A028234 -id:A028234 - cp's OEIS frontend

A270428 Exponentially odious numbers: 1 together with positive integers n such that all exponents in prime factorization of n are odious numbers (A000069).

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84

Offset: 1

Antti Karttunen, May 26 2016

A268385 maps each term of this sequence to a unique term of A268335, and vice versa.

The asymptotic density of this sequence is Product_{p prime} f(1/p) = 0.87686263163054480657..., where f(x) = 1 - x + (1 - (1-x) * Product_{k>=0} (1-x^(2^k)))/2. - Amiram Eldar, Oct 27 2023

Antti Karttunen, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175 (2016), 385-395.

Apart from 1, a subsequence of A270420.
Indices of ones in A270419.
Sequence A270436 sorted into ascending order.
Cf. A010060, A028234, A067029, A355825 (characteristic function).
Cf. also A262675, A268335, A268385.
Differs from its subsequence A138302 for the first time at n=113, where a(113) = 128 = 2^7, a value which does not occur in A138302.

odiousQ[n_] := OddQ[DigitCount[n, 2, 1]]; Select[Range[100], AllTrue[FactorInteger[#][[;;, 2]], odiousQ] &] (* Amiram Eldar, May 18 2023 *)

A355825(n) = factorback(apply(e->(hammingweight(e)%2), factor(n)[, 2]));
isA270428(n) = A355825(n); \\ Antti Karttunen, Jul 21 2022

;; Requires Antti Karttunen's IntSeq-library.
(define A270428 (ZERO-POS 1 1 (COMPOSE sub1 A270419)))

;; Requires Antti Karttunen's IntSeq-library.
(definec (chA270428 n) (cond ((= 1 n) 1) (else (* (A010060 (A067029 n)) (chA270428 (A028234 n))))))
(define A270428 (NONZERO-POS 1 1 chA270428))

A005064 Sum of cubes of primes dividing n.

Original entry on oeis.org

0, 8, 27, 8, 125, 35, 343, 8, 27, 133, 1331, 35, 2197, 351, 152, 8, 4913, 35, 6859, 133, 370, 1339, 12167, 35, 125, 2205, 27, 351, 24389, 160, 29791, 8, 1358, 4921, 468, 35, 50653, 6867, 2224, 133, 68921, 378, 79507, 1339, 152, 12175, 103823, 35, 343, 133, 4940, 2205, 148877, 35, 1456, 351, 6886, 24397, 205379, 160

Offset: 1

N. J. A. Sloane

nonn

The set of these terms is A213519. - Bernard Schott, Feb 11 2022

Inverse Möbius transform of n^3 * c(n), where c(n) is the prime characteristic (A010051). - Wesley Ivan Hurt, Jun 22 2024

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000578, A005067, A005072, A005076, A005080, A005084, A059841, A079978.
Cf. A010051, A213519.
Sum of the k-th powers of the primes dividing n for k=0..10 : A001221 (k=0), A008472 (k=1), A005063 (k=2), this sequence (k=3), A005065 (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), this sequence (k=8), A351197 (k=9), A351198 (k=10).

Array[DivisorSum[#, #^3 &, PrimeQ] &, 60] (* Michael De Vlieger, Jul 11 2017 *)
f[p_, e_] := p^3; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 20 2022 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,1]^3); \\ Michel Marcus, Jul 11 2017

from sympy import primefactors
def a(n): return sum(p**3 for p in primefactors(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 11 2017

(define (A005064 n) (if (= 1 n) 0 (+ (A000578 (A020639 n)) (A005064 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

Additive with a(p^e) = p^3.
G.f.: Sum_{k>=1} prime(k)^3*x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Dec 24 2016
From Antti Karttunen, Jul 11 2017: (Start)
a(n) = A005067(n) + 8*A059841(n).
a(n) = A005080(n) + A005084(n) + 8*A059841(n).
a(n) = A005072(n) + A005076(n) + 27*A079978(n).
(End)
Dirichlet g.f.: primezeta(s-3)*zeta(s). - Benedict W. J. Irwin, Jul 11 2018
a(n) = Sum_{p|n, p prime} p^3. - Wesley Ivan Hurt, Feb 04 2022
a(n) = Sum_{d|n} d^3 * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024

More terms from Antti Karttunen, Jul 10 2017

A056624 Number of unitary square divisors of n.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Aug 08 2000

Unitary analog of A046951.

