A374459 -id:A374459 - cp's OEIS frontend

A374461 Numbers k such that k and k+1 are both nonsquarefree exponentially odd numbers (A374459).

135, 296, 343, 351, 375, 512, 999, 1160, 1375, 1431, 1592, 1624, 2079, 2295, 2375, 2456, 2727, 2943, 3104, 3159, 3429, 3591, 3624, 3752, 3992, 4023, 4184, 4616, 4832, 4887, 4913, 5048, 5144, 5319, 5480, 5535, 6183, 6344, 6375, 6655, 6858, 7047, 7263, 7479, 7624

Offset: 1

Amiram Eldar, Jul 09 2024

nonn

The numbers of terms that do not exceed 10^k, for k = 3, 4, ..., are 7, 59, 556, 5539, 55329, 553188, 5531116, 55311354, ... . Apparently, the asymptotic density of this sequence exists and equals 0.005531... .

			135 is a term since both 135 = 3^3 * 5 and 136 = 2^3 * 17 are nonsquarefree exponentially odd numbers.

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

Intersection of A068781 and A367695.

Subsequence of A268335 and A374459.

q[n_] := q[n] = Module[{e = FactorInteger[n][[;; , 2]]}, AllTrue[e, OddQ] && ! AllTrue[e, # == 1 &]]; Select[Range[10000], q[#] && q[# + 1] &]

is1(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!(e[i] %2), return(0))); for(i = 1, #e, if(e[i] >1, return(1))); 0;}
lista(kmax) = {my(q1 = is1(1), q2); for(k = 2, kmax, q2 = is1(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

A301517 Numbers whose ratio (sum of nonsquarefree divisors)/(sum of squarefree divisors) is a positive integer.

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 88, 96, 104, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 216, 224, 232, 243, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 352, 375, 376, 378, 384, 408, 416, 424, 440, 456, 459, 472, 480, 486, 488, 512, 513, 520

Offset: 1

Michel Lagneau, Mar 23 2018

nonn

Or numbers m such that r = A162296(m) / A048250(m) is a positive integer.
Conjecture: if r = A162296(a(n)) / A048250(a(n)) is a perfect square, r belongs to A001248.
The corresponding sequence b(n) = {r} begins with {4, 4, 9, 20, 4, 9, 4, 4, 20, 4, 4, 25, 84, 9, 4, 4, 20, 4, 4, 9, 49, 20, 4, 90, 4, 25, ... }. A majority of numbers of b(n) are perfect squares.
The numbers 2^(2n+1) with k > 0 are in the sequence (A004171).
The numbers prime(n)^3 are in the sequence (A030078).
The numbers 8*prime(n) with n > 1 are in the sequence.
Note that "positive integer", in the definition, eliminates squarefree numbers (A005117) from this sequence. - Michel Marcus, Mar 24 2018
From Robert Israel, Mar 29 2018: (Start)
If n is in the sequence, then so is n*p for any prime p coprime to n.
If m and n are in the sequence and are coprime, then m*n is in the sequence. (End)
The exponentially odd numbers (A268335) that are not squarefree are in the sequence. - Amiram Eldar, Jul 04 2020
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 9, 99, 972, 9672, 96630, 966119, 9660732, 96606486, 966062725, ... . Apparently the asymptotic density of this sequence is 0.096606... . Note that most of the terms are in its subsequence A374459 whose asymptotic density is A065463 - A059956 = 0.096515099145... . - Amiram Eldar, Feb 20 2025

			27 is in the sequence because A162296(27) / A048250(27) = 36/4 = 9.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A004171, A005117, A030078, A048250, A162296, A268335, A335989, A374459.
Contains A056824.
Cf. A059956, A065463.

filter:= proc(n) local S,N; uses numtheory;
  S, N:= selectremove(issqrfree, divisors(n));
  N <> {} and type(convert(N,`+`)/convert(S,`+`),integer)
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 29 2018

lst={};Do[If[DivisorSigma[1,n]-Total[Select[Divisors[n],SquareFreeQ]]>0&&IntegerQ[(DivisorSigma[1,n]-Total[Select[Divisors[n],SquareFreeQ]])/Total[Select[Divisors[n],SquareFreeQ]]],AppendTo[lst,n]],{n,520}];lst
rpiQ[n_]:=Module[{d=Divisors[n],sf,ot,ra},sf=Select[d,SquareFreeQ];ot=Complement[ d, sf];ra= Total[ ot]/Total[sf];ra>0&&IntegerQ[ra]]; Select[Range[600],rpiQ] (* Harvey P. Dale, Mar 19 2019 *)
f[p_, e_] := (p^(e + 1) - 1)/(p^2 - 1); ratio[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[2, 520], (r = ratio[#]) > 1 && IntegerQ[r] &] (* Amiram Eldar, Jul 04 2020 *)

isok(n) = my(s = sumdiv(n, d, !issquarefree(d)*d)); s && !(s % (sigma(n) - s)); \\ Michel Marcus, Mar 24 2018

A381312 Numbers whose powerful part (A057521) is a power of a prime with an odd exponent >= 3 (A056824).

