A268335 -id:A268335 - cp's OEIS frontend

A374458 Squares of exponentially odd numbers (A268335).

1, 4, 9, 25, 36, 49, 64, 100, 121, 169, 196, 225, 289, 361, 441, 484, 529, 576, 676, 729, 841, 900, 961, 1024, 1089, 1156, 1225, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 2116, 2209, 2601, 2809, 2916, 3025, 3136, 3249, 3364, 3481, 3721, 3844, 4225, 4356, 4489

Offset: 1

Amiram Eldar, Jul 09 2024

Numbers whose exponents in their prime factorization are all congruent to 2 (mod 4).

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

Cf. A000290, A016825, A013662, A268335.

Similar sequences: A030140, A062503, A085986, A153158, A374291.

Select[Range[100], AllTrue[FactorInteger[#][[;;, 2]], OddQ] &]^2

is(k) = issquare(k) && if(k == 1, 1, my(e = factor(k)[, 2]); for(i = 1, #e, if(e[i] % 4 != 2, return(0))); 1);

a(n) = A000290(A268335(n)) = A268335(n)^2.
Sum_{n>=1} 1/a(n) = zeta(4) * Product_{p prime} (1 + 1/p^2 - 1/p^4) = 1.54211628314015874165... .
Sum_{n>=1} 1/a(n)^s = zeta(4*s) * Product_{p prime} (1 + 1/p^(2*s) - 1/p^(4*s)), for s > 1/2.

A374538 a(n) is the sum of the squares of the unitary divisors of n that are exponentially odd numbers (A268335).

Original entry on oeis.org

1, 5, 10, 1, 26, 50, 50, 65, 1, 130, 122, 10, 170, 250, 260, 1, 290, 5, 362, 26, 500, 610, 530, 650, 1, 850, 730, 50, 842, 1300, 962, 1025, 1220, 1450, 1300, 1, 1370, 1810, 1700, 1690, 1682, 2500, 1850, 122, 26, 2650, 2210, 10, 1, 5, 2900, 170, 2810, 3650, 3172

Offset: 1

Amiram Eldar, Jul 11 2024

The number of unitary divisors of n that are exponentially odd is A055076(n) and their sum is A358346(n).

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

Cf. A000290, A005117, A034676, A055076, A268335, A350389, A358346, A374537.

Cf. A002117, A013661, A183699.

f[p_, e_] := 1 + If[OddQ[e], p^(2*e), 0]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + if(f[i, 2]%2,  f[i, 1]^(2*f[i, 2]), 0));}

a(n) = A034676(A350389(n)).
a(n) >= 1 with equality if and only if n is a square (A000290).
a(n) <= A374537(n) with equality if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = p^(2*e) + 1 if e is odd, and 1 otherwise.
Dirichlet g.f.: zeta(s) * zeta(2*s-4) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-4) - 1/p^(2*s-2)).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2) * zeta(3) * Product_{p prime} (1 - 2/p^2 + 1/p^3 - 1/p^4 + 1/p^5) = 0.79482441214759383925... .

A382660 The unitary totient function applied to the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 1, 2, 4, 2, 6, 7, 4, 10, 12, 6, 8, 16, 18, 12, 10, 22, 14, 12, 26, 28, 8, 30, 31, 20, 16, 24, 36, 18, 24, 28, 40, 12, 42, 22, 46, 32, 52, 26, 40, 42, 36, 28, 58, 60, 30, 48, 20, 66, 44, 24, 70, 72, 36, 60, 24, 78, 40, 82, 64, 42, 56, 70, 88, 72, 60, 46, 72

Offset: 1

Amiram Eldar, Apr 02 2025

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

Cf. A013662, A047994, A065463, A268335.

Similar sequences: A358346, A363825, A366438, A366439, A366534, A366535, A367417, A368711, A368979, A374456, A382661.

f[p_, e_] := p^e-1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; expOddQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ]; uphi /@ Select[Range[100], expOddQ]

uphi(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2]-1);}
isexpodd(n) = {my(f = factor(n)); for(i=1, #f~, if(!(f[i, 2] % 2), return (0))); 1;}
list(lim) = apply(uphi, select(isexpodd, vector(lim, i, i)));

a(n) = A047994(A268335(n)).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(4)/(2*d^2)) * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 2/p^4 + 1/p^5) = 0.504949539649594981601..., and d = A065463 is the asymptotic density of the exponentially odd numbers.

