A085987 -id:A085987 - cp's OEIS frontend

A275387 Numbers of ordered pairs of divisors d < e of n such that gcd(d, e) > 1.

0, 0, 0, 1, 0, 2, 0, 3, 1, 2, 0, 8, 0, 2, 2, 6, 0, 8, 0, 8, 2, 2, 0, 18, 1, 2, 3, 8, 0, 15, 0, 10, 2, 2, 2, 24, 0, 2, 2, 18, 0, 15, 0, 8, 8, 2, 0, 32, 1, 8, 2, 8, 0, 18, 2, 18, 2, 2, 0, 44, 0, 2, 8, 15, 2, 15, 0, 8, 2, 15, 0, 49, 0, 2, 8, 8, 2, 15, 0, 32, 6, 2

Offset: 1

Michel Lagneau, Aug 03 2016

nonn

Number of elements in the set {(x, y): x|n, y|n, x < y, gcd(x, y) > 1}.
Every element of the sequence is repeated indefinitely, for instance:
a(n)=0 if n prime;
a(n)=1 if n = p^2 for p prime (A001248);
a(n)=2 if n is a squarefree semiprime (A006881);
a(n)=3 if n = p^3 for p prime (A030078);
a(n)=6 if n = p^4 for p prime (A030514);
a(n)=8 if n is a number which is the product of a prime and the square of a different prime (A054753);
a(n)=10 if n = p^5 for p prime (A050997);
a(n)=15 if n is in the set {A007304} union {64} = {30, 42, 64, 66, 70,...} = {Sphenic numbers} union {64};
a(n)=18 if n is the product of the cube of a prime (A030078) and a different prime (see A065036);
a(n)=21 if n = p^7 for p prime (A092759);
a(n)=24 if n is square of a squarefree semiprime (A085986);
a(n)=32 if n is the product of the 4th power of a prime (A030514) and a different prime (see A178739);
a(n)=36 if n = p^9 for p prime (A179665);
a(n)=44 if n is the product of exactly four primes, three of which are distinct (A085987);
a(n)=45 if n is a number with 11 divisors (A030629);
a(n)=49 if n is of the form p^2*q^3, where p,q are distinct primes (A143610);
a(n)=50 if n is the product of the 5th power of a prime (A050997) and a different prime (see A178740);
a(n)=55 if n if n = p^11 for p prime(A079395);
a(n)=72 if n is a number with 14 divisors (A030632);
a(n)=80 if n is the product of four distinct primes (A046386);
a(n)=83 if n is a number with 15 divisors (A030633);
a(n)=89 if n is a number with prime factorization pqr^3 (A189975);
a(n)=96 if n is a number that are the cube of a product of two distinct primes (A162142);
a(n)=98 if n is the product of the 7th power of a prime and a distinct prime (p^7*q) (A179664);
a(n)=116 if n is the product of exactly 2 distinct squares of primes and a different prime (p^2*q^2*r) (A179643);
a(n)=126 if n is the product of the 5th power of a prime and different distinct prime of the 2nd power (p^5*q^2) (A179646);
a(n)=128 if n is the product of the 8th power of a prime and a distinct prime (p^8*q) (A179668);
a(n)=150 if n is the product of the 4th power of a prime and 2 different distinct primes (p^4*q*r) (A179644);
a(n)=159 if n is the product of the 4th power of a prime and a distinct prime of power 3 (p^4*q^3) (A179666).
It is possible to continue with a(n) = 162, 178, 209, 224, 227, 238, 239, 260, 289, 309, 320, 333,...

			a(12) = 8 because the divisors of 12 are {1, 2, 3, 4, 6, 12} and GCD(d_i, d_j)>1 for the 8 following pairs of divisors: (2,4), (2,6), (2,12), (3,6), (3,12), (4,6), (4,12) and (6,12).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001248, A006881, A007304, A030078, A030514, A030632, A046386, A050997, A054753, A063647, A065036, A066446, A079395, A085986, A085987, A092759, A143610, A162142, A178739, A178740, A179644, A179646, A179664, A189975.