The number of exponential divisors (A322791) of n that are cubefree (A004709). - Amiram Eldar, Jun 03 2025

			n=256, it has 5 square divisors of which only 2,{1,256} are unitary, 3 divisors are not.
n=124 has 2 (1 and 4) square divisors, both of them unitary a(124) = 2.
n=108 has 12 divisors, 4 square divisors: {1,4,9,36} of which 1 and 4 are unitary, 9 and 36 are not. So a(108)=2. The largest unitary square divisor of 108 is 4 with 1 prime divisor so a(108) = 2^1 = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A000188, A004709, A008833, A034444, A046952, A055229, A056626, A059841, A162641, A322791, A350388.

isA056624 := (n, d) -> igcd(n, d) = d and igcd(n/d, d) = d and igcd(n/d^2, d) = 1:
a := n -> nops(select(k -> isA056624(n, k), [seq(1..n)])):  # Peter Luschny, Jun 13 2025

Table[DivisorSum[n, 1 &, And[IntegerQ@ Sqrt@ #, CoprimeQ[#, n/#]] &], {n, 105}] (* Michael De Vlieger, Jul 28 2017 *)
f[p_, e_] := 2^(1 - Mod[e, 2]); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 03 2022 *)

a(n) = sumdiv(n, d, if(gcd(d, n/d)==1, issquare(d))); \\ Michel Marcus, Jul 28 2017

from sympy import factorint
def A056624(n): return 1<Chai Wah Wu, Aug 03 2024

def is_A056624(n, d): return gcd(n, d) == d and gcd(n//d, d) == d and gcd(n//(d*d), d) == 1
def a(n): return len([k for k in range(1, n+1) if is_A056624(n, k)])
print([a(n) for n in range(1, 106)])  # Peter Luschny, Jun 13 2025

(define (A056624 n) (if (= 1 n) n (* (A000079 (A059841 (A067029 n))) (A056624 (A028234 n))))) ;; Antti Karttunen, Jul 28 2017

a(n) = 2^r, where r is the number of prime factors of the largest unitary square divisor of n.
Multiplicative with a(p^e) = 2^(1-(e mod 2)). - Vladeta Jovovic, Dec 13 2002
Dirichlet g.f.: zeta(s)*zeta(2*s)/zeta(3*s). - Werner Schulte, Apr 03 2018
Sum_{k=1..n} a(k) ~ n*Pi^2/(6*zeta(3)) + sqrt(n)*zeta(1/2)/zeta(3/2). - Vaclav Kotesovec, Feb 07 2019
a(n) = 2^A162641(n). - Amiram Eldar, Sep 26 2022
a(n) = A034444(A350388(n)). - Amiram Eldar, Sep 09 2023

More terms from Vladeta Jovovic, Dec 13 2002

A293442 Multiplicative with a(p^e) = A019565(e).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 6, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 12, 3, 4, 6, 6, 2, 8, 2, 10, 4, 4, 4, 9, 2, 4, 4, 12, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 12, 4, 12, 4, 4, 2, 12, 2, 4, 6, 15, 4, 8, 2, 6, 4, 8, 2, 18, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4, 4, 12, 2, 12, 4, 6, 4, 4, 4, 20, 2, 6, 6, 9, 2, 8, 2, 12, 8

