A119847 -id:A119847 - cp's OEIS frontend

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

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 *)

A119842 Number of alternating linear extensions of the divisor lattice of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 6, 1, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 2, 1, 0

Offset: 1

Antti Karttunen, Jun 04 2006

nonn

For prime powers there is only one solution. For integers with prime signature p1^2 * p2 there's exactly one solution, for p1^4 * p2 there are two and in general for p1^(2k) * p2 there are A000108(k) solutions. - Mitch Harris, Apr 27 2006

			In other words, the number of ways to arrange the divisors of n in such a way that no divisor has any of its own divisors following it AND the divisors d_i, d_j, d_k, etc. are arranged so that values bigomega(d_i) (cf. A001222), bigomega(d_j), bigomega(d_k) are alternatively even and odd. E.g., a(12)=1, as of the five arrangements shown in A114717, here the only one allowed is 1,2,4,3,6,12, with A001222(1)=0, A001222(2)=1, A001222(4)=2, A001222(3)=1, A001222(6)=2, A001222(12)=3. a(36) = 2, as there are two solutions for 36: 1,2,4,3,6,12,9,18,36 and 1,3,9,2,6,18,4,12,36.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
T. Y. Chow, H. Eriksson, C. K. Fan, Chess Tableaux, The Electronic Journal of Combinatorics, vol. 11(2), 2004.
T. Y. Chow, H. Eriksson, C. K. Fan, Chess Tableaux and Chess Problems, slides for MIT Combinatorics Seminar, 20 October 2004.
Index entries for sequences computed from exponents in factorization of n

a(n) <= A114717(n). Cf. A119844, A119846, A119847, A119849.

with(numtheory):
b:= proc(s, t) option remember; `if`(nops(s)<1, 1, add(
      `if`(irem(bigomega(x), 2)=1-t and nops(select(y->
      irem(y, x)=0, s))=1, b(s minus {x}, 1-t), 0), x=s))
    end:
a:= proc(n) option remember; local l, m;
      l:= sort(ifactors(n)[2], (x, y)-> x[2]>y[2]);
      m:= mul(ithprime(i)^l[i][2], i=1..nops(l));
      b(divisors(m) minus {1, m}, irem(bigomega(m), 2))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 26 2016

b[s_, t_] := b[s, t] = If[Length[s] < 1, 1, Sum[If[Mod[PrimeOmega[x], 2] == 1-t && Length[Select[s, Mod[#, x] == 0&]] == 1, b[s ~Complement~ {x}, 1-t ], 0], {x, s}]]; a[n_] := a[n] = Module[{l, m}, l = Sort[FactorInteger[n ], #1[[2]] > #2[[2]]&]; m = Product[Prime[i]^l[[i]][[2]], {i, 1, Length[ l]}]; b[Divisors[m][[2 ;; -2]], Mod[PrimeOmega[m], 2]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 27 2016, after Alois P. Heinz *)

A229153 Numbers of the form c * m^2, where m > 0 and c is composite and squarefree.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 24, 26, 30, 33, 34, 35, 38, 39, 40, 42, 46, 51, 54, 55, 56, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 102, 104, 105, 106, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126, 129, 130, 132, 133, 134, 135, 136, 138, 140, 141, 142, 143, 145, 146, 150

Offset: 1

Chris Boyd, Sep 15 2013

nonn

Subsequence of A048943. According to Gerard P. Michon, one of the criteria for N to belong to A048943 is that it has at least two prime factors with odd multiplicities. By definition, the composite factor c in any term of A229153 conforms to this criterion.

From a(1) to a(63), identical to the given terms of A119847, except for the single term a(55) = 120.

Chris Boyd, Table of n, a(n) for n = 1..10000

Complement of A265640.

Cf. A048943, A229125.

iscomposite(n)={if(!isprime(n)&&n!=1,return(1));}
test(n)={if(iscomposite(core(n)),return(1));}
for(n=1,200,if(test(n)==1,print1(n",")))

lista(nn) = {for(n=1,nn, if(!ispseudoprime(core(n)) && !issquare(n), print1(n, ", ")));} \\ Altug Alkan, Feb 04 2016

list(lim)=my(v=List()); forsquarefree(c=6,lim\=1, if(#c[2]~ > 1, for(m=1,sqrtint(lim\c[1]), listput(v, c[1]*m^2)))); Set(v) \\ Charles R Greathouse IV, Jan 09 2022

from math import isqrt
from sympy import primepi, mobius
def A229153(n):
    def f(x):
        c = n+x+(a:=isqrt(x))
        for y in range(1,a+1):
            m = x//y**2
            c += primepi(m)-sum(mobius(k)*(m//k**2) for k in range(1, isqrt(m)+1))
        return c
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Jan 30 2025

A119848 Positions where A119842 is not zero.

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, 31, 32, 36, 37, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 108, 109, 112, 113, 116, 117, 120, 121

Offset: 1

Antti Karttunen, Jun 04 2006

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..3323

Complement: A119847. Cf. A119850.

b[s_, t_] := b[s, t] = If[Length[s] < 1, 1, Sum[If[Mod[PrimeOmega[x], 2] == 1 - t && Length[Select[s, Mod[#, x] == 0 &]] == 1, b[s~Complement~{x}, 1 - t], 0], {x, s}]]; c[n_] := Module[{l, m}, l = Sort[FactorInteger[n], #1[[2]] > #2[[2]] &]; m = Product[Prime[i]^l[[i]][[2]], {i, 1, Length[l]} ]; b[Divisors[m][[2 ;; -2]], Mod[PrimeOmega[m], 2]]]; A119848 = Position[ Table[c[n], {n, 1, 200}], ?Positive] // Flatten (* _Jean-François Alcover, Mar 03 2016, after Alois P. Heinz *)

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

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A119842 Number of alternating linear extensions of the divisor lattice of n.

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A229153 Numbers of the form c * m^2, where m > 0 and c is composite and squarefree.

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A119848 Positions where A119842 is not zero.

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