A027748 -id:A027748 - cp's OEIS frontend

A051377 a(1)=1; for n > 1, a(n) = sum of exponential divisors (or e-divisors) of n.

1, 2, 3, 6, 5, 6, 7, 10, 12, 10, 11, 18, 13, 14, 15, 22, 17, 24, 19, 30, 21, 22, 23, 30, 30, 26, 30, 42, 29, 30, 31, 34, 33, 34, 35, 72, 37, 38, 39, 50, 41, 42, 43, 66, 60, 46, 47, 66, 56, 60, 51, 78, 53, 60, 55, 70, 57, 58, 59, 90, 61, 62, 84, 78, 65, 66, 67, 102, 69, 70, 71

Offset: 1

Yasutoshi Kohmoto

The e-divisors (or exponential divisors) of x=Product p(i)^r(i) are all numbers of the form Product p(i)^s(i) where s(i) divides r(i) for all i.

a(n) = n if and only if n is squarefree. - Jon Perry, Nov 13 2012

			a(8)=10 because 2 and 2^3 are e-divisors of 8 and 2+2^3=10.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
J. Fabrykowski and M. V. Subbarao, The maximal order and the average order of multiplicative function sigma^(e)(n), in Théorie des nombres/Number theory (Quebec, PQ, 1987), 201-206, de Gruyter, Berlin, 1989.
Nicussor Minculete, Concerning some arithmetic functions which use exponential divisors, Acta Universitatis Apulensis, No. 28/2011, pp. 125-133.
Y.-F. S. Pétermann and J. Wu, On the sum of exponential divisors of an integer, Acta Math. Hungar. 77 (1997), 159-175.
Eric Weisstein's World of Mathematics, e-Divisor

Cf. A051378, A049419 (number of e-divisors).
Cf. A027748, A124010, A027750.
Row sums of A322791.
See A307042 and A275480 where the formula and constant appear.

A051377:=n->Product(List(Collected(Factors(n)), p -> Sum(DivisorsInt(p[2]),d->p[1]^d))); List([1..10^4], n -> A051377(n)); # Muniru A Asiru, Oct 29 2017

a051377 n = product $ zipWith sum_e (a027748_row n) (a124010_row n) where
   sum_e p e = sum [p ^ d | d <- a027750_row e]
-- Reinhard Zumkeller, Mar 13 2012

A051377 := proc(n)
    local a,pe,p,e;
    a := 1;
    for pe in ifactors(n)[2] do
        p := pe[1] ;
        e := pe[2] ;
        add(p^d,d=numtheory[divisors](e)) ;
        a := a*% ;
    end do:
    a ;
end proc:
seq(A051377(n),n=1..100) ; # R. J. Mathar, Oct 05 2017

a[n_] := Times @@ (Sum[ First[#]^d, {d, Divisors[Last[#]]}] & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 71}] (* Jean-François Alcover, Apr 06 2012 *)

a(n)=my(f=factor(n));prod(i=1,#f[,1],sumdiv(f[i,2],d,f[i,1]^d)) \\ Charles R Greathouse IV, Nov 22 2011

ediv(n,f=factor(n))=my(v=List(),D=apply(divisors,f[,2]~),t=#f~); forvec(u=vector(t,i,[1,#D[i]]), listput(v,prod(j=1,t,f[j,1]^D[j][u[j]]))); Set(v)
a(n)=vecsum(ediv(n)) \\ Charles R Greathouse IV, Oct 29 2018

Multiplicative with a(p^e) = Sum_{d|e} p^d. - Vladeta Jovovic, Apr 23 2002
a(n) = A126164(n)+n. - R. J. Mathar, Oct 05 2017
The average order of a(n) is Dn + O(n^e) for any e > 0, due to Fabrykowski & Subbarao, where D is about 0.568. (D >= 0.5 since a(n) >= n.) - Charles R Greathouse IV, Sep 22 2023