Offset: 1

Antti Karttunen, Oct 31 2017

From Peter Munn, Apr 06 2021: (Start)
a(n) is determined by the prime signature of n.
Compare with the multiplicative, self-inverse A225546, which also maps 2^e to the squarefree number A019565(e). However, this sequence maps p^e to the same squarefree number for every prime p, whereas A225546 maps the e-th power of progressively larger primes to progressively greater powers of A019565(e).
Both sequences map powers of squarefree numbers to powers of squarefree numbers.
(End)

Sequences used in a definition of this sequence: A000188, A003961, A019565, A028234, A059895, A067029, A162642.
Sequences with related definitions: A225546, A293443, A293444.
Cf. also A293214.
Sequences used to express relationship between terms of this sequence: A006519, A007913, A008833, A064989, A334747.
Sequences related via this sequence: (A001222, A048675, A064547), (A007814, A162642), (A087207, A267116), (A248663, A268387).

f[n_] := If[n == 1, 1, Apply[Times, Prime@ Flatten@ Position[Reverse@ IntegerDigits[Last@ #, 2], 1]] * f[n/Apply[Power, #]] &@ FactorInteger[n][[1]]]; Array[f, 105] (* Michael De Vlieger, Oct 31 2017 *)

a(1) = 1; for n > 1, a(n) = A019565(A067029(n)) * a(A028234(n)).
Other identities. For all n >= 1:
a(a(n)) = A293444(n).
A048675(a(n)) = A001222(n).
A001222(a(n)) = A064547(n) = A048675(A293444(n)).
A007814(a(n)) = A162642(n).
A087207(a(n)) = A267116(n).
A248663(a(n)) = A268387(n).
From Peter Munn, Mar 14 2021: (Start)
Alternative definition: a(1) = 1; a(2) = 2; a(n^2) = A003961(a(n)); a(A003961(n)) = a(n); if A059895(n, k) = 1, a(n*k) = a(n) * a(k).
For n >= 3, a(n) < n.
a(2n) = A334747(a(A006519(n))) * a(n/A006519(n)), where A006519(n) is the largest power of 2 dividing n.
a(2n+1) = a(A064989(2n+1)).
a(n) = a(A007913(n)) * a(A008833(n)) = 2^A162642(n) * A003961(a(A000188(n))).
(End)

A332823 A 3-way classification indicator generated by the products of two consecutive primes and the cubes of primes. a(n) is -1, 0, or 1 such that a(n) == A048675(n) (mod 3).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen and Peter Munn, Feb 25 2020

sign

Completely additive modulo 3.
The equivalent sequence modulo 2 is A096268 (with offset 1), which produces the {A003159, A036554} classification.
Let H be the multiplicative subgroup of the positive rational numbers generated by the products of two consecutive primes and the cubes of primes. a(n) indicates the coset of H containing n. a(n) = 0 if n is in H. a(n) = 1 if n is in 2H. a(n) = -1 if n is in (1/2)H.
The properties of this classification can usefully be compared to two well-studied classifications. With the {A026424, A028260} classes, multiplying a member of one class by a prime gives a member of the other class. With the {A000028, A000379} classes, adding a factor to the Fermi-Dirac factorization of a member of one class gives a member of the other class. So, if 4 is not a Fermi-Dirac factor of k, k and 4k will be in different classes of the {A000028, A000379} set; but k and 4k will be in the same class of the {A026424, A028260} set. For two numbers to necessarily be in different classes when they differ in either of the 2 ways described above, 3 classes are needed.
With the classes defined by this sequence, no two of k, 2k and 4k are in the same class. This is a consequence of the following stronger property: if k is any positive integer and m is a member of A050376 (often called Fermi-Dirac primes), then no two of k, k * m, k * m^2 are in the same class. Also, if p and q are consecutive primes, then k * p and k * q are in different classes.
Further properties are given in the sequences that list the classes: A332820, A332821, A332822.
The scaled imaginary part of the Eisenstein integer-valued function, f, defined in A353445. - Peter Munn, Apr 27 2022

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A332813 (0,1,2 version of this sequence), A353350.
Cf. A353354 (inverse Möbius transform, gives another 3-way classification indicator function).
Cf. A102283, A048675.
Cf. A332820, A332821, A332822 for positions of 0's, 1's and -1's in this sequence; also A003159, A036554 for the modulo 2 equivalents.
Comparable functions: A008836, A064179, A096268, A332814.
A000035, A003961, A028234, A055396, A067029, A097248, A225546, A297845, A331590 are used to express relationship between terms of this sequence.
The formula section also details how the sequence maps the terms of A000040, A332461, A332462.

