A188999 -id:A188999 - cp's OEIS frontend

A369205 Numbers m such that A188999(A034448(m)) = k*m for some k, where A034448 and A188999 are respectively the unitary and the bi-unitary sigma function.

1, 2, 9, 10, 15, 18, 21, 30, 40, 42, 60, 120, 288, 567, 630, 720, 756, 1023, 1134, 1428, 2046, 2160, 2268, 2520, 3024, 3276, 3570, 4092, 6048, 8184, 8925, 9240, 11424, 11550, 15345, 17850, 18144, 30690, 35700, 46200, 57120, 85680, 147312, 285600, 491040, 556920

Offset: 1

Tomohiro Yamada, Jan 16 2024

nonn

			A034448(18) = 4 * 10 = 40 and A188999(40) = 15 * 6 = 90 = 5 * 18, so 18 is a term with k = 5.

Amiram Eldar, Table of n, a(n) for n = 1..105

Cf. A038843 (analog for A034448(A034448(m))), A318175 (analog for A188999(A188999(m))).

Cf. A369204 (analog for A034448(A188999(m))).

a034448(n) = {my(f,i,p,e);f=factor(n);for(i=1,#f~,p=f[i,1];e=f[i,2];f[i,1]=p^e+1;f[i,2]=1);factorback(f)};
a188999(n) = {my(f,i,p,e);f=factor(n);for(i=1,#f~,p=f[i,1];e=f[i,2];f[i,1]=if(e%2,(p^(e+1)-1)/(p-1),(p^(e+1)-1)/(p-1)-p^(e/2));f[i,2]=1);factorback(f)};
isok(n) = (a188999(a034448(n))%n) == 0;

A294029 Values of bsigma(k) = bsigma(k+1), where bsigma is the sum of the bi-unitary divisors (A188999).

Original entry on oeis.org

24, 40, 60, 720, 960, 1440, 2160, 2640, 2400, 3000, 4320, 4320, 4320, 5280, 7400, 11520, 11880, 12960, 14400, 20160, 30240, 26640, 34560, 25200, 34560, 49920, 51840, 60480, 63360, 60480, 65280, 62400, 61560, 115200, 93600, 114912, 100800, 120960, 120960

Offset: 1

Amiram Eldar, Oct 22 2017

nonn

The sum of bi-unitary divisors of numbers n such that n and n+1 have the same sum (A293183).

The bi-unitary version of A053215.

			24 is in the sequence since 24 = bsigma(14) = bsigma(15).

Amiram Eldar, Table of n, a(n) for n = 1..2148

Cf. A053215, A290303, A188999, A293183.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] := DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; a = {}; b1 = 0; For[k = 0, k < 10^6, k++; b2 = bsigma[k]; If[b1 == b2, a = AppendTo[a, b1]]; b1 = b2]; a (* after Michael De Vlieger at A188999 *)

a(n) = A188999(A293183(n)).

A309568 Bi-unitary k-hyperperfect numbers: numbers m such that m = 1 + k * (bsigma(m) - m - 1) where bsigma(m) is the sum of bi-unitary divisors of m (A188999) and k >= 1 is an integer.

Original entry on oeis.org

6, 21, 52, 60, 90, 301, 657, 697, 1333, 1909, 2041, 2133, 3901, 15025, 24601, 26977, 96361, 130153, 163201, 176661, 250321, 275833, 296341, 389593, 486877, 495529, 542413, 808861, 1005421, 1005649, 1055833, 1063141, 1232053, 1246417, 1284121, 1357741, 1403221

Offset: 1

Amiram Eldar, Aug 08 2019

nonn

The bi-unitary version of A034897.
The only bi-unitary 1-hyperperfect numbers are 6, 60, and 90 (the bi-unitary perfect numbers).
The corresponding k values are 1, 2, 3, 1, 1, 6, 8, 12, 18, 18, 12, 2, 30, 24, 60, 48, 132, 132, 192, 2, 168, 108, 66, 252, 78, 132, 342, 366, 390, 168, 348, 282, 498, 552, 540, 30, 546, ...

			21 is in the sequence since bsigma(21) = 32 and 21 = 1 + 2 * (32 - 21 - 1).

Amiram Eldar, Table of n, a(n) for n = 1..1000
József Sándor and Mihály Bencze, On modified hyperperfect numbers, Research report collection, Vol. 8, No. 2 (2005).