A382661 The unitary Jordan totient function applied to the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 3, 8, 24, 24, 48, 63, 72, 120, 168, 144, 192, 288, 360, 384, 360, 528, 504, 504, 728, 840, 576, 960, 1023, 960, 864, 1152, 1368, 1080, 1344, 1512, 1680, 1152, 1848, 1584, 2208, 2304, 2808, 2184, 2880, 3024, 2880, 2520, 3480, 3720, 2880, 4032, 2880, 4488, 4224

Offset: 1

Amiram Eldar, Apr 02 2025

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

Cf. A013664, A065463, A191414, A268335.

Similar sequences: A358346, A363825, A366438, A366439, A366534, A366535, A367417, A368711, A368979, A374456, A382660.

f[p_, e_] := p^(2*e)-1; uj2[1] = 1; uj2[n_] := Times @@ f @@@ FactorInteger[n]; expOddQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ]; uj2 /@ Select[Range[100], expOddQ]

uj2(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^(2*f[i, 2])-1);}
isexpodd(n) = {my(f = factor(n)); for(i=1, #f~, if(!(f[i, 2] % 2), return (0))); 1;}
list(lim) = apply(uj2, select(isexpodd, vector(lim, i, i)));

a(n) = A191414(A268335(n)).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(6)/(3*d^3)) * Product_{p prime} (1 - 1/p^2 + 1/p^5 - 2/p^6 + 1/p^7) = 0.59726984314764530141..., and d = A065463 is the asymptotic density of the exponentially odd numbers.

A384557 The number of exponential unitary (or e-unitary) divisors of n that are exponentially odd numbers (A268335).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 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, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 03 2025

First differs from A359411 at n = 2097152 = 2^21: a(2097152) = 4, while A359411(2097152) = 2.
First differs from A368979 at n = 512 = 2^9: a(512) = 2, while A368979(512) = 3.
First differs from A367516 at n = 128 = 2^7: a(128) = 2, while A367516(128) = 1.
First differs from A382291 at n = 128 = 2^7: a(128) = 2, while A382291(128) = 4.
First differs from A368168 at n = 64 = 2^6: a(64) = 2, while A368168(64) = 1.
The sum of these divisors is A384559(n), and the largest of them is A331737(n).
The number of exponential unitary (or e-unitary) divisors of n is A278908(n) and the number of divisors of n that are exponentially odd numbers is A322483(n).
All the terms are powers of 2. The first term that is greater than 2 is a(32768) = 4.

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

Cf. A068068, A138302, A268335, A278908, A322483, A331737, A359411, A367516, A368168, A368979, A382291, A384559.

f[p_, e_] := 2^PrimeNu[e/2^IntegerExponent[e, 2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> 2^omega(x >> valuation(x, 2)) , factor(n)[, 2]));

Multiplicative with a(p^e) = A068068(e).
a(n) >= 1, with equality if and only if n is in A138302.
a(n) <= A278908(n), with equality if and only if n is an exponentially odd number.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=2} (d(k) - d(k-1))/p^k) = 1.13551542615965557947..., where d(k) is the number of odd unitary divisors of k (A068068).

A384558 The sum of the exponential divisors of n that are exponentially odd numbers (A268335).

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 10, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 30, 5, 26, 30, 14, 29, 30, 31, 34, 33, 34, 35, 6, 37, 38, 39, 50, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 60, 55, 70, 57, 58, 59, 30, 61, 62, 21, 10, 65, 66, 67, 34, 69

Offset: 1

Amiram Eldar, Jun 03 2025

First differs from A384559 at n = 512: a(512) = 522, while A384559(512) = 514.
The number of these divisors is A368979(n), and the largest of them is A331737(n).
The indices of records of a(n)/n are the primorial numbers (A002110) cubed, i.e., 1 and the terms of A115964.

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

Cf. A002110, A005117, A051377, A072587, A082020, A115964, A268335, A331737, A368979, A374459, A384559.