More terms from Jud McCranie, May 29 2000

Definition corrected by Jaroslav Krizek, Feb 27 2009

A035316 Sum of the square divisors of n.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 5, 1, 1, 1, 21, 1, 10, 1, 5, 1, 1, 1, 5, 26, 1, 10, 5, 1, 1, 1, 21, 1, 1, 1, 50, 1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 21, 50, 26, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 85, 1, 1, 1, 5, 1, 1, 1, 50, 1, 1, 26, 5, 1, 1, 1, 21, 91, 1, 1, 5, 1, 1, 1, 5, 1, 10, 1, 5, 1

Offset: 1

N. J. A. Sloane

The Dirichlet generating function is zeta(s)*zeta(2s-2). The sequence is the Dirichlet convolution of A000012 with the sequence defined by n*A010052(n). - R. J. Mathar, Feb 18 2011

Inverse Möbius transform of n * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 20 2024

Nick Hobson, Table of n, a(n) for n = 1..1000
A. Dixit, B. Maji, and A. Vatwani, Voronoi summation formula for the generalized divisor function sigma_z^k(n), arXiv:2303.09937 [math.NT], 2023, sigma_(z=2,k=2,n).
R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions, arXiv:1106.4038 [math.NT] (2011), Remark 15.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

Cf. A001157, A010052, A027748, A124010, A113061 (sum cube divs).
Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), this sequence (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).
Cf. A008836, A000203.

a035316 n = product $
   zipWith (\p e -> (p ^ (e + 2 - mod e 2) - 1) `div` (p ^ 2 - 1))
           (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Jul 28 2014

A035316 := proc(n)
    local a,pe,p,e;
    a := 1;
    for pe in ifactors(n)[2] do
        p := pe[1] ;
        e := pe[2] ;
        if type(e,'even') then
            e := e+2 ;
        else
            e := e+1 ;
        end if;
        a := a*(p^e-1)/(p^2-1) ;
    end do:
    a ;
end proc:
seq(A035316(n),n=1..100) ; # R. J. Mathar, Oct 10 2017

Table[ Plus @@ Select[ Divisors@ n, IntegerQ@ Sqrt@ # &], {n, 93}] (* Robert G. Wilson v, Feb 19 2011 *)
f[p_, e_] := (p^(2*(1 + Floor[e/2])) - 1)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 01 2020 *)

vector(93, n, sumdiv(n, d, issquare(d)*d))

a(n)=my(f=factor(n));prod(i=1,#f[,1],(f[i,1]^(f[i,2]+2-f[i,2]%2)-1)/(f[i,1]^2-1)) \\ Charles R Greathouse IV, May 20 2013

Multiplicative with a(p^e)=(p^(e+2)-1)/(p^2-1) for even e and a(p^e)=(p^(e+1)-1)/(p^2-1) for odd e. - Vladeta Jovovic, Dec 05 2001
G.f.: Sum_{k>0} k^2*x^(k^2)/(1-x^(k^2)). - Vladeta Jovovic, Dec 13 2002
a(n^2) = A001157(n). - Michel Marcus, Jan 14 2014
L.g.f.: -log(Product_{k>=1} (1 - x^(k^2))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017
Sum_{k=1..n} a(k) ~ Zeta(3/2)*n^(3/2)/3 - n/2. - Vaclav Kotesovec, Feb 04 2019
a(n) = Sum_{k=1..n} k * (floor(sqrt(k)) - floor(sqrt(k-1))) * (1 - ceiling(n/k) + floor(n/k)). - Wesley Ivan Hurt, Jun 13 2021
a(n) = Sum_{d|n} d * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 20 2024
a(n) = Sum_{d|n} lambda(d)*d*sigma(n/d), where lambda = A008836. - Ridouane Oudra, Jul 18 2025

A008475 If n = Product (p_j^k_j) then a(n) = Sum (p_j^k_j) (a(1) = 0 by convention).

Original entry on oeis.org

0, 2, 3, 4, 5, 5, 7, 8, 9, 7, 11, 7, 13, 9, 8, 16, 17, 11, 19, 9, 10, 13, 23, 11, 25, 15, 27, 11, 29, 10, 31, 32, 14, 19, 12, 13, 37, 21, 16, 13, 41, 12, 43, 15, 14, 25, 47, 19, 49, 27, 20, 17, 53, 29, 16, 15, 22, 31, 59, 12, 61, 33, 16, 64, 18, 16, 67, 21, 26, 14, 71, 17, 73

Offset: 1

Olivier Gérard

For n>1, a(n) is the minimal number m such that the symmetric group S_m has an element of order n. - Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 26 2001

If gcd(u,w) = 1, then a(u*w) = a(u) + a(w); behaves like logarithm; compare A001414 or A056239. - Labos Elemer, Mar 31 2003

			a(180) = a(2^2 * 3^2 * 5) = 2^2 + 3^2 + 5 = 18.

F. J. Budden, The Fascination of Groups, Cambridge, 1972; pp. 322, 573.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter IV, p. 147.
T. Z. Xuan, On some sums of large additive number theoretic functions (in Chinese), Journal of Beijing normal university, No. 2 (1984), pp. 11-18.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
John Bamberg, Grant Cairns and Devin Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, Vol. 110, No. 3 (March 2003), pp. 202-209.
Roger B. Eggleton and William P. Galvin, Upper Bounds on the Sum of Principal Divisors of an Integer, Mathematics Magazine, Vol. 77, No. 3 (Jun., 2004), pp. 190-200.

Cf. A001414, A000961, A005117, A051613, A072691, A081402, A081403, A081404, A027748, A124010, A001221, A028233, A034684, A053585, A159077, A023888, A078771, A092509, A286875.

See A222416 for the variant with a(1)=1.

a008475 1 = 0
a008475 n = sum $ a141809_row n
-- Reinhard Zumkeller, Jan 29 2013, Oct 10 2011

A008475 := proc(n) local e,j; e := ifactors(n)[2]:
add(e[j][1]^e[j][2], j=1..nops(e)) end:
seq(A008475(n), n=1..60); # Peter Luschny, Jan 17 2010

f[n_] := Plus @@ Power @@@ FactorInteger@ n; f[1] = 0; Array[f, 73]

for(n=1,100,print1(sum(i=1,omega(n), component(component(factor(n),1),i)^component(component(factor(n),2),i)),","))

a(n)=local(t);if(n<1,0,t=factor(n);sum(k=1,matsize(t)[1],t[k,1]^t[k,2])) /* Michael Somos, Oct 20 2004 */

A008475(n) = { my(f=factor(n)); vecsum(vector(#f~,i,f[i,1]^f[i,2])); }; \\ Antti Karttunen, Nov 17 2017

from sympy import factorint
def a(n):
    f=factorint(n)
    return 0 if n==1 else sum([i**f[i] for i in f]) # Indranil Ghosh, May 20 2017

Additive with a(p^e) = p^e.
a(A000961(n)) = A000961(n); a(A005117(n)) = A001414(A005117(n)).
a(n) = Sum_{k=1..A001221(n)} A027748(n,k) ^ A124010(n,k) for n>1. - Reinhard Zumkeller, Oct 10 2011
a(n) = Sum_{k=1..A001221(n)} A141809(n,k) for n > 1. - Reinhard Zumkeller, Jan 29 2013
Sum_{k=1..n} a(k) ~ (Pi^2/12)* n^2/log(n) + O(n^2/log(n)^2) (Xuan, 1984). - Amiram Eldar, Mar 04 2021

A087207 A binary representation of the primes that divide a number, shown in decimal.

Original entry on oeis.org

0, 1, 2, 1, 4, 3, 8, 1, 2, 5, 16, 3, 32, 9, 6, 1, 64, 3, 128, 5, 10, 17, 256, 3, 4, 33, 2, 9, 512, 7, 1024, 1, 18, 65, 12, 3, 2048, 129, 34, 5, 4096, 11, 8192, 17, 6, 257, 16384, 3, 8, 5, 66, 33, 32768, 3, 20, 9, 130, 513, 65536, 7, 131072, 1025, 10, 1, 36, 19, 262144, 65, 258

Offset: 1

Mitch Cervinka (puritan(AT)planetkc.com), Oct 26 2003

The binary representation of a(n) shows which prime numbers divide n, but not the multiplicities. a(2)=1, a(3)=10, a(4)=1, a(5)=100, a(6)=11, a(10)=101, a(30)=111, etc.
For n > 1, a(n) gives the (one-based) index of the column where n is located in array A285321. A008479 gives the other index. - Antti Karttunen, Apr 17 2017
From Antti Karttunen, Jun 18 & 20 2017: (Start)
A268335 gives all n such that a(n) = A248663(n); the squarefree numbers (A005117) are all the n such that a(n) = A285330(n) = A048675(n).
For all n > 1 for which the value of A285331(n) is well-defined, we have A285331(a(n)) <= floor(A285331(n)/2), because then n is included in the binary tree A285332 and a(n) is one of its ancestors (in that tree), and thus must be at least one step nearer to its root than n itself.
Conjecture: Starting at any n and iterating the map n -> a(n), we will always reach 0 (see A288569). This conjecture is equivalent to the conjecture that at any n that is neither a prime nor a power of two, we will eventually hit a prime number (which then becomes a power of two in the next iteration). If this conjecture is false then sequence A285332 cannot be a permutation of natural numbers. On the other hand, if the conjecture is true, then A285332 must be a permutation of natural numbers, because all primes and powers of 2 occur in definite positions in that tree. This conjecture also implies the conjectures made in A019565 and A285320 that essentially claim that there are neither finite nor infinite cycles in A019565.
If there are any 2-cycles in this sequence, then both terms of the cycle should be present in A286611 and the larger one should be present in A286612.
(End)
Binary rank of the distinct prime indices of n, where the binary rank of an integer partition y is given by Sum_i 2^(y_i-1). For all prime indices (with multiplicity) we have A048675. - Gus Wiseman, May 25 2024

			a(38) = 129 because 38 = 2*19 = prime(1)*prime(8) and 129 = 2^0 + 2^7 (in binary 10000001).
a(140) = 13, binary 1101 because 140 is divisible by the first, third and fourth primes and 2^(1-1) + 2^(3-1) + 2^(4-1) = 13.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000 [First 1000 terms from _T. D. Noe_]
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization

For partial sums see A288566.
Cf. A000040, A000120, A001221, A005117, A008479, A019565, A055396, A285320, A285321, A285329, A285330, A285332.
Sequences with related definitions: A007947, A008472, A027748, A048675, A248663, A276379 (same sequence shown in base 2), A288569, A289271, A297404.
Cf. A286608 (numbers n for which a(n) < n), A286609 (n for which a(n) > n), and also A286611, A286612.
A003986, A003961, A059896 are used to express relationship between terms of this sequence.
Related to A267116 via A225546.
Positions of particular values are: A000079\{1} (1), A000244\{1} (2), A033845 (3), A000351\{1} (4), A033846 (5), A033849 (6), A143207 (7), A000420\{1} (8), A033847 (9), A033850 (10), A033851 (12), A147576 (14), A147571 (15), A001020\{1} (16), A033848 (17).
A048675 gives binary rank of prime indices.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Binary indices (listed A048793):
- length A000120, complement A023416
- min A001511, opposite A000012
- sum A029931, product A096111
- max A029837 or A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
- opposite A371572, sum A230877
Cf. A000720, A005940, A018819, A023506, A071814, A225620, A277319, A277905, A304818, A372689, A372890.

a087207 = sum . map ((2 ^) . (subtract 1) . a049084) . a027748_row
-- Reinhard Zumkeller, Jul 16 2013

a[n_] := Total[ 2^(PrimePi /@ FactorInteger[n][[All, 1]] - 1)]; a[1] = 0; Table[a[n], {n, 1, 69}] (* Jean-François Alcover, Dec 12 2011 *)

a(n) = {if (n==1, 0, my(f=factor(n), v = []); forprime(p=2, vecmax(f[,1]), v = concat(v, vecsearch(f[,1], p)!=0);); fromdigits(Vecrev(v), 2));} \\ Michel Marcus, Jun 05 2017

A087207(n)=vecsum(apply(p->1<M. F. Hasler, Jun 23 2017

from sympy import factorint, primepi
def a(n):
    return sum(2**primepi(i - 1) for i in factorint(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 06 2017

(definec (A087207 n) (if (= 1 n) 0 (+ (A000079 (+ -1 (A055396 n))) (A087207 (A028234 n))))) ;; This uses memoization-macro definec
(define (A087207 n) (A048675 (A007947 n))) ;; Needs code from A007947 and A048675. - Antti Karttunen, Jun 19 2017

Additive with a(p^e) = 2^(i-1) where p is the i-th prime. - Vladeta Jovovic, Oct 29 2003
a(n) gives the m such that A019565(m) = A007947(n). - Naohiro Nomoto, Oct 30 2003
A000120(a(n)) = A001221(n); a(n) = Sum(2^(A049084(p)-1): p prime-factor of n). - Reinhard Zumkeller, Nov 30 2003
G.f.: Sum_{k>=1} 2^(k-1)*x^prime(k)/(1-x^prime(k)). - Franklin T. Adams-Watters, Sep 01 2009
From Antti Karttunen, Apr 17 2017, Jun 19 2017 & Dec 06 2018: (Start)
a(n) = A048675(A007947(n)).
a(1) = 0; for n > 1, a(n) = 2^(A055396(n)-1) + a(A028234(n)).
A000035(a(n)) = 1 - A000035(n). [a(n) and n are of opposite parity.]
A248663(n) <= a(n) <= A048675(n). [XOR-, OR- and +-variants.]
a(A293214(n)) = A218403(n).
a(A293442(n)) = A267116(n).
A069010(a(n)) = A287170(n).
A007088(a(n)) = A276379(n).
A038374(a(n)) = A300820(n) for n >= 1.
(End)
From Peter Munn, Jan 08 2020: (Start)
a(A059896(n,k)) = a(n) OR a(k) = A003986(a(n), a(k)).
a(A003961(n)) = 2*a(n).
a(n^2) = a(n).
a(n) = A267116(A225546(n)).
a(A225546(n)) = A267116(n).
(End)

More terms from Don Reble, Ray Chandler and Naohiro Nomoto, Oct 28 2003

Name clarified by Antti Karttunen, Jun 18 2017

A206369 a(p^k) = p^k - p^(k-1) + p^(k-2) - ... +- 1, and then extend by multiplicativity.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 6, 5, 7, 4, 10, 6, 12, 6, 8, 11, 16, 7, 18, 12, 12, 10, 22, 10, 21, 12, 20, 18, 28, 8, 30, 21, 20, 16, 24, 21, 36, 18, 24, 20, 40, 12, 42, 30, 28, 22, 46, 22, 43, 21, 32, 36, 52, 20, 40, 30, 36, 28, 58, 24, 60, 30, 42, 43, 48, 20, 66, 48, 44, 24, 70, 35

Offset: 1

N. J. A. Sloane, Feb 06 2012

For more information see the Comments in A061020.
a(n) is the number of integers j such that 1 <= j <= n and gcd(n,j) is a perfect square. For example, a(12) = 6 because |{1,4,5,7,8,11}|=6 and the respective GCDs with 12 are 1,4,1,1,4,1, which are squares. - Geoffrey Critzer, Feb 16 2015
If m is squarefree (A005117), then a(m) = A000010(m) where A000010 is the Euler totient function. - Michel Marcus, Nov 08 2017
Also it appears that the primorials (A002110) is the sequence of indices of minimum records for a(n)/n, and these records are A038110(n)/A060753(n). - Michel Marcus, Nov 09 2017
Also called rho(n). When rho(n) | n, then n is called k-imperfect, with k = n/rho(n), cf. A127724. - M. F. Hasler, Feb 13 2020

P. J. McCarthy, Introduction to Arithmetical Functions, Springer Verlag, 1986, page 25.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
László Tóth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014.

Cf. A061020, A206368.
Cf. A027748 row, A124010, A206475 (first differences).
Cf. A078429.
Cf. A127724 (k-imperfect), A127725 (2-imperfect), A127726 (3-imperfect).

a206369 n = product $
   zipWith h (a027748_row n) (map toInteger $ a124010_row n) where
           h p e = sum $ take (fromInteger e + 1) $
                         iterate ((* p) . negate) (1 - 2 * (e `mod` 2))
-- Reinhard Zumkeller, Feb 08 2012

a:= n-> mul(add(i[1]^(i[2]-j)*(-1)^j, j=0..i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 03 2017

Table[Length[Select[Range[n], IntegerQ[GCD[n, #]^(1/2)] &]], {n, 72}] (* Geoffrey Critzer, Feb 16 2015 *)
a[n_] := n*DivisorSum[n, LiouvilleLambda[#]/#&]; Array[a, 72] (* Jean-François Alcover, Dec 04 2017, after Enrique Pérez Herrero *)
f[p_,e_] := Sum[(-1)^(e-k)*p^k, {k,0,e}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jan 01 2020 *)

a(n) = sum(k=1, n, issquare(gcd(n, k)));

ak(p,e)=my(s=1); for(i=1,e, s=s*p + (-1)^i); s
a(n)=my(f=factor(n)); prod(i=1,#f~, ak(f[i,1],f[i,2])) \\ Charles R Greathouse IV, Dec 27 2016

a(n) = sumdiv(n, d, eulerphi(n/d) * issquare(d)); \\ Daniel Suteu, Jun 27 2018

apply( {A206369(n)=vecprod([f[1]^(f[2]+1)\/(f[1]+1)|f<-factor(n)~])}, [1..99]) \\ M. F. Hasler, Feb 13 2020

from math import prod
from sympy import factorint
def A206369(n): return prod((lambda x:x[0]+int((x[1]<<1)>=p+1))(divmod(p**(e+1),p+1)) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 05 2024

a(n) = abs(A061020(n)).
a(n) = n*Sum_{d|n} lambda(d)/d, where lambda(n) is A008836(n). - Enrique Pérez Herrero, Sep 23 2012
Dirichlet g.f.: zeta(s - 1)*zeta(2*s)/zeta(s). - Geoffrey Critzer, Feb 25 2015
From Michel Marcus, Nov 05 2017: (Start)
a(2^n) = A001045(n+1);
a(3^n) = A015518(n+1);
a(5^n) = A015531(n+1);
a(7^n) = A015552(n+1);
a(11^n) = A015592(n+1). (End)
a(p^k) = p^k - a(p^(k - 1)) for k > 0 and prime p. - David A. Corneth, Nov 09 2017
a(n) = Sum_{d|n, d is a perfect square} phi(n/d), where phi(k) is the Euler totient function. - Daniel Suteu, Jun 27 2018
a(p^k) = A071324(p^k), for k >= 0 and prime p. - Michel Marcus, Aug 11 2018
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / 30. - Vaclav Kotesovec, Feb 07 2019
G.f.: Sum_{k>=1} lambda(k)*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, May 23 2019
a(n) = Sum_{i=1..n} A010052(gcd(n,i)). - Ridouane Oudra, Nov 24 2019
a(p^k) = round(p^(k+1)/(p+1)). - M. F. Hasler, Feb 13 2020

A304038 Irregular triangle T(n,k) read by rows: first row is 0, n-th row (n > 1) lists indices of distinct primes dividing n.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 2, 4, 1, 2, 1, 3, 5, 1, 2, 6, 1, 4, 2, 3, 1, 7, 1, 2, 8, 1, 3, 2, 4, 1, 5, 9, 1, 2, 3, 1, 6, 2, 1, 4, 10, 1, 2, 3, 11, 1, 2, 5, 1, 7, 3, 4, 1, 2, 12, 1, 8, 2, 6, 1, 3, 13, 1, 2, 4, 14, 1, 5, 2, 3, 1, 9, 15, 1, 2, 4, 1, 3, 2, 7, 1, 6, 16, 1, 2, 3, 5, 1, 4, 2, 8, 1, 10, 17, 1, 2, 3, 18, 1, 11

Offset: 1

Ilya Gutkovskiy, May 05 2018

			The irregular triangle begins:
1:  {0}
2:  {1}
3:  {2}
4:  {1}
5:  {3}
6:  {1, 2}
7:  {4}
8:  {1}
9:  {2}
10: {1, 3}
11: {5}
12: {1, 2}

Eric Weisstein's World of Mathematics, Distinct Prime Factors
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A000720, A001221 (row lengths), A027748, A055396, A061395, A066328 (row sums), A112798, A156061 (row products), A302170.

Flatten[Table[PrimePi[FactorInteger[n][[All, 1]]], {n, 1, 62}]]

T(n,k) = A000720(A027748(n,k)).
T(n,1) = A055396(n).
T(n,A001221(n)) = A061395(n).

A002479 Numbers of the form x^2 + 2*y^2.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 9, 11, 12, 16, 17, 18, 19, 22, 24, 25, 27, 32, 33, 34, 36, 38, 41, 43, 44, 48, 49, 50, 51, 54, 57, 59, 64, 66, 67, 68, 72, 73, 75, 76, 81, 82, 83, 86, 88, 89, 96, 97, 98, 99, 100, 102, 107, 108, 113, 114, 118, 121, 123, 128, 129, 131

Offset: 1

N. J. A. Sloane

A positive number k belongs to this sequence if and only if every prime p == 5, 7 (mod 8) dividing k occurs to an even power. - Sharon Sela (sharonsela(AT)hotmail.com), Mar 23 2002
Norms of numbers in Z[sqrt(-2)]. - Alonso del Arte, Sep 23 2014
Euler (E256) shows that these numbers are closed under multiplication, according to the Euler Archive. - Charles R Greathouse IV, Jun 16 2016
In addition to the previous comment: The proof was already given 1100 years before Euler by Brahmagupta's identity (a^2 + m*b^2)*(c^2 + m*d^2) = (a*c - m*b*d)^2 + m*(a*d + b*c)^2. - Klaus Purath, Oct 07 2023

L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 421.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 59.
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).

N. J. A. Sloane, Table of n, a(n) for n = 1..3148 (first 1000 terms from T. D. Noe)
L. Euler, Vollstaendige Anleitung zur Algebra, Zweiter Teil.
L. Euler, (E256) Specimen de usu observationum in mathesi pura, Novi Commentarii academiae scientiarum Petropolitanae 6 (1761), pp. 185-230.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Complement of A097700. Subsequence of A000408. For primes see A033203.

Cf. A035251, A027748, A124010, A003628.

a002479 n = a002479_list !! (n-1)
a002479_list = 0 : filter f [1..] where
   f x = all (even . snd) $ filter ((`elem` [5,7]) . (`mod` 8) . fst) $
                            zip (a027748_row x) (a124010_row x)
-- Reinhard Zumkeller, Feb 20 2014

[n: n in [0..131] | NormEquation(2, n) eq true]; // Arkadiusz Wesolowski, May 11 2016

lis:={}; M:=50; M2:=M^2;
for x from 0 to M do for y from 0 to M do
if x^2+2*y^2 <= M2 then lis:={op(lis),x^2+2*y^2}; fi; od: od:
sort(convert(lis,list)); # N. J. A. Sloane, Apr 30 2015

q = 16; imax = q^2; Select[Union[Flatten[Table[x^2 + 2y^2, {y, 0, q/Sqrt[2]}, {x, 0, q}]]], # <= imax &] (* Vladimir Joseph Stephan Orlovsky, Apr 20 2011 *)
Union[#[[1]]+2#[[2]]&/@Tuples[Range[0,10]^2,2]] (* Harvey P. Dale, Nov 24 2014 *)

is(n)=my(f=factor(n));for(i=1,#f[,1],if(f[i,1]%8>4 && f[i,2]%2, return(0)));1 \\ Charles R Greathouse IV, Nov 20 2012

list(lim)=my(v=List()); for(a=0,sqrtint(lim\=1), for(b=0,sqrtint((lim-a^2)\2), listput(v,a^2+2*b^2))); Set(v) \\ Charles R Greathouse IV, Jun 16 2016

from itertools import count, islice
from sympy import factorint
def A002479_gen(): # generator of terms
    return filter(lambda n:all(p & 7 < 5 or e & 1 == 0 for p, e in factorint(n).items()),count(0))
A002479_list = list(islice(A002479_gen(),30)) # Chai Wah Wu, Jun 27 2022

A013939 Partial sums of sequence A001221 (number of distinct primes dividing n).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 15, 17, 19, 20, 21, 23, 24, 26, 28, 30, 31, 33, 34, 36, 37, 39, 40, 43, 44, 45, 47, 49, 51, 53, 54, 56, 58, 60, 61, 64, 65, 67, 69, 71, 72, 74, 75, 77, 79, 81, 82, 84, 86, 88, 90, 92, 93, 96, 97, 99, 101, 102, 104, 107, 108, 110, 112

Offset: 1

N. J. A. Sloane, Henri Lifchitz

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Distinct Prime Factors

Cf. A005187, A006218, A011371, A013936.
Cf. A093614, A048865, A006218.
Cf. A027748, A000040, A082287.
Cf. A022559.
Cf. A077761.
Cf. A346617.
Cf. A001222, A048803.

a013939 n = a013939_list !! (n-1)
a013939_list = scanl1 (+) $ map a001221 [1..]
-- Reinhard Zumkeller, Feb 16 2012

[(&+[Floor(n/NthPrime(k)): k in [1..n]]): n in [1..70]]; // G. C. Greubel, Nov 24 2018

A013939 := proc(n) option remember;  `if`(n = 1, 0, a(n) + iquo(n+1, ithprime(n+1))) end:
seq(A013939(i), i = 1..69);  # Peter Luschny, Jul 16 2011

a[n_] := Sum[Floor[n/Prime[k]], {k, 1, n}]; Table[a[n], {n, 1, 69}] (* Jean-François Alcover, Jun 11 2012, from 2nd formula *)
Accumulate[PrimeNu[Range[120]]] (* Harvey P. Dale, Jun 05 2015 *)

t=0;vector(99,n,t+=omega(n)) \\ Charles R Greathouse IV, Jan 11 2012

a(n)=my(s);forprime(p=2,n,s+=n\p);s \\ Charles R Greathouse IV, Jan 11 2012

a(n) = sum(k=1, sqrtint(n), k * (primepi(n\k) - primepi(n\(k+1)))) + sum(k=1, n\(sqrtint(n)+1), if(isprime(k), n\k, 0)); \\ Daniel Suteu, Nov 24 2018

from sympy.ntheory import primefactors
print([sum(len(primefactors(k)) for k in range(1,n+1)) for n in range(1, 121)]) # Indranil Ghosh, Mar 19 2017

from sympy import primerange
def A013939(n): return sum(n//p for p in primerange(n+1)) # Chai Wah Wu, Oct 06 2024

[sum(floor(n/nth_prime(k)) for k in (1..n)) for n in (1..70)] # G. C. Greubel, Nov 24 2018

a(n) = Sum_{k <= n} omega(k).
a(n) = Sum_{k = 1..n} floor( n/prime(k) ).
a(n) = a(n-1) + A001221(n).
a(n) = A093614(n) - A048865(n); see also A006218.
A027748(a(A000040(n))+1) = A000040(n), A082287(a(n)+1) = n.
a(n) = Sum_{k=1..n} pi(floor(n/k)). - Vladeta Jovovic, Jun 18 2006
a(n) = n log log n + O(n). - Charles R Greathouse IV, Jan 11 2012
a(n) = n*(log log n + B) + o(n), where B = 0.261497... is the Mertens constant A077761. - Arkadiusz Wesolowski, Oct 18 2013
G.f.: (1/(1 - x))*Sum_{k>=1} x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Jan 02 2017
a(n) = Sum_{k=1..floor(sqrt(n))} k * (pi(floor(n/k)) - pi(floor(n/(k+1)))) + Sum_{p prime <= floor(n/(1+floor(sqrt(n))))} floor(n/p). - Daniel Suteu, Nov 24 2018
a(n) = Sum_{k>=1} k * A346617(n,k). - Alois P. Heinz, Aug 19 2021