Cf. A333976 (same with d <= e).

with(numtheory):nn:=100:
for n from 1 to nn do:
x:=divisors(n):n0:=nops(x):it:=0:
for i from 1 to n0 do:
  for j from i+1 to n0 do:
   if gcd(x[i],x[j])>1
    then
    it:=it+1:
    else
   fi:
  od:
od:
  printf(`%d, `,it):
od:

Table[Sum[Sum[(1 - KroneckerDelta[GCD[i, k], 1]) (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k - 1}], {k, n}], {n, 100}] (* Wesley Ivan Hurt, Jan 01 2021 *)

a(n)=my(d=divisors(n)); sum(i=2,#d, sum(j=1,i-1, gcd(d[i],d[j])>1)) \\ Charles R Greathouse IV, Aug 03 2016

a(n)=my(f=factor(n)[,2],t=prod(i=1,#f,f[i]+1)); t*(t-1)/2 - (prod(i=1,#f,2*f[i]+1)+1)/2 \\ Charles R Greathouse IV, Aug 03 2016

a(n) = A066446(n) - A063647(n).

a(n) = Sum_{d1|n, d2|n, d1Wesley Ivan Hurt, Jan 01 2021

A086975 Numbers of the form p^2 * q * r with primes p < q < r.

Original entry on oeis.org

60, 84, 132, 140, 156, 204, 220, 228, 260, 276, 308, 315, 340, 348, 364, 372, 380, 444, 460, 476, 492, 495, 516, 532, 564, 572, 580, 585, 620, 636, 644, 693, 708, 732, 740, 748, 765, 804, 812, 819, 820, 836, 852, 855, 860, 868, 876, 884, 940, 948, 988

Offset: 1

Reinhard Zumkeller, Jul 25 2003

nonn

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

Subsequence of A086974 and of A085987.

Cf. A000005, A001221, A001222, A020639.

Select[Range[1000], FactorInteger[#][[;;, 2]] == {2, 1, 1} &] (* Amiram Eldar, Sep 15 2024 *)

is(k) = factor(k)[, 2] == [2, 1, 1]~; \\ Amiram Eldar, Sep 15 2024

A001221(a(n)) = 3.
A001222(a(n)) = 4.
A020639(a(n))^2 divides a(n).
A000005(a(n)) = 12. - Amiram Eldar, Sep 15 2024

A179746 Numbers of the form p^4q^2r^2 where p, q, and r are distinct primes.

Original entry on oeis.org

3600, 7056, 8100, 15876, 17424, 19600, 22500, 24336, 39204, 41616, 48400, 51984, 54756, 67600, 76176, 86436, 93636, 94864, 99225, 115600, 116964, 121104, 122500, 132496, 138384, 144400, 171396, 197136, 211600, 226576, 240100, 242064, 245025

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 25 2010

nonn

Numbers k such that tau(k^2)/tau(k) = 5 where tau(n) is the number of divisors of n (A000005). - Bernard Schott, Nov 27 2020

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, List of Prime Signatures
Index to sequences related to prime signature

Subsequence of A217584.

Cf. A189988 (tau(k^2)/tau(k) = 3).

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,2,4}; Select[Range[200000],f]

list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\36)^(1/4), t1=p^4;forprime(q=2, sqrt(lim\t1), if(p==q, next);t2=t1*q^2;forprime(r=q+1, sqrt(lim\t2), if(p==r,next);listput(v,t2*r^2)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A179746(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x+sum((t:=primepi(s:=isqrt(y:=x//r**2)))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(isqrt(x)+1))+sum(primepi(x//p**3) for p in primerange(integer_nthroot(x,3)[0]+1))-primepi(integer_nthroot(x,4)[0])
    return bisection(f,n,n)**2 # Chai Wah Wu, Mar 27 2025

Sum_{n>=1} 1/a(n) = (P(2)^2*P(4) - P(4)^2)/2 - P(2)*P(6) + P(8) = 0.00125114..., where P is the prime zeta function. - Amiram Eldar, Jul 03 2022

a(n) = A085987(n)^2. - R. J. Mathar, May 05 2023

A356413 Numbers with an equal sum of the even and odd exponents in their prime factorizations.