A332823(n) = { my(f = factor(n),u=(sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3); if(2==u,-1,u); };

a(n) = A102283(A048675(n)) = -1 + (1 + A048675(n)) mod 3.
a(1) = 0; for n > 1, a(n) = A102283[(A067029(n) * (2-(A000035(A055396(n))))) + a(A028234(n))].
For all n >= 1, k >= 1: (Start)
a(n * k) == a(n) + a(k) (mod 3).
a(A331590(n,k)) == a(n) + a(k) (mod 3).
a(n^2) = -a(n).
a(A003961(n)) = -a(n).
a(A297845(n,k)) = a(n) * a(k).
(End)
For all n >= 1: (Start)
a(A000040(n)) = (-1)^(n-1).
a(A225546(n)) = a(n).
a(A097248(n)) = a(n).
a(A332461(n)) = a(A332462(n)) = A332814(n).
(End)
a(n) = A332814(A332462(n)). [Compare to the formula above. For a proof, see A353350.] - Antti Karttunen, Apr 16 2022

A007948 Largest cubefree number dividing n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 4, 9, 10, 11, 12, 13, 14, 15, 4, 17, 18, 19, 20, 21, 22, 23, 12, 25, 26, 9, 28, 29, 30, 31, 4, 33, 34, 35, 36, 37, 38, 39, 20, 41, 42, 43, 44, 45, 46, 47, 12, 49, 50, 51, 52, 53, 18, 55, 28, 57, 58, 59, 60, 61, 62, 63, 4, 65, 66, 67, 68, 69, 70, 71, 36, 73

Offset: 1

R. Muller

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative functions.
Florentin Smarandache, Only Problems, Not Solutions!, 1993.

Cf. A003557, A004709, A007947, A027748, A058035, A062378, A065465, A124010, A197863.

a007948 = last . filter ((== 1) . a212793) . a027750_row
-- Reinhard Zumkeller, May 27 2012, Jan 06 2012

Table[Apply[Times, FactorInteger[n] /. {p_, e_} /; p > 0 :> p^Min[e, 2]], {n, 73}] (* Michael De Vlieger, Jul 18 2017 *)

a(n) = my(f=factor(n)); for (i=1, #f~, f[i,2] = min(f[i, 2], 2)); factorback(f); \\ Michel Marcus, Jun 09 2014
(Scheme, with memoization-macro definec) (definec (A007948 n) (if (= 1 n) n (* (expt (A020639 n) (min 2 (A067029 n))) (A007948 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

Multiplicative with a(p^e) = p^(min(e, 2)). - David W. Wilson, Aug 01 2001
a(n) = max{A212793(A027750(n,k)) * A027750(n,k): k=1..A000005(n)}. - Reinhard Zumkeller, May 27 2012
a(n) = A071773(n)*A007947(n). - observed by Velin Yanev, Aug 20 2017, confirmed by Antti Karttunen, Nov 28 2017
a(n) = n / A062378(n) = n / A003557(A003557(n)). - Antti Karttunen, Nov 28 2017
Sum_{k=1..n} a(k) ~ (1/2) * c * n^2, where c = Product_{p prime} (1 - 1/(p^2*(p+1))) = 0.881513... (A065465). - Amiram Eldar, Oct 13 2022

More terms from Henry Bottomley, Jun 18 2001

A053164 4th root of largest 4th power dividing n.

