A188999 -id:A188999 - cp's OEIS frontend

A072587 Numbers having at least one prime factor with an even exponent.

4, 9, 12, 16, 18, 20, 25, 28, 36, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 90, 92, 98, 99, 100, 108, 112, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 180, 188, 192, 196, 198, 200, 204

Offset: 1

Reinhard Zumkeller, Jun 23 2002

nonn

Complement of the union of {1} and A002035. [Correction, Nov 15 2012]
A162645 is a subsequence and this sequence is a subsequence of A162643. - Reinhard Zumkeller, Jul 08 2009
The asymptotic density of this sequence is 1 - A065463 = 0.2955577990... - Amiram Eldar, Jul 21 2020
A number k is a term iff its core (A007913) properly divides its kernel (A007947), that is iff A336643(k) > 1. - David James Sycamore, Sep 18 2023

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

Cf. A000037, A000203, A002035, A065463, A072588, A124010, A162643, A162645, A188999.

Cf. A007913, A007947, A336643.

a072587 n = a072587_list !! (n-1)
a072587_list = tail $ filter (any even . a124010_row) [1..]
-- Reinhard Zumkeller, Nov 15 2012

Select[Range[210], MemberQ[EvenQ[Transpose[FactorInteger[#]][[2]]], True] &] (* Harvey P. Dale, Apr 03 2012 *)

is(n)=n>3 && Set(factor(n)[,2]%2)[1]==0 \\ Charles R Greathouse IV, Oct 16 2015

from itertools import count, islice
from sympy import factorint
def A072587_gen(startvalue=1): # generator of terms
    return filter(lambda n:not all(map(lambda m:m&1,factorint(n).values())),count(max(startvalue,1)))
A072587_list = list(islice(A072587_gen(),30)) # Chai Wah Wu, Jan 04 2023

Thanks to Zak Seidov, who noticed that 1 had to be removed. - Reinhard Zumkeller, Nov 15 2012

A348271 a(n) is the sum of noninfinitary divisors of n.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 8, 0, 0, 0, 14, 0, 9, 0, 12, 0, 0, 0, 0, 5, 0, 0, 16, 0, 0, 0, 12, 0, 0, 0, 41, 0, 0, 0, 0, 0, 0, 0, 24, 18, 0, 0, 56, 7, 15, 0, 28, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 24, 42, 0, 0, 0, 36, 0, 0, 0, 45, 0, 0, 20, 40, 0, 0, 0, 84, 39

Offset: 1

Amiram Eldar, Oct 09 2021

nonn

			a(12) = 8 since 12 has 2 noninfinitary divisors, 2 and 6, and 2 + 6 = 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Infinitary Divisor.

Cf. A000203, A036537, A049417, A077609, A162643, A327633.

Similar sequences: A034448, A048146, A051377, A188999.

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; a[n_]:= DivisorSigma[1,n] - isigma[n]; Array[a, 100]

a(n) = A000203(n) - A049417(n).
a(n) = 0 if and only if the number of divisors of n is a power of 2, (i.e., n is in A036537).
a(n) > 0 if and only if the number of divisors of n is not a power of 2, (i.e., n is in A162643).

A286325 Bi-unitary harmonic numbers.

Original entry on oeis.org

1, 6, 45, 60, 90, 270, 420, 630, 672, 2970, 5460, 8190, 9072, 9100, 10080, 15925, 22680, 22848, 27300, 30240, 40950, 45360, 54600, 81900, 95550, 99792, 136500, 163800, 172900, 204750, 208656, 245700, 249480, 312480, 332640, 342720, 385560, 409500, 472500, 491400

Offset: 1

Michel Marcus, May 07 2017

nonn

A number m is a term if the sum of its bi-unitary divisors, A188999(m) divides the product of m by the number of its bi-unitary divisors A286324(m).

Numbers k whose harmonic mean of their bi-unitary divisors, A361782(k)/A361783(k), is an integer. - Amiram Eldar, Mar 24 2023

Amiram Eldar, Table of n, a(n) for n = 1..313 (all terms below 10^10, first 40 terms from Michel Marcus)
Jozsef Sandor, On bi-unitary harmonic numbers, arXiv:1105.0294 [math.NT], 2011. See pp. 13-20, but beware of typos and possible errors.

Cf. A222266, A188999, A286324, A361782, A361783, A361784.

Cf. A001599 (Ore harmonic), A006086 (unitary harmonic).

f[p_, e_] := p^e * If[OddQ[e], (e + 1)*(p - 1)/(p^(e + 1) - 1), e/((p^(e + 1) - 1)/(p - 1) - p^(e/2))]; bhQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; bhQ[1] = True; Select[Range[10^5], bhQ] (* Amiram Eldar, Mar 24 2023 *)

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));
isok(n) = my(v=biudivs(n)); denominator(n*#v/vecsum(v))==1;

A292980 Smaller of bi-unitary amicable pair.