Original entry on oeis.org

1, 60, 84, 90, 126, 132, 140, 150, 156, 198, 204, 220, 228, 234, 260, 276, 294, 306, 308, 315, 340, 342, 348, 350, 364, 372, 380, 414, 444, 460, 476, 490, 492, 495, 516, 522, 525, 532, 550, 558, 564, 572, 580, 585, 620, 636, 644, 650, 666, 693, 708, 726, 732, 735

Offset: 1

Amiram Eldar, Aug 06 2022

nonn

Numbers k such that A350386(k) = A350387(k).

A085987 is a subsequence. Terms that are not in A085987 are 1, 2160, 3024, ...

			60 is a term since A350386(60) = A350387(60) = 2.

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

Cf. A350386, A350387.
Subsequence of A028260.
Subsequences: A085987, A179698, A190109, A190110.
Similar sequences: A048109, A187039, A348097.

f[p_, e_] := (-1)^e*e; q[1] = True; q[n_] := Plus @@ f @@@ FactorInteger[n] == 0; Select[Range[1000], q]

isok(n) = {my(f = factor(n)); sum(i = 1, #f~, (-1)^f[i,2]*f[i,2]) == 0};

A336530 Number of triples of divisors d_i < d_j < d_k of n such that gcd(d_i, d_j, d_k) > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 0, 4, 0, 5, 0, 5, 0, 0, 0, 23, 0, 0, 1, 5, 0, 12, 0, 10, 0, 0, 0, 36, 0, 0, 0, 23, 0, 12, 0, 5, 5, 0, 0, 62, 0, 5, 0, 5, 0, 23, 0, 23, 0, 0, 0, 87, 0, 0, 5, 20, 0, 12, 0, 5, 0, 12, 0, 120, 0, 0, 5, 5, 0, 12, 0, 62, 4

Offset: 1

Michel Lagneau, Oct 04 2020

nonn

Number of elements in the set {(x, y, z): x|n, y|n, z|n, x < y < z, GCD(x, y, z) > 1}.
Every element of the sequence is repeated indefinitely, for instance:
a(n) = 0 for n = 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, ... (Numbers with at most 2 prime factors (counted with multiplicity). See A037143);
a(n) = 1 for n = 8, 27, 125, 343, 1331, 2197, 4913,... (cubes of primes. See A030078);
a(n) = 4 for n = 16, 81, 625, 2401, 14641, 28561, ... (prime(n)^4. See A030514);
a(n) = 5 for n = 12, 18, 20, 28, 44, 45, ... (Numbers which are the product of a prime and the square of a different prime (p^2 * q). See A054753);
a(n) = 12 for n = 30, 42, 66, 70, 78, 102, 105, 110,... (Sphenic numbers: products of 3 distinct primes. See A007304);
a(n) = 20 for n = 64, 729, 15625, 117649, ... (Numbers with 7 divisors. 6th powers of primes. See A030516);
a(n) = 23 for n = 24, 40, 54, 56, 88, 104, 135, 136, ... (Product of the cube of a prime (A030078) and a different prime. See A065036);
a(n) = 36 for n = 36, 100, 196, 225, 441, 484, 676,... (Squares of the squarefree semiprimes (p^2*q^2). See A085986);
a(n) = 62 for n = 48, 80, 112, 162, 176, 208, 272, ... (Product of the 4th power of a prime (A030514) and a different prime (p^4*q). See A178739);
a(n) = 87 for n = 60, 84, 90, 126, 132, 140, 150, 156, ... (Product of exactly four primes, three of which are distinct (p^2*q*r). See A085987);
a(n) = 120 for n = 72, 108, 200, 392, 500, 675, 968, ... (Numbers of the form p^2*q^3, where p,q are distinct primes. See A143610);
It is possible to continue with a(n) = 130, 235, 284, 289, 356, ...