Original entry on oeis.org

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

Offset: 1

Henry Bottomley, Feb 29 2000

Multiplicative with a(p^e) = p^[e/4]. - Mitch Harris, Apr 19 2005

			a(32) = 2 since 2 = 16^(1/4) and 16 is the largest 4th power dividing 32.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.

Cf. A000188, A000190, A008835, A053150.

A053164 := proc(n) local a,f,e,p ; for f in ifactors(n)[2] do e:= op(2,f) ; p := op(1,f) ; a := a*p^floor(e/4) ; end do ; a ; end proc: # R. J. Mathar, Jan 11 2012

f[list_] := list[[1]]^Quotient[list[[2]], 4]; Table[Apply[Times, Map[f,FactorInteger[n]]], {n, 1, 81}] (* Geoffrey Critzer, Jan 21 2015 *)

a(n) = A000188(A000188(n)) = A008835(n)^(1/4).
Multiplicative with a(p^e) = p^[e/4].
Dirichlet g.f.: zeta(4s-1)*zeta(s)/zeta(4s). - R. J. Mathar, Apr 09 2011
Sum_{k=1..n} a(k) ~ 90*zeta(3)*n/Pi^4 + 3*zeta(1/2)*sqrt(n)/Pi^2. - Vaclav Kotesovec, Dec 01 2020
a(n) = Sum_{d^4|n} phi(d). - Ridouane Oudra, Dec 31 2020
G.f.: Sum_{k>=1} phi(k) * x^(k^4) / (1 - x^(k^4)). - Ilya Gutkovskiy, Aug 20 2021

More terms from Antti Karttunen, Sep 13 2017

A046073 Number of squares in multiplicative group modulo n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 3, 2, 5, 1, 6, 3, 2, 2, 8, 3, 9, 2, 3, 5, 11, 1, 10, 6, 9, 3, 14, 2, 15, 4, 5, 8, 6, 3, 18, 9, 6, 2, 20, 3, 21, 5, 6, 11, 23, 2, 21, 10, 8, 6, 26, 9, 10, 3, 9, 14, 29, 2, 30, 15, 9, 8, 12, 5, 33, 8, 11, 6, 35, 3, 36, 18, 10, 9, 15, 6, 39, 4, 27, 20, 41, 3, 16, 21

Offset: 1

Eric W. Weisstein

a(n) is the number of different diagonal elements in Cayley table for multiplicative group modulo n. But the fact that the same number of different elements are on the diagonal of the Cayley table does not mean in every case that these groups are isomorphic. - Artur Jasinski, Jul 03 2010

The number of quadratic residues modulo n that are coprime to n. These residues are listed in A096103. - Peter Munn, Mar 10 2021

Daniel Shanks, Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 95, 1993.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Steven R. Finch and Pascal Sebah, Square and Cubes Modulo n, arXiv:math/0604465 [math.NT], 2006-2016.
Eric Weisstein's World of Mathematics, Modulo Multiplication Group.
Eric Weisstein's World of Mathematics, Quadratic Residue.

Cf. A046072, A007735, A060594, A000010, A087692, A000224.
Row lengths of A096103.
Positions of ones: A018253.

F:= n -> nops({seq}(`if`(igcd(t,n)=1,t^2 mod n,NULL), t=1..floor(n/2))):
1, seq(F(n), n=2..100); # Robert Israel, Jan 04 2015
# 2nd program
A046073 := proc(n)
    local a,p,e,pf;
    a := 1;
    for pf in ifactors(n)[2] do
        p := op(1,pf) ;
        e := op(2,pf) ;
        if p = 2 then
            a := a*p^max(e-3,0) ;
        else
            a := a*(p-1)/2*p^(e-1) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Oct 03 2016

Table[EulerPhi[n]/Sum[Boole[Mod[k^2, n] == 1] + Boole[n == 1], {k, n}], {n, 86}] (* or *)
Table[Apply[Times, FactorInteger[n] /. {p_, e_} /; p > 0 :> Which[p == 1, 1, p == 2, 2^Max[e - 3, 0], True, (p - 1) p^(e - 1)/2]], {n, 86}] (* Michael De Vlieger, Jul 18 2017 *)

A060594(n) = if(n<=2, 1, 2^#znstar(n)[3]); \\ This function from Joerg Arndt, Jan 06 2015
A046073(n) = eulerphi(n)/A060594(n); \\ Antti Karttunen, Jul 17 2017, after Sharon Sela's Mar 09 2002 formula.