Cf. A034897, A188999, A225150.

fun[p_, e_] := If[OddQ[e], (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); hpnQ[n_] := (c = bsigma[n]-n-1) > 0 && Divisible[n-1, c]; Select[Range[10^5],  hpnQ]

A318242 a(n) is the least k such that A188999(A188999(k)) = n*k, where A188999 is the bi-unitary sigma function, or 0 if no such k exists.

Original entry on oeis.org

1, 2, 8, 15, 24, 42, 240, 648, 168, 480, 321408, 4320, 57120, 103680, 1827840, 23591520, 898128000, 374250240

Offset: 1

Michel Marcus, Aug 22 2018

It is also known that a(20) = 11975040.
Then for higher indices n, we have:
  a(19) <= 5235707393280;
  a(21) <= 49110437376000;
  a(22) <= 106780561395056640;
  a(24) <= 1099525819392000;
  a(25) <= 41252767395840;
  a(26) <= 202768780032000.

Tomohiro Yamada, 2 and 9 are the only biunitary superperfect numbers, arXiv:1705.00189 [math.NT], 2017. See Table 1.
Tomohiro Yamada, 2 and 9 are the only biunitary superperfect numbers, Annales Univ. Sci. Budapest., Sec. Comp., Volume 48 (2018). See Table 1.

Cf. A188999, A318175.

Cf. A272930 (analog for sigma), A318272 (analog for infinitary sigma).

A318781 A188999(m)/m for the integers m such that A188999(m) is divisible by m, where A188999 is the bi-unitary sigma function.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 3, 3, 4, 3, 4, 3, 4, 3, 3, 4, 4, 3, 3, 3, 4, 4, 4, 3, 3, 4, 3, 4, 4, 3, 4, 3, 3, 4, 4, 4, 4, 3

Offset: 1

Michel Marcus, Sep 03 2018, following a suggestion from Felix Fröhlich

10496266260480 is a term of A189000 and it is the smallest known value x such that A188999(x)/x is 5.

Cf. A188999 (bi-unitary sigma), A189000 (multiply perfect for bi-unitary sigma).

Cf. A054030 (analog for sigma), A007691 (multiply perfect for sigma).

a188999(n) = my(f = factor(n)); for (i=1, #f~, p = f[i, 1]; e = f[i, 2]; f[i, 1] = if (e % 2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)); f[i, 2] = 1; ); factorback(f) \\ after Michel Marcus in A189000
is_a189000(n) = ! frac(a188999(n)/n) \\ after Michel Marcus in A189000
for(n=1, oo, if(is_a189000(n), print1(a188999(n)/n, ", "))) \\ Felix Fröhlich, Sep 03 2018

a(n) = A188999(A189000(n))/A189000(n).

a(33)-a(42) from Giovanni Resta, Sep 03 2018

A322162 Numbers k such that bsigma(k) = 2k + 2, where bsigma(k) is the sum of bi-unitary divisors of k (A188999).

Original entry on oeis.org

80, 104, 832, 1952, 7424, 62464, 522752, 8382464, 33357824, 134193152, 267649024, 17167286272, 549754241024

Offset: 1

Amiram Eldar, Nov 29 2018

The bi-unitary version of A088831.

If m is a term of A050414, i.e., 2^m - 3 is prime, then 2^(2*m-2) * (2^m-3) is in this sequence, and also 2^(m-1) * (2^m-3) if m is even.

			80 is in this sequence since its sum of bi-unitary divisors is 162 = 2 * 80 + 2.

Cf. A050414, A088831, A188999.

fun[p_, e_] := If[OddQ[e], (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)]; Select[Range[2,10000], Times@@(fun @@@ FactorInteger[#]) == 2#+2 &]

bsigma(n,f=factor(n))=prod(i=1,#f~, my(p=f[i,1], e=f[i, 2]); if (e%2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)));
forfactored(n=1,10^8, if(bsigma(n[1],n[2])==2*n[1]+2, print1(n[1]", "))) \\ Charles R Greathouse IV, Nov 29 2018

a(13) from Giovanni Resta, Dec 01 2018

A034448 usigma(n) = sum of unitary divisors of n (divisors d such that gcd(d, n/d)=1); also called UnitarySigma(n).

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 9, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 36, 26, 42, 28, 40, 30, 72, 32, 33, 48, 54, 48, 50, 38, 60, 56, 54, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 84, 72, 72, 80, 90, 60, 120, 62, 96, 80, 65, 84, 144, 68, 90, 96, 144

Offset: 1

N. J. A. Sloane, Dec 11 1999

Row sums of the triangle in A077610. - Reinhard Zumkeller, Feb 12 2002

Multiplicative with a(p^e) = p^e+1 for e>0. - Franklin T. Adams-Watters, Sep 11 2005

			Unitary divisors of 12 are 1, 3, 4, 12. Or, 12=3*2^2 hence usigma(12)=(3+1)*(2^2+1)=20.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.