			a(12) = 5 because the divisors of 12 are {1, 2, 3, 4, 6, 12} and GCD(d_i, d_j, d_k) > 1 for the 5 following triples of divisors: (2,4,6), (2,4,12), (2,6,12), (3,6,12) and (4,6,12).

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A275387.

with(numtheory):nn:=100:
for n from 1 to nn do:
it:=0:d:=divisors(n):n0:=nops(d):
  for i from 1 to n0-2 do:
   for j from i+1 to n0-1 do:
     for k from j+1 to n0 do:
    if igcd(d[i],d[j],d[k])> 1
       then
       it:=it+1:
       else
      fi:
     od:
     od:
     od:
    printf(`%d, `,it):
   od:

Array[Count[GCD @@ # & /@ Subsets[Divisors[#], {3}], ?(# > 1 &)] &, 81] (* _Michael De Vlieger, Oct 05 2020 *)

a(n) = my(d=divisors(n)); sum(i=1, #d-2, sum (j=i+1, #d-1, sum (k=j+1, #d, gcd([d[i], d[j], d[k]]) > 1))); \\ Michel Marcus, Oct 31 2020

a(n) = {my(f = factor(n), vp = vecprod(f[,1]), d = divisors(vp), res = 0);
for(i = 2, #d, res-=binomial(numdiv(n/d[i]), 3)*(-1)^omega(d[i])); res} \\ David A. Corneth, Nov 01 2020

Name clarified by editors, Oct 31 2020

A231122 Numbers k >= 0 such that 2^k is number of ways to write n as n = x*y, where x, y = squarefree numbers, 1 <= x <= n, 1 <= y <= n, or -1 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, -1, 0, 1, 0, 0, 0, 1, 1, -1, 0, 0, 0, 0, 1, 1, 0, -1, 0, 1, -1, 0, 0, 2, 0, -1, 1, 1, 1, 0, 0, 1, 1, -1, 0, 2, 0, 0, 0, 1, 0, -1, 0, 0, 1, 0, 0, -1, 1, -1, 1, 1, 0, 1, 0, 1, 0, -1, 1, 2, 0, 0, 1, 2, 0, -1, 0, 1, 0, 0, 1, 2, 0, -1, -1, 1, 0, 1, 1

Offset: 1

Gerasimov Sergey, Nov 04 2013

sign

			1(1*1), 2(1*2), 3(1*3), 4(2*2), 5(1*5), 6(1*6, 2*3), 7(1*7), 8(-), 9(3*3), 10(1*10, 2*5), 11(1*11), 12(2*6), 13(1*13), 14(1*14, 2*7), 15(1*15, 3*5), 16(-), 17(1*17), 18(3*6), 19(1*19), 20(2*10), 21(1*21, 3*7), 22(1*22, 2*11), 23(1*23), 24(-), 25(5*5), 26(1*26, 2*13), 27(-).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A046099(k<0), A130091(k<=0), A004709(k>=0), A230843(k=0), (A006881 U A085987 U ...)(k=1).

a(n)=if(n==1,return(0)); my(f=factor(n)[,2]); if(vecmax(f)>2, return(-1)); max(sum(i=1,#f,2-f[i])-1, 0) \\ Charles R Greathouse IV, Nov 04 2013

a(80) corrected by Charles R Greathouse IV, Nov 04 2013

A275345 Characteristic polynomials of a square matrix based on A051731 where A051731(1,N)=1 and A051731(N,N)=0 and where N=size of matrix, analogous to the Redheffer matrix.