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 88, 96, 104, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 224, 232, 243, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 352, 375, 376, 378, 384, 408, 416, 424, 440, 456, 459, 472, 480, 486, 488, 512, 513, 520, 536, 544

Offset: 1

Amiram Eldar, Feb 19 2025

Subsequence of A301517 and A374459 and first differs from them at n = 21. A301517(21) = A374459(21) = 216 is not a term of this sequence.
Numbers having exactly one non-unitary prime factor and its multiplicity is odd.
Numbers whose prime signature (A118914) is of the form {1, 1, ..., 2*m+1} with m >= 1, i.e., any number (including zero) of 1's and then a single odd number > 1.
The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} 1/((p-1)*(p+1)^2) = 0.093382464285953613312...

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

Complement of A381311 within A190641.
Cf. A057521, A118914.
Subsequence of A268335, A301517 and A374459.
Subsequences: A030078, A050997, A056824, A065036, A079395, A092759, A138031, A178740, A179664, A179665, A179667, A179670, A179692, A179696, A179704, A189975, A189984, A190378, A190383, A190473.

q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;; , 2]]]}, e[[1]] > 1 && OddQ[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] > 1 && (#e == 1 || e[2] == 1));

A374460 Indices of the nonsquarefree terms in the sequence of exponentially odd numbers (A268335).

Original entry on oeis.org

7, 18, 20, 24, 31, 39, 41, 63, 69, 74, 86, 89, 91, 97, 98, 109, 115, 121, 131, 135, 154, 161, 167, 174, 177, 179, 189, 194, 200, 211, 212, 223, 234, 243, 244, 249, 250, 265, 266, 268, 273, 290, 296, 302, 314, 325, 328, 338, 343, 348, 350, 366, 367, 373, 382, 388, 393

Offset: 1

Amiram Eldar, Jul 09 2024

The asymptotic density of this sequence is 1 - A059956 / A065463 = 0.13700925215474602945... .

			The first 7 exponentially odd numbers are 1, 2, 3, 5, 6, 7, and 8. A268335(7) = 8 = 3^3 is the least nonsquarefree term. Therefore a(1) = 7.

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

Cf. A005117, A059956, A065463, A268335.

Similar sequences: A361936, A363189, A371186, A371188.

Position[Select[Range[120], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &], _?(!SquareFreeQ[#] &), Heads -> False] // Flatten

isexpodd(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!(e[i] % 2), return(0))); 1;}
lista(kmax) = {my(f, c = 0); for(k = 1, kmax, if(isexpodd(k), c++; if(!issquarefree(k), print1(c, ", "))));}

A268335(a(n)) = A374459(n).

A384559 The sum of the exponential unitary (or e-unitary) 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 A384558 at n = 512: a(512) = 514, while A384558(512) = 522.

The number of these divisors is A384557(n), and the largest of them is A331737(n).

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

Cf. A005117, A051377, A072587, A268335, A331737, A368979, A374459, A384557, A384558.

f[p_, e_] := DivisorSum[e, p^# &, OddQ[#] && CoprimeQ[#, e/#] &]; 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) * (gcd(d, f[i,2]/d) == 1) * f[i,1]^d));}

Multiplicative with a(p^e) = Sum_{d|e, d odd, gcd(d, e/d) = 1} 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.

A384520 Numbers whose powerful part (A057521) is greater than 1 and is equal to a squarefree number raised to an odd power (A384518).