T. D. Noe, Table of n, a(n) for n = 1..10000
Octavio A. Agustín-Aquino, Prime injections and quasipolarities, Matematiche 69 (2014) 159-168
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 50.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Math. Comp., to appear (2014).
Tim Trudgian, The sum of the unitary divisor function, Publications de l'Institut Mathématique 2015 Vol. 97, Issue 111, pp. 175-180.
Eric Weisstein's World of Mathematics, Unitary Divisor Function
Wikipedia, Unitary divisor

Cf. A000203, A034444, A034460, A047994, A048250, A064000, A064609, A191414.
Cf. A063937 (squares > 1).
Cf. A188999, A301981, A301982.

a034448 = sum . a077610_row  -- Reinhard Zumkeller, Feb 12 2012
(Python 3.8+)
from math import prod
from sympy import factorint
def A034448(n): return prod(p**e+1 for p, e in factorint(n).items()) # Chai Wah Wu, Jun 20 2021

A034448 := proc(n) local ans, i:ans := 1: for i from 1 to nops(ifactors(n)[ 2 ]) do ans := ans*(1+ifactors(n)[ 2 ][ i ][ 1 ]^ifactors(n)[ 2 ] [ i ] [ 2 ]): od: RETURN(ans) end:
a := proc(n) local i; numtheory[divisors](n); select(d -> igcd(d,n/d)=1, %); add(i,i=%) end; # Peter Luschny, May 03 2009

usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]]; Table[ usigma[n], {n, 71}] (* Robert G. Wilson v, Aug 28 2004 *)
Table[DivisorSum[n, # &, CoprimeQ[#, n/#] &], {n, 70}] (* Michael De Vlieger, Mar 01 2017 *)
usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; Array[usigma, 100] (* faster since avoids generating divisors, Giovanni Resta, Apr 23 2017 *)

A034448(n)=sumdiv(n,d,if(gcd(d,n/d)==1,d)) \\ Rick L. Shepherd

A034448(n) = {my(f=factorint(n)); prod(k=1, #f[,2], f[k,1]^f[k,2]+1)} \\ Andrew Lelechenko, Apr 22 2014

a(n)=sumdivmult(n,d,if(gcd(d,n/d)==1,d)) \\ Charles R Greathouse IV, Sep 09 2014

If n = Product p_i^e_i, usigma(n) = Product (p_i^e_i + 1). - Vladeta Jovovic, Apr 19 2001
Dirichlet generating function: zeta(s)*zeta(s-1)/zeta(2s-1). - Franklin T. Adams-Watters, Sep 11 2005
Conjecture: a(n) = sigma(n^2/rad(n))/sigma(n/rad(n)), where sigma = A000203 and rad = A007947. - Velin Yanev, Aug 20 2017
This conjecture is easily verified since all the functions involved are multiplicative and proving it for prime powers is straightforward. - Juan José Alba González, Mar 19 2021
From Amiram Eldar, May 29 2020: (Start)
Sum_{d|n, gcd(d, n/d) = 1} a(d) * (-1)^omega(n/d) = n.
a(n) <= sigma(n) = A000203(n), with equality if and only if n is squarefree (A005117). (End)
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / (12*zeta(3)). - Vaclav Kotesovec, May 20 2021
a(n) = uphi(n^2)/uphi(n) = A191414(n)/uphi(n), where uphi(n) = A047994(n). - Amiram Eldar, Sep 21 2024

More terms from Erich Friedman

A286324 a(n) is the number of bi-unitary divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 8, 2, 6, 4, 4, 4, 4, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 8, 4, 8, 4, 4, 2, 8, 2, 4, 4, 6, 4, 8, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 4, 8, 2, 8, 4, 4, 2, 8, 4, 4, 4, 8, 2, 8

Offset: 1

Michel Marcus, May 07 2017

a(n) is the number of terms of the n-th row of A222266.