Original entry on oeis.org

1, 1, -1, -1, -1, 1, -1, 0, 2, -1, 0, 0, 2, -3, 1, -1, 2, 1, -5, 4, -1, 1, -3, 5, -8, 9, -5, 1, -1, 4, -4, -5, 15, -14, 6, -1, 0, -1, 6, -17, 29, -31, 20, -7, 1, 0, 0, 2, -13, 36, -55, 50, -27, 8, -1, 1, -7, 23, -50, 84, -112, 112, -78, 35, -9, 1

Offset: 0

Mats Granvik, Jul 24 2016

From Mats Granvik, Sep 30 2017: (Start)
Conjecture: The largest absolute value of the eigenvalues of these characteristic polynomials appear to have the same prime signature in the factorization of the matrix sizes N.
In other words: Let b(N) equal the sequence of the largest absolute values of the eigenvalues of the characteristic polynomials of the matrices of size N. b(N) is then a sequence of truncated eigenvalues starting:
b(N=1..infinity)
= 1.00000, 1.61803, 1.61803, 2.00000, 1.61803, 2.20557, 1.61803, 2.32472, 2.00000, 2.20557, 1.61803, 2.67170, 1.61803, 2.20557, 2.20557, 2.61803, 1.61803, 2.67170, 1.61803, 2.67170, 2.20557, 2.20557, 1.61803, 3.08032, 2.00000, 2.20557, 2.32472, 2.67170, 1.61803, 2.93796, 1.61803, 2.89055, 2.20557, 2.20557, 2.20557, 3.21878, 1.61803, 2.20557, 2.20557, 3.08032, 1.61803, 2.93796, 1.61803, 2.67170, 2.67170, 2.20557, 1.61803, 3.45341, 2.00000, 2.67170, 2.20557, 2.67170, 1.61803, 3.08032, 2.20557, 3.08032, 2.20557, 2.20557, 1.61803, 3.53392, 1.61803, 2.20557, 2.67170, ...
It then appears that for n = 1,2,3,4,5,...,infinity we have the table:
Prime signature: b(Axxxxxx(n)) = Largest abs(eigenvalue):
p^0      : b(1)          = 1.0000000000000000000000000000...
p        : b(A000040(n)) = 1.6180339887498949025257388711...
p^2      : b(A001248(n)) = 2.0000000000000000000000000000...
p*q      : b(A006881(n)) = 2.2055694304005917238953315973...
p^3      : b(A030078(n)) = 2.3247179572447480566665944934...
p^2*q    : b(A054753(n)) = 2.6716998816571604358216518448...
p^4      : b(A030514(n)) = 2.6180339887498917939012699207...
p^3*q    : b(A065036(n)) = 3.0803227214906021558249449299...
p*q*r    : b(A007304(n)) = 2.9379558827528557962693867011...
p^5      : b(A050997(n)) = 2.8905508875432590620846440288...
p^2*q^2  : b(A085986(n)) = 3.2187765853016649941764626419...
p^4*q    : b(A178739(n)) = 3.4534111136673804054453285061...
p^2*q*r  : b(A085987(n)) = 3.5339198574905377192578725953...
p^6      : b(A030516(n)) = 3.1478990357047909043330946587...
p^3*q^2  : b(A143610(n)) = 3.7022736187975437971431347250...
p^5*q    : b(A178740(n)) = 3.8016448153137023524550386355...
p^3*q*r  : b(A189975(n)) = 4.0600260453688532535920785448...
p^7      : b(A092759(n)) = 3.3935083220984414431597997463...
p^4*q^2  : b(A189988(n)) = 4.1453038440113498808159420150...
p^2*q^2*r: b(A179643(n)) = 4.2413382309993874486053755390...
p^6*q    : b(A189987(n)) = 4.1311805192254587026923218218...
p*q*r*s  : b(A046386(n)) = 3.8825338629275134572083061357...
...
b(Axxxxxx(1)) in the sequences above, is given by A025487.
(End)
First column in the coefficients of the characteristic polynomials is the Möbius function A008683.
Row sums of coefficients start: 0, -1, 0, 0, 0, 0, 0, 0, 0, ...
Third diagonal is a signed version of A000096.
Most of the eigenvalues are equal to 1. The number of eigenvalues equal to 1 are given by A075795 for n>1.
The first three of the eigenvalues above can be calculated as nested radicals. The fourth eigenvalue 2.205569430400590... minus 1 = 1.205569430400590... is also a nested radical.