Original entry on oeis.org

114, 594, 1140, 3608, 4698, 5940, 6232, 7704, 9520, 10744, 12285, 13500, 41360, 44772, 46980, 60858, 62100, 67095, 67158, 73360, 79650, 79750, 105976, 118500, 141664, 142310, 177750, 185368, 193392, 217840, 241024, 298188, 308220, 308992, 356408, 399200

Offset: 1

Amiram Eldar, Sep 27 2017

nonn

Analogous to amicable numbers with bi-unitary sigma (A188999) instead of sigma (A000203).
Hagis found all the bi-unitary amicable pairs with smaller members below 10^6.
The larger members are in A292981.

			3608 is in the sequence since A188999(3608) - 3608 = 3952 and A188999(3952) - 3952 = 3608.

Amiram Eldar, Table of n, a(n) for n = 1..6368 (all terms below 2*10^11, from David Moews' site).
Peter Hagis, Jr., Bi-unitary amicable and multiperfect numbers, Fibonacci Quarterly, Vol. 25, No. 2 (1987), pp. 144-150.
David Moews, Perfect, amicable and sociable numbers.

Cf. A002025, A188999, A292981.

fun[p_,e_]:=If[Mod[e,2]==1,(p^(e+1)-1)/(p-1),(p^(e+1)-1)/(p-1)-p^(e/2)];
bsigma[n_] := If[n==1,1,Times @@ (fun @@@ FactorInteger[n])]; Do[s = bsigma[n]; If[s > 2 n && bsigma[s - n] == s, Print[n]],{n,1,10000}] (* Amiram Eldar, Sep 29 2018 *)

A292981 Larger of bi-unitary amicable pair.

Original entry on oeis.org

126, 846, 1260, 3952, 5382, 8460, 6368, 8496, 13808, 10856, 14595, 17700, 51952, 49308, 53820, 83142, 62700, 71145, 73962, 97712, 107550, 88730, 108224, 131100, 153176, 168730, 196650, 203432, 195408, 287600, 309776, 306612, 365700, 332528, 399592, 419800

Offset: 1

Amiram Eldar, Sep 27 2017

nonn

Analogous to amicable numbers with bi-unitary sigma (A188999) instead of sigma (A000203).
Hagis found all the bi-unitary amicable pairs with smaller members below 10^6.
The smaller members are in A292980.
The terms are ordered according to their lesser counterparts.

			3952 is in the sequence since A188999(3608) - 3608 = 3952 and A188999(3952) - 3952 = 3608.

Amiram Eldar, Table of n, a(n) for n = 1..6368 (all terms with lesser member below 2*10^11, from David Moews' site).
Peter Hagis, Jr., Bi-unitary amicable and multiperfect numbers, Fibonacci Quarterly, Vol. 25, No. 2 (1987), pp. 144-150.
David Moews, Perfect, amicable and sociable numbers.

Cf. A002046, A188999, A292980.

fun[p_,e_]:=If[Mod[e,2]==1,(p^(e+1)-1)/(p-1),(p^(e+1)-1)/(p-1)-p^(e/2)];
bsigma[n_] := If[n==1,1,Times @@ (fun @@@ FactorInteger[n])]; Do[s = bsigma[n]; If[s > 2 n && bsigma[s - n] == s, Print[s-n]],{n,1,10000}] (* Amiram Eldar, Sep 29 2018 *)

A318167 Numbers k such that both k and k+1 are bi-unitary abundant numbers.

Original entry on oeis.org

21735, 21944, 43064, 49664, 58695, 76544, 106784, 135135, 144584, 160544, 188055, 209055, 227744, 256095, 262184, 300104, 345344, 348704, 382304, 387584, 407295, 409184, 414855, 437535, 498015, 520695, 560384, 567944, 611415, 679455, 687015, 705375, 709695

Offset: 1

Amiram Eldar, Aug 20 2018

nonn

The bi-unitary version of A096399.

			21735 is in the sequence since both 21735 and 21736 are bi-unitary abundant numbers.

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

Cf. A096399 (analog for sigma), A188999 (bi-unitary sigma).

Cf. A292982 (bi-unitary abundant), A293186 (odd bi-unitary abundant).

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] := DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; bAbundantQ[n_] := bsigma[n] > 2 n; seq={}; n=1; While[Length[seq]<32,If[bAbundantQ[n] && bAbundantQ [n+1],AppendTo[seq,n]];n++];seq

a188999(n) = {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(n) > 2*n) && (a188999(n+1) > 2*(n+1)); \\ Michel Marcus, Aug 21 2018

A292986 Bi-unitary weird numbers: bi-unitary abundant numbers (A292982) that are not bi-unitary pseudoperfect (A292985).

Original entry on oeis.org

70, 4030, 5390, 5830, 7192, 7400, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 11830, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17272, 17570, 17990, 18410, 18830, 18970, 19390, 19670, 19810

Offset: 1

Amiram Eldar, Sep 27 2017

nonn

Analogous to weird numbers (A006037) with bi-unitary sigma (A188999) instead of sigma (A000203).