			From _Michael De Vlieger_, May 07 2017: (Start)
a(1) = 1 since 1 is the empty product; all divisors of 1 (i.e., 1) have a greatest common unitary divisor that is 1. 1 is a unitary divisor of all numbers n.
a(p) = 2 since 1 and p have greatest common unitary divisor 1.
a(6) = 4 since the divisor pairs {1, 6} and {2, 3} have greatest common unitary divisor 1.
a(24) = 8 since {1, 24}, {2, 12}, {3, 8}, {4, 6} have greatest unitary divisors {1, {1, 3, 8, 24}}, {{1, 2}, {1, 3, 4, 12}}, {{1, 3}, {1, 8}}, {{1, 4}, {1, 2, 3, 6}}: 1 is the greatest common unitary divisor among all 4 pairs. (End)

Antti Karttunen, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)
D. Suryanarayana, The number of bi-unitary divisors of an integer, in The Theory of Arithmetic Functions pp 273-282, Lecture Notes in Mathematics book series (LNM, volume 251).
Eric Weisstein's World of Mathematics, Biunitary Divisor.

Cf. A222266, A188999, A293185 (indices of records), A340232, A350390.

Cf. A000005, A034444 (unitary), A037445 (infinitary).

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; Table[DivisorSum[n, 1 &, Last@ Intersection[f@ #, f[n/#]] == 1 &], {n, 90}] (* Michael De Vlieger, May 07 2017 *)
f[p_, e_] := If[OddQ[e], e + 1, e]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 120] (* Amiram Eldar, Dec 19 2018 *)

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));
a(n) = #biudivs(n);

a(n)={my(f=factor(n)[,2]); prod(i=1, #f, my(e=f[i]); e + e % 2)} \\ Andrew Howroyd, Aug 05 2018

for(n=1, 100, print1(direuler(p=2, n, (X^3 - X^2 + X + 1) / ((X-1)^2 * (X+1)))[n], ", ")) \\ Vaclav Kotesovec, Jan 11 2024

Multiplicative with a(p^e) = e + (e mod 2). - Andrew Howroyd, Aug 05 2018
a(A340232(n)) = 2*n. - Bernard Schott, Mar 12 2023
a(n) = A000005(A350390(n)) (the number of divisors of the largest exponentially odd number dividing n). - Amiram Eldar, Sep 01 2023
From Vaclav Kotesovec, Jan 11 2024: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - (p^s - 1)/((p^s + 1)*p^(2*s))).
Let f(s) = Product_{p prime} (1 - (p^s - 1)/((p^s + 1)*p^(2*s))).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - (p-1)/((p+1)*p^2)) = A306071 = 0.80733082163620503914865427993003113402584582508155664401800520770441381...,
f'(1) = f(1) * Sum_{p prime} 2*(p^2 - p - 1) * log(p) /(p^4 + 2*p^3 + 1) = f(1) * 0.40523703144422392508596509911218523410441417240419849262346362977537989... = f(1) * A306072
and gamma is the Euler-Mascheroni constant A001620. (End)

A222266 Irregular triangle which lists the bi-unitary divisors of n in row n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 9, 1, 2, 5, 10, 1, 11, 1, 3, 4, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 2, 8, 16, 1, 17, 1, 2, 9, 18, 1, 19, 1, 4, 5, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 4, 6, 8, 12, 24, 1, 25, 1, 2, 13, 26, 1, 3, 9, 27, 1, 4, 7, 28, 1, 29, 1, 2, 3, 5, 6, 10, 15, 30, 1, 31, 1, 2, 4, 8, 16, 32, 1, 3, 11, 33, 1, 2, 17, 34, 1, 5, 7, 35

Offset: 1

R. J. Mathar, May 05 2013

The bi-unitary divisors of n are the divisors of n such that the largest common unitary divisor of d and n/d is 1, indicated by A165430.
The first difference from the triangle A077609 is in row n=16.
The concept of bi-unitary divisors was introduced by Suryanarayana (1972). - Amiram Eldar, Mar 09 2024

			The table starts
  1;
  1, 2;
  1, 3;
  1, 4;
  1, 5;
  1, 2, 3, 6;
  1, 7;
  1, 2, 4, 8;
  1, 9;
  1, 2, 5, 10;
  1, 11;
  1, 3, 4, 12;
  1, 13;
  1, 2, 7, 14;
  1, 3, 5, 15;
  1, 2, 8, 16;
  1, 17;

Michael De Vlieger, Table of n, a(n) for n = 1..13171 (rows 1 <= n <= 2000).
D. Suryanarayana, The number of bi-unitary divisors of an integer, in: A. A. Gioia and D. L. Goldsmith (eds.), The Theory of Arithmetic Functions, Lecture Notes in Mathematics, Vol 251, Springer, Berlin, Heidelberg, 1972.

Cf. A077609, A165430, A188999 (row sums), A286324 (row lengths).

# Return set of unitary divisors of n.
A077610_row := proc(n)
    local u,d ;
    u := {} ;
    for d in numtheory[divisors](n) do
        if igcd(n/d,d) = 1 then
            u := u union {d} ;
        end if;
    end do:
    u ;
end proc:
# true if d is a bi-unitary divisor of n.
isbiudiv := proc(n,d)
    if n mod d = 0 then
        A077610_row(d) intersect A077610_row(n/d) ;
        if % = {1} then
            true;
        else
            false;
        end if;
    else
        false;
    end if;
end proc:
# Return set of bi-unitary divisors of n
biudivs := proc(n)
    local u,d ;
    u := {} ;
    for d in numtheory[divisors](n) do
        if isbiudiv(n,d) then
            u := u union {d} ;
        end if;
    end do:
    u ;
end proc:
for n from 1 to 35 do
    print(op(biudivs(n))) ;
end do:

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; Table[Function[d, Union@ Flatten@ Select[Transpose@ {d, n/d}, Last@ Intersection[f@ #1, f@ #2] == 1 & @@ # &]]@ Select[Divisors@ n, # <= Floor@ Sqrt@ n &], {n, 35}] (* Michael De Vlieger, May 07 2017 *)

isbdiv(f, d) = {for (i=1, #f~, if(f[i, 2]%2 == 0 && valuation(d, f[i, 1]) == f[i, 2]/2, return(0))); 1;}
row(n) = {my(d = divisors(n), f = factor(n), bdiv = []); for(i=1, #d, if(isbdiv(f, d[i]), bdiv = concat(bdiv, d[i]))); bdiv; } \\ Amiram Eldar, Mar 24 2023

A002035 Numbers that contain primes to odd powers only.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 96, 97, 101

Offset: 1

N. J. A. Sloane

Complement of the union of {1} and A072587. - Reinhard Zumkeller, Nov 15 2012, corrected version from Jun 23 2002
A036537 is a subsequence and this sequence is a subsequence of A162644. - Reinhard Zumkeller, Jul 08 2009
The asymptotic density of this sequence is A065463. - Amiram Eldar, Sep 18 2022

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
Eckford Cohen, Quadratic congruences with an odd number of summands, Amer. Math. Monthly, 73 (1966), 138-143.

Cf. A000203, A036537, A065463, A072586, A072587, A124010, A162644, A188999, A295316.

a002035 n = a002035_list !! (n-1)
a002035_list = filter (all odd . a124010_row) [1..]
-- Reinhard Zumkeller, Nov 14 2012

isA002035 := proc(n)
    local pe;
    for pe in ifactors(n)[2] do
        if type(pe[2],'even') then
            return false;
        end if;
    end do:
    true ;
end proc:
A002035 := proc(n)
    option remember;
    if n =1 then
        2;
    else
        for a from procname(n-1)+1 do
            if isA002035(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A002035(n),n=1..100) ; # R. J. Mathar, Nov 27 2017

ok[n_] := And @@ OddQ /@ FactorInteger[n][[All, 2]];
Select[Range[2, 101], ok]
(* Jean-François Alcover, Apr 22 2011 *)
Select[Range[2,110],AllTrue[FactorInteger[#][[All,2]],OddQ]&] (* Harvey P. Dale, Nov 02 2022 *)

is(n)=Set(factor(n)[,2]%2)==[1] \\ Charles R Greathouse IV, Feb 07 2017

More terms from Reinhard Zumkeller, Jun 23 2002

cp's OEIS Frontend

A369205 Numbers m such that A188999(A034448(m)) = k*m for some k, where A034448 and A188999 are respectively the unitary and the bi-unitary sigma function.

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A294029 Values of bsigma(k) = bsigma(k+1), where bsigma is the sum of the bi-unitary divisors (A188999).

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A309568 Bi-unitary k-hyperperfect numbers: numbers m such that m = 1 + k * (bsigma(m) - m - 1) where bsigma(m) is the sum of bi-unitary divisors of m (A188999) and k >= 1 is an integer.

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A318242 a(n) is the least k such that A188999(A188999(k)) = n*k, where A188999 is the bi-unitary sigma function, or 0 if no such k exists.

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A318781 A188999(m)/m for the integers m such that A188999(m) is divisible by m, where A188999 is the bi-unitary sigma function.

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A322162 Numbers k such that bsigma(k) = 2k + 2, where bsigma(k) is the sum of bi-unitary divisors of k (A188999).

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A034448 usigma(n) = sum of unitary divisors of n (divisors d such that gcd(d, n/d)=1); also called UnitarySigma(n).

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Haskell

Maple

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PARI

PARI

PARI

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A286324 a(n) is the number of bi-unitary divisors of n.

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