A384558:=proc(n)
    local a, pe,p,e,af,d;
    a := 1;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        af := 0 ;
        for d in numtheory[divisors](e) do
            if type(d,'odd') then
                af := af+p^d ;
            end if;
        end do:
        a := a*af ;
    end do;
    a
end proc:
seq(A384558(n), n=1..100); # R. J. Mathar, Jun 04 2025

f[p_, e_] := DivisorSum[e, p^# &, OddQ[#] &]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, sumdiv(f[i,2], d, (d % 2) * f[i,1]^d));}

Multiplicative with a(p^e) = Sum_{d|e, d odd} p^d.
a(n) = n if and only if n is squarefree (A005117).
a(n) < n if and only if n is in A072587.
a(n) > n if and only if n is in A374459.
limsup_{n->oo} a(n)/n = Product_{p prime} (1 + 1/p^2) = 15/Pi^2 (A082020).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p*(p^2-1)) - 1/(p^2-1) + (1-1/p) * Sum_{k>=1} p^(2*k+1)/(p^(4*k+2)-1)) = 0.80824764393216997768... .

A385416 The number of unordered factorizations of n into exponentially odd numbers (A268335).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 5, 1, 3, 2, 2, 2, 3, 1, 2, 2, 4, 1, 5, 1, 2, 2, 2, 1, 5, 1, 2, 2, 2, 1, 4, 2, 4, 2, 2, 1, 6, 1, 2, 2, 4, 2, 5, 1, 2, 2, 5, 1, 5, 1, 2, 2, 2, 2, 5, 1, 5, 2, 2, 1, 6, 2, 2, 2

Offset: 1

Amiram Eldar, Jun 28 2025

nonn

a(n) depends only on the prime signature of n (A118914).

The number of unordered factorizations of n into exponentially odd prime powers (A246551) is A050361(n).

			a(6) = 2 since 6 has 2 factorizations: 2 * 3 and 6.
a(24) = 4 since 24 has 4 factorizations: 2 * 2 * 2 * 3, 2 * 2 * 6, 3 * 8, and 24.

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

Cf. A001055, A005117, A050361, A118914, A246551, A268335, A385417 (indices of records).

T[n_, k_] := T[n, k] = If[n == 1 || k == 1, {{}}, Flatten[Table[Map[Prepend[#, d] &, T[d, k/d]], {d, Select[Divisors[k], 1 < # <= n &]}], 1]]; f[n_] := T[n, n];
expOdd[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ]; a[n_] := Count[f[n], _?(And @@ (expOdd /@ #) &)]; Array[a, 100]

a(n) <= A001055(n), with equality if and only if n is squarefree (A005117).

A385417 Numbers with a record number of unordered factorizations into exponentially odd numbers (A268335).

Original entry on oeis.org

1, 6, 24, 30, 60, 96, 120, 210, 240, 420, 480, 720, 840, 1680, 2520, 3360, 5040, 6720, 9240, 10080, 13440, 15120, 18480, 27720, 30240, 36960, 55440, 73920, 110880, 147840, 166320, 221760, 332640, 443520, 665280, 720720, 887040, 960960, 1108800, 1330560, 1441440

Offset: 1

Amiram Eldar, Jun 28 2025

nonn

Indices of records of A385416.

The corresponding record values are 1, 2, 4, 5, 6, 7, 11, 15, 16, 21, 22, 25, 37, 59, 62, 83, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..86
Amiram Eldar, Table of n, a(n), A385416(a(n)) for n = 1..86.

Subsequence of A025487.

Cf. A268335, A385416.

T[n_, k_] := T[n, k] = If[n == 1 || k == 1, {{}}, Flatten[Table[Map[Prepend[#, d] &, T[d, k/d]], {d, Select[Divisors[k], 1 < # <= n &]}], 1]]; f[n_] := T[n, n];
expOdd[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ]; s[n_] := Count[f[n], _?(And @@ (expOdd /@ #) &)];
With[{lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]}, sm = -1; seq = {}; Do[s1 = s[lps[[i]]]; If[s1 > sm, sm = s1; AppendTo[seq, lps[[i]]]], {i, 1, 300}]; seq]

The factorizations of the first 5 terms:
  n | a(n)  | A385416(a(n)) | factorizations
  --+-------+---------------|------------------------------------------------------------------
  1 |     1 |             1 | 1
  2 |     6 |             2 | 2*3, 6
  3 |    24 |             4 | 2 * 2 * 2 * 3, 2 * 2 * 6, 3 * 8, 24
  4 |    30 |             5 | 2 * 3 * 5, 2 * 15, 3 * 10, 5 * 6, 30
  5 |    60 |             6 | 2 * 2 * 3 * 5, 2 * 2 * 15, 2 * 3 * 10, 2 * 5 * 6, 2 * 30, 6 * 10

A335216 Bi-unitary Zumkeller numbers (A335215) that are not exponentially odd numbers (A268335).