			{
{ 1},
{ 1, -1},
{-1, -1,  1},
{-1,  0,  2,  -1},
{ 0,  0,  2,  -3,  1},
{-1,  2,  1,  -5,  4,   -1},
{ 1, -3,  5,  -8,  9,   -5,   1},
{-1,  4, -4,  -5, 15,  -14,   6,  -1},
{ 0, -1,  6, -17, 29,  -31,  20,  -7,  1},
{ 0,  0,  2, -13, 36,  -55,  50, -27,  8, -1},
{ 1, -7, 23, -50, 84, -112, 112, -78, 35, -9, 1}
}

Mats Granvik, Mathematica program to verify that eigenvalues determine prime signature
OEIS Wiki, Prime signatures
Eric Weisstein, Prime signature
Index to sequences related to prime signature

Cf. A051731, A008683, A000040, A001248, A006881, A030078, A030514, A054753, A000096, A001622, A272874, A075795.

Cf. A025487. - Mats Granvik, Sep 30 2017

Clear[x, AA, nn, s]; Monitor[AA = Flatten[Table[A = Table[Table[If[Mod[n, k] == 0, 1, 0], {k, 1, nn}], {n, 1, nn}]; MatrixForm[A]; a = A[[1, nn]]; A[[1, nn]] = A[[nn, nn]]; A[[nn, nn]] = a; CoefficientList[CharacteristicPolynomial[A, x], x], {nn, 1, 10}]], nn]

A307341 Products of four primes, not all distinct.

Original entry on oeis.org

16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, 104, 126, 132, 135, 136, 140, 150, 152, 156, 184, 189, 196, 198, 204, 220, 225, 228, 232, 234, 248, 250, 260, 276, 294, 296, 297, 306, 308, 315, 328, 340, 342, 344, 348, 350, 351, 364, 372, 375, 376, 380, 414

Offset: 1

Kalle Siukola, Apr 02 2019

Numbers with exactly four prime factors (counted with multiplicity) but fewer than four distinct prime factors.

Numbers n such that bigomega(n) = 4 and omega(n) < 4.

Kalle Siukola, Table of n, a(n) for n = 1..10000

Setwise difference of A014613 and A046386.

Union of A030514, A065036, A085986 and A085987.

isok(n) = (bigomega(n) == 4) && (omega(n) < 4); \\ Michel Marcus, Apr 03 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) < 4])

A343511 a(n) = 1 + Sum_{d|n, d < n} a(d)^2.

Original entry on oeis.org

1, 2, 2, 6, 2, 10, 2, 42, 6, 10, 2, 146, 2, 10, 10, 1806, 2, 146, 2, 146, 10, 10, 2, 23226, 6, 10, 42, 146, 2, 314, 2, 3263442, 10, 10, 10, 42814, 2, 10, 10, 23226, 2, 314, 2, 146, 146, 10, 2, 542731938, 6, 146, 10, 146, 2, 23226, 10, 23226, 10, 10, 2, 141578, 2, 10, 146, 10650056950806, 10

Offset: 1

Ilya Gutkovskiy, Apr 17 2021

nonn

a(n) depends only on the prime signature of n (see formulas). - Bernard Schott, Apr 24 2021