A046073(n)=if(n>4,(n=znstar(n))[1]>>#n[3],1) \\ Avoids duplicate computation of phi(n). - M. F. Hasler, Nov 27 2017, typo fixed Mar 11 2021

from sympy import factorint, prod
def a(n): return 1 if n==1 else prod([2**max(e - 3, 0) if p==2 else (p - 1)*p**(e - 1)//2 for p, e in factorint(n).items()])
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 17 2017

(define (A046073 n) (cond ((= 1 n) n) ((even? n) (* (A000079 (max (- (A007814 n) 3) 0)) (A046073 (A028234 n)))) (else (* (/ 1 2) (- (A020639 n) 1) (/ (A028233 n) (A020639 n)) (A046073 (A028234 n)))))) ;; Antti Karttunen, Jul 17 2017, after the given multiplicative formula.

a(n) * A060594(n) = A000010(n) = phi(n) (This gives a formula for a(n) using the one in A060594(n) ). - Sharon Sela (sharonsela(AT)hotmail.com), Mar 09 2002
Multiplicative with a(2^e) = 2^max(e-3,0), a(p^e) = (p-1)*p^(e-1)/2 for p an odd prime.
Sum_{k=1..n} a(k) ~ c * n^2/sqrt(log(n)), where c = (43/(80*sqrt(Pi))) * Product_{p prime} (1+1/(2*p))*sqrt(1-1/p) = 0.24627260085060864229... (Finch and Sebah, 2006). - Amiram Eldar, Oct 18 2022

Edited and verified by Franklin T. Adams-Watters, Nov 07 2006

A092261 Sum of unitary, squarefree divisors of n, including 1.

Original entry on oeis.org

1, 3, 4, 1, 6, 12, 8, 1, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 4, 1, 42, 1, 8, 30, 72, 32, 1, 48, 54, 48, 1, 38, 60, 56, 6, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 3, 72, 8, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72, 1, 74, 114, 4, 20, 96, 168, 80

Offset: 1

Steven Finch, Feb 20 2004

Unitary convolution of the sequence of n*mu^2(n) (absolute values of A055615) and A000012. - R. J. Mathar, May 30 2011

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Zeitschr. 74 (1960) 66-80, sequence sigma'(n).
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]

Cf. A000203, A003557, A007947, A048250, A055231, A056671, A057521, A065465, A295294, A295295, A329728.

Table[Plus @@ Select[Divisors@ n, Max @@ Last /@ FactorInteger@ # == 1 && GCD[#, n/#] == 1 &], {n, 1, 79}] (* Michael De Vlieger, Mar 08 2015 *)
f[p_, e_] := If[e==1, p+1, 1]; a[1]=1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 79] (* Amiram Eldar, Mar 01 2019 *)

a(n) = sumdiv(n, d, d*issquarefree(d)*(gcd(d, n/d) == 1)); \\ Michel Marcus, Mar 06 2015

for(n=1, 100, print1(direuler(p=2, n, (1 + p^2*X^3 - p*X^2 - p^2*X^2)/(1-X)/(1-p*X))[n], ", ")) \\ Vaclav Kotesovec, Aug 20 2021

;; This implementation utilizes the memoization-macro definec for which an implementation is available at http://oeis.org/wiki/Memoization#Scheme
;; The other functions, A020639, A067029 and A028234 can be found under the respective entries, and should likewise defined with definec:
(definec (A092261 n) (if (= 1 n) 1 (* (+ 1 (if (> (A067029 n) 1) 0 (A020639 n))) (A092261 (A028234 n))))) ;; Antti Karttunen, Nov 25 2017