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 88, 96, 104, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 216, 224, 232, 243, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 352, 375, 376, 378, 384, 408, 416, 424, 440, 456, 459, 472, 480, 486, 488, 512, 513, 520, 536

Offset: 1

Amiram Eldar, Jun 01 2025

nonn

Subsequence of A301517 and A374459 and first differs from them at n = 85: A374459(85) = A374459(85) = 864 = 2^5 * 3^3 is not a term of this sequence.
First differs from its subsequence A381312 at n = 21: a(21) = 216 = 2^3 * 3^3 is not a term of A381312.
Numbers whose prime factorization has one distinct exponent that is larger than 1 and it is odd.
Numbers that are a product of a squarefree number (A005117) and a coprime nonsquarefree number that is a squarefree number raised to an odd power (A384518).
The asymptotic density of this sequence is Sum_{k>=1} (d(2*k+1)-1)/zeta(2) = 0.095609588748823080455..., where d(k) = (zeta(2*k)/zeta(k)) * Product_{p prime} (1 + 2/p^k + Sum_{i=k+1..2*k-1} (-1)^(i+1)/p^i).

Intersection of A268335 and A375142.
Intersection of A295661 and A375142.
Intersection of A376142 and A375142.
Equals A375142 \ A384519.
Subsequence of A301517 and A374459.		
Subsequences: A381312, A384518.
Cf. A005117, A013661, A057521.

q[n_] := Module[{u = Union[Select[FactorInteger[n][[;; , 2]], # > 1 &]]}, Length[u] == 1 && OddQ[u[[1]]]]; Select[Range[250], q]

isok(k) = {my(e = select(x -> (x > 1), Set(factor(k)[, 2]))); #e == 1 && e[1] % 2;}

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... .

A374536 a(n) is the least exponentially odd number that is nonsquarefree and is followed by exactly n successive exponentially odd numbers that are squarefree, or -1 if no such number exists.

Original entry on oeis.org

135, 24, 120, 27, 96, 88, 32, 40, 328, 168, 136, 104, 1288, 1161, 352, 488, 8, 783, 189, 952, 4520, 56, 11576, 67384, 5088, 1336, 35768, 16173, 53768, 80328, 128169, 28576, 247375, 208552, 2556192, 1486568, 3099368, 1653032, 910568, 7864008, 34242976, 14484152

Offset: 0

Amiram Eldar, Jul 11 2024

nonn

			a(0) = 135 because 135 and 136 are successive nonsquarefree exponentially odd numbers with no squarefree number between them.
a(1) = 24 because 24 and 27 are successive nonsquarefree exponentially odd numbers with one squarefree number between them, 26.
a(2) = 120 because 120 and 125 are successive nonsquarefree exponentially odd numbers with two squarefree number between them, 122 and 123.

Amiram Eldar, Table of n, a(n) for n = 0..53

Cf. A005117, A045882, A268335, A270344, A374459, A374460.

sq[k_] := Module[{e = FactorInteger[k][[;;, 2]]}, If[AnyTrue[e, EvenQ], 0, If[k == 1 || Max[e] == 1, 2, 1]]]; seq[len_, kmax_ : Infinity] := Module[{v = Table[0, {len}], c = 0, k = 1, k0 = 0, m, i = 1}, While[c < len && k < kmax, m = sq[k]; If[m > 0, If[m == 2, i++, If[k0 > 0, If[i <= len && v[[i]] == 0, c++; v[[i]] = k0]; i = 1];	k0 = k]]; k++]; v]; seq[10]

issq(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!(e[i] % 2), return(0))); if(k == 1 || vecmax(e) == 1, 2, 1);}
lista(len, kmax = oo) = {my(v = vector(len), c = 0, k = 1, k0 = 0, m, i = 1); while(c < len && k < kmax, m = issq(k); if(m > 0, if(m == 2, i++, if(k0 > 0, if(i <= len && v[i] == 0, c++; v[i] = k0); i = 1); k0 = k)); k++); v; }

a(n) = A268335(A374460(k)), where k is the least number such that A374459(k+1) - A374459(k) = n + 1.

cp's OEIS Frontend

A374461 Numbers k such that k and k+1 are both nonsquarefree exponentially odd numbers (A374459).

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A301517 Numbers whose ratio (sum of nonsquarefree divisors)/(sum of squarefree divisors) is a positive integer.

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A381312 Numbers whose powerful part (A057521) is a power of a prime with an odd exponent >= 3 (A056824).

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A374460 Indices of the nonsquarefree terms in the sequence of exponentially odd numbers (A268335).

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A384559 The sum of the exponential unitary (or e-unitary) divisors of n that are exponentially odd numbers (A268335).

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A384520 Numbers whose powerful part (A057521) is greater than 1 and is equal to a squarefree number raised to an odd power (A384518).

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

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A374536 a(n) is the least exponentially odd number that is nonsquarefree and is followed by exactly n successive exponentially odd numbers that are squarefree, or -1 if no such number exists.