Alois P. Heinz, Table of n, a(n) for n = 1..3000
Index entries for sequences related to prime signature

Cf. A025487, A006881, A007018, A007304, A067824, A082588, A333120.

a:= proc(n) option remember;
      1+add(a(d)^2, d=numtheory[divisors](n) minus {n})
    end:
seq(a(n), n=1..65);  # Alois P. Heinz, Apr 17 2021

a[n_] := a[n] = 1 + Sum[If[d < n, a[d]^2, 0], {d, Divisors[n]}]; Table[a[n], {n, 65}]

lista(nn) = {my(va = vector(nn)); for (n=1, nn, va[n] = 1 + sumdiv(n, d, if (dMichel Marcus, Apr 18 2021

from functools import lru_cache
from sympy import divisors
@lru_cache(maxsize=None)
def A343511(n): return 1+sum(A343511(d)**2 for d in divisors(n) if d < n) # Chai Wah Wu, Apr 17 2021

G.f.: x / (1 - x) + Sum_{n>=1} a(n)^2 * x^(2*n) / (1 - x^n).
a(p^k) = A007018(k) for p prime.
From Bernard Schott, Apr 24 2021: (Start)
a(A006881(n)) = 10 for signature [1, 1].
a(A054753(n)) = 146 for signature [2, 1].
a(A007304(n)) = 314 for signature [1, 1, 1].
a(A065036(n)) = 23226 for signature [3, 1].
a(A085986(n)) = 42814 for signature [2, 2].
a(A085987(n)) = 141578 for signature [2, 1, 1]. (End)

A381311 Numbers whose powerful part (A057521) is a power of a prime with an even exponent >= 2.

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 75, 76, 80, 81, 84, 90, 92, 98, 99, 112, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 188, 192, 198, 204, 207, 208, 212, 220, 228, 234, 236

Offset: 1

Amiram Eldar, Feb 19 2025

Numbers k whose largest unitary divisor that is a square, A350388(k), is a prime power (A246655), or equivalently, A350388(k) is in A056798 \ {1}.
Numbers having exactly one non-unitary prime factor and its multiplicity is even.
Numbers whose prime signature (A118914) is of the form {1, 1, ..., 2*m} with m >= 1, i.e., any number (including zero) of 1's and then a single even number.
The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} p/((p-1)*(p+1)^2) = 0.24200684327095676029... .

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

Complement of A381312 within A190641.
Cf. A057521, A118914, A246655, A350388.
Subsequence of: A377816.
Subsequences: A001248, A030514, A030516, A030629, A030631, A054753, A056798 \ {1}, A085987, A178739, A179644, A179645, A179668, A179672, A179693, A179747, A189982, A189983, A189985, A189987, A190110, A190292, A190388.

q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;;,2]]]}, EvenQ[e[[1]]] && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000],q]

isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); !(e[1] % 2) && (#e == 1 || e[2] == 1));

cp's OEIS Frontend

A275387 Numbers of ordered pairs of divisors d < e of n such that gcd(d, e) > 1.

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A086975 Numbers of the form p^2 * q * r with primes p < q < r.

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A179746 Numbers of the form p^4*q^2*r^2 where p, q, and r are distinct primes.

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A356413 Numbers with an equal sum of the even and odd exponents in their prime factorizations.

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A336530 Number of triples of divisors d_i < d_j < d_k of n such that gcd(d_i, d_j, d_k) > 1.

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A231122 Numbers k >= 0 such that 2^k is number of ways to write n as n = x*y, where x, y = squarefree numbers, 1 <= x <= n, 1 <= y <= n, or -1 if no such k exists.

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A275345 Characteristic polynomials of a square matrix based on A051731 where A051731(1,N)=1 and A051731(N,N)=0 and where N=size of matrix, analogous to the Redheffer matrix.

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A179746 Numbers of the form p^4q^2r^2 where p, q, and r are distinct primes.