Multiplicative with a(p) = p+1 and a(p^e) = 1 for e > 1. - Vladeta Jovovic, Feb 22 2004
From Álvar Ibeas, Mar 06 2015: (Start)
a(n) = a(A055231(n)) = A000203(A055231(n)).
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + p^(1-s) - p^(1-2s)).
(End)
From Antti Karttunen, Nov 25 2017: (Start)
a(n) = A048250(A055231(n)).
a(n) = A000203(n) / A295294(n).
a(n) = A048250(n) / A295295(n) = A048250(n) / A048250(A057521(n)), where A057521(n) = A064549(A003557(n)).
(End)
Lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k = Product_{p prime}(1 - 1/(p^2*(p+1))) = 0.881513... (A065465). - Amiram Eldar, Jun 10 2020
Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 + p^(2-3*s) - p^(1-2*s) - p^(2-2*s)). - Vaclav Kotesovec, Aug 20 2021
a(n) = Sum_{d|n, gcd(d,n/d)=1} d * mu(d)^2. - Wesley Ivan Hurt, May 26 2023

A005065 Sum of 4th powers of primes dividing n.

Original entry on oeis.org

0, 16, 81, 16, 625, 97, 2401, 16, 81, 641, 14641, 97, 28561, 2417, 706, 16, 83521, 97, 130321, 641, 2482, 14657, 279841, 97, 625, 28577, 81, 2417, 707281, 722, 923521, 16, 14722, 83537, 3026, 97, 1874161, 130337, 28642, 641, 2825761, 2498, 3418801, 14657, 706, 279857, 4879681, 97, 2401, 641, 83602, 28577, 7890481, 97

Offset: 1

N. J. A. Sloane

nonn

Primes are taken without multiplicity, e.g., 12 = 2*2*3, and a(12) = 2^4+3^4 = 97. - Harvey P. Dale, Jul 16 2014

Inverse Möbius transform of n^4 * c(n), where c(n) is the prime characteristic (A010051). - Wesley Ivan Hurt, Jun 22 2024

Antti Karttunen, Table of n, a(n) for n = 1..10000

Column k=4 of A322080.
Cf. A000583, A005068, A005073, A005077, A005081, A005085, A008472, A059841, A079978.
Sum of the k-th powers of the primes dividing n for k=0..10 : A001221 (k=0), A008472 (k=1), A005063 (k=2), A005064 (k=3), this sequence (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), this sequence (k=8), A351197 (k=9), A351198 (k=10).
Cf. A010051.

A005065 := proc(n)
        add(d^4, d= numtheory[factorset](n)) ;
end proc;
seq(A005065(n),n=1..40) ; # R. J. Mathar, Nov 08 2011

Join[{0},Table[Total[Transpose[FactorInteger[n]][[1]]^4],{n,2,40}]] (* Harvey P. Dale, Jul 16 2014 *)
Array[DivisorSum[#, #^4 &, PrimeQ] &, 54] (* Michael De Vlieger, Jul 11 2017 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,1]^4); \\ Michel Marcus, Jul 11 2017

from sympy import primefactors
def a(n): return sum(p**4 for p in primefactors(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 11 2017

(define (A005065 n) (if (= 1 n) 0 (+ (A000583 (A020639 n)) (A005065 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

Additive with a(p^e) = p^4.
From Antti Karttunen, Jul 11 2017: (Start)
a(n) = A005068(n) + 16*A059841(n).
a(n) = A005081(n) + A005085(n) + 16*A059841(n).
a(n) = A005073(n) + A005077(n) + 81*A079978(n).
(End)
G.f.: Sum_{k>=1} prime(k)^4*x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Dec 24 2018
a(n) = Sum_{p|n, p prime} p^4. - Wesley Ivan Hurt, Feb 04 2022
a(n) = Sum_{d|n} d^4 * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024

More terms from Antti Karttunen, Jul 10 2017

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