Original entry on oeis.org

48, 60, 72, 80, 90, 150, 162, 192, 240, 288, 294, 320, 336, 360, 420, 432, 448, 504, 528, 540, 560, 576, 600, 624, 630, 648, 660, 720, 726, 756, 768, 780, 792, 800, 810, 816, 832, 880, 912, 924, 936, 960, 990, 1008, 1014, 1020, 1040, 1050, 1092, 1104, 1134, 1140

Offset: 1

Amiram Eldar, May 27 2020

nonn

Zumkeller numbers (A083207) that are exponentially odd (A268335) are also bi-unitary Zumkeller numbers (A335215), since all of their divisors are bi-unitary.

			48 is a term since it is not exponentially odd number (48 = 2^4 * 3 and 4 is even), and its set of bi-unitary divisors, {1, 2, 3, 6, 8, 16, 24, 48}, can be partitioned into 2 disjoint sets, whose sum is equal: 1 + 2 + 3 + 8 + 16 + 24 = 6 + 48.

Subsequence of A335215.

Cf. A083207, A268335, A290466.

uDivs[n_] := Select[Divisors[n], CoprimeQ[#, n/#] &]; bDivs[n_] := Select[Divisors[n], Last @ Intersection[uDivs[#], uDivs[n/#]] == 1 &]; bzQ[n_] := Module[{d = bDivs[n], sum, x}, sum = Plus @@ d; If[sum < 2*n || OddQ[sum], False, CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]]; expOddQ[n_] := AllTrue[Last /@ FactorInteger[n], OddQ]; Select[Range[1000], !expOddQ[#] && bzQ[#] &]

A335939 Bi-unitary weird numbers (A292986) that are not exponentially odd numbers (A268335).

Original entry on oeis.org

5390, 7400, 11830, 20230, 24010, 25270, 37030, 58870, 67270, 95830, 117670, 129430, 154630, 196630, 243670, 260470, 314230, 352870, 373030, 436870, 482230, 491744, 507232, 554470, 658630, 714070, 742630, 801430, 831670, 893830, 1129030, 1201270, 1313830, 1352470

Offset: 1

Amiram Eldar, Jun 30 2020

nonn

Weird numbers (A006037) that are exponentially odd (A268335) are also bi-unitary weird numbers (A292986), since all of their divisors are bi-unitary.

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

Intersection of A072587 and A292986.

Cf. A006037, A222266, A268335, A335936, A335938.

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]); biuabQ[n_] := bsigma[n] > 2*n; f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bdiv[m_] := Select[Divisors[m], Last@Intersection[f@#, f[m/#]] == 1 &]; bweirdQ[n_] := biuabQ[n] && Module[{d = Most @ bdiv[n], x}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] == 0]; expOddQ[n_] := AllTrue[Last /@ FactorInteger[n], OddQ]; Select[Range[1000], !expOddQ[#] && bweirdQ[#] &]

More terms from Amiram Eldar, Mar 25 2023

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A374458 Squares of exponentially odd numbers (A268335).

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A374538 a(n) is the sum of the squares of the unitary divisors of n that are exponentially odd numbers (A268335).

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A382660 The unitary totient function applied to the exponentially odd numbers (A268335).

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A382661 The unitary Jordan totient function applied to the exponentially odd numbers (A268335).

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A384557 The number of exponential unitary (or e-unitary) divisors of n that are exponentially odd numbers (A268335).

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A384558 The sum of the exponential divisors of n that are exponentially odd numbers (A268335).

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A385416 The number of unordered factorizations of n into exponentially odd numbers (A268335).

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A385417 Numbers with a record number of unordered factorizations into exponentially odd numbers (A268335).

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