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

Cf. A006037, A064114, A188999, A292982, A292985.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bdiv[m_] := Select[Divisors[m], Last@Intersection[f@#, f[m/#]] == 1 &]; bsigma[m_] := DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; bAbundantQ[n_] := bsigma[n] > 2 n; a = {}; n = 0; While[Length[a] < 5, n++; If[!bAbundantQ[n], Continue[]]; d = Most[bdiv[n]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c <= 0, AppendTo[a, n]]]; a (* after T. D. Noe at A005835 and Michael De Vlieger at A188999 *)

A353900 a(n) is the sum of divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 7, 13, 18, 12, 28, 14, 24, 24, 23, 18, 39, 20, 42, 32, 36, 24, 28, 31, 42, 13, 56, 30, 72, 32, 23, 48, 54, 48, 91, 38, 60, 56, 42, 42, 96, 44, 84, 78, 72, 48, 92, 57, 93, 72, 98, 54, 39, 72, 56, 80, 90, 60, 168, 62, 96, 104, 23, 84, 144

Offset: 1

Amiram Eldar, May 10 2022

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

Cf. A000203, A004709, A138302, A353898.

Similar sequences: A034448, A048146, A051377, A188999.

f[p_, e_] := 1 + Sum[p^(2^k), {k, 0, Floor[Log2[e]]}]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + sum(k = 0, logint(f[i,2], 2), f[i,1]^(2^k)));} \\ Amiram Eldar, Nov 19 2022

Multiplicative with a(p^e) = 1 + Sum_{k=0..floor(log_2(e))} p^(2^k).
a(n) = A000203(n) if and only if n is cubefree (A004709).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((1-1/p)*(1 + Sum_{k>=1} (Sum_{j=0..floor(log_2(k))} p^(2^j)/p^(2*k)))) = 0.7176001667... . - Amiram Eldar, Nov 19 2022

A292985 Bi-unitary pseudoperfect numbers: numbers that are equal to the sum of a subset of their aliquot bi-unitary divisors.

Original entry on oeis.org

6, 24, 30, 40, 42, 48, 54, 56, 60, 66, 72, 78, 80, 88, 90, 96, 102, 104, 114, 120, 138, 150, 160, 162, 168, 174, 186, 192, 210, 216, 222, 224, 240, 246, 258, 264, 270, 280, 282, 288, 294, 312, 318, 320, 330, 336, 352, 354, 360, 366, 378, 384, 390, 402, 408

Offset: 1

Amiram Eldar, Sep 27 2017

nonn

Analogous to pseudoperfect numbers (A005835) with bi-unitary sigma (A188999) instead of sigma (A000203).

			48 is in the sequence since its bi-unitary divisors are 1, 2, 3, 6, 8, 16, 24, 48 and 48 = 8 + 16 + 24.

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

Cf. A005835, A188999.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bdiv[m_] := Select[Divisors[m], Last@Intersection[f@#, f[m/#]] == 1 &]; a = {}; n = 0; While[n < 1000, n++; d = Most[bdiv[n]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c > 0, AppendTo[a, n]]];a (* after T. D. Noe at A005835 and Michael De Vlieger at A188999 *)

A362852 The number of divisors of n that are both bi-unitary and exponential.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 05 2023

First differs from A061704 at n = 128, and from A304327 and abs(A307428) at n = 64.
If e > 0 is the exponent of the highest power of p dividing n (where p is a prime), then for each divisor d of n that is both a bi-unitary and an exponential divisor, the exponent of the highest power of p dividing d is a number k such that k | e but k != e/2.
The least term that is higher than 2 is a(64) = 3.
This sequence is unbounded. E.g., a(2^(2^prime(n))) = prime(n).

			a(8) = 2 since 8 has 2 divisors that are both bi-unitary and exponential: 2 and 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Biunitary Divisor.
Eric Weisstein's World of Mathematics, e-Divisor.

Cf. A000005, A004709, A049419, A188999, A222266, A286324, A322791, A359411, A362853 (indices of records), A362854.

Cf. A061704, A304327, A307428.

f[p_, e_] := DivisorSigma[0, e] - If[OddQ[e], 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, numdiv(f[i, 2]) - !(f[i, 2] % 2));}

Multiplicative with a(p^e) = d(e) if e is odd, and d(e)-1 if e is even, where d(k) is the number of divisors of k (A000005).
a(n) = 1 if and only if n is cubefree (A004709).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=1} (d(k)+(k mod 2)-1)/p^k) = 1.1951330849... .

cp's OEIS Frontend

A072587 Numbers having at least one prime factor with an even exponent.

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A348271 a(n) is the sum of noninfinitary divisors of n.

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A286325 Bi-unitary harmonic numbers.

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A292980 Smaller of bi-unitary amicable pair.

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A292981 Larger of bi-unitary amicable pair.

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A318167 Numbers k such that both k and k+1 are bi-unitary abundant numbers.

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A292986 Bi-unitary weird numbers: bi-unitary abundant numbers (A292982) that are not bi-unitary pseudoperfect (A292985).

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A353900 a(n) is the sum of divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

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