {2,2,3}
   48: {1,1,1,1,2}
   51: {2,7}
   52: {1,1,6}
   54: {1,2,2,2}
   56: {1,1,1,4}
   58: {1,10}

A324967(n) = A324966(n).

Cf. A001221, A026010, A028260, A045931, A063886, A097613, A112798, A130780, A239241, A241638, A325698, A325699.

Select[Range[100],0==Total[(-1)^PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]

A320891 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into squarefree semiprimes.

Original entry on oeis.org

4, 9, 16, 24, 25, 40, 49, 54, 56, 64, 81, 88, 96, 104, 121, 135, 136, 144, 152, 160, 169, 184, 189, 224, 232, 240, 248, 250, 256, 289, 296, 297, 324, 328, 336, 344, 351, 352, 361, 375, 376, 384, 400, 416, 424, 459, 472, 486, 488, 513, 528, 529, 536, 544, 560

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

A squarefree semiprime (A006881) is a product of any two distinct primes.

Also numbers with an even number x of prime factors, whose greatest prime multiplicity exceeds x/2.

			A complete list of all factorizations of 24 is:
  (2*2*2*3),
  (2*2*6), (2*3*4),
  (2*12), (3*8), (4*6),
  (24).
All of these contain at least one number that is not a squarefree semiprime, so 24 belongs to the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001055, A001358, A005117, A006881, A007717, A028260, A318871, A318953, A320655, A320656, A320892, A320893, A320894.

semfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[semfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
Select[Range[100],And[EvenQ[PrimeOmega[#]],semfacs[#]=={}]&]

A320894 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into distinct squarefree semiprimes.

Original entry on oeis.org

4, 9, 16, 24, 25, 36, 40, 49, 54, 56, 64, 81, 88, 96, 100, 104, 121, 135, 136, 144, 152, 160, 169, 184, 189, 196, 216, 224, 225, 232, 240, 248, 250, 256, 289, 296, 297, 324, 328, 336, 344, 351, 352, 360, 361, 375, 376, 384, 400, 416, 424, 441, 459, 472, 484

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

A squarefree semiprime (A006881) is a product of any two distinct primes.

			A complete list of all strict factorizations of 24 is: (2*3*4), (2*12), (3*8), (4*6), (24). All of these contain at least one number that is not a squarefree semiprime, so 24 belongs to the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001055, A001358, A005117, A006881, A007717, A028260, A318871, A318953, A320655, A320656, A320891, A320892, A320893.

strsqfsemfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strsqfsemfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
Select[Range[100],And[EvenQ[PrimeOmega[#]],strsqfsemfacs[#]=={}]&]

cp's OEIS Frontend

A320892 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into distinct semiprimes.

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A046337 Odd numbers with an even number of prime factors (counted with multiplicity).

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A235992 Numbers with an even arithmetic derivative, cf. A003415.

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A340387 Numbers whose sum of prime indices is twice their number, counted with multiplicity in both cases.

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A119899 Integers i such that bigomega(i) (A001222) and tau(i) (A000005) are both even.

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A320912 Numbers with an even number of prime factors (counted with multiplicity) that can be factored into distinct semiprimes.

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A344609 Numbers whose alternating sum of prime indices is >= 0.

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A325700 Numbers with as many distinct even as distinct odd prime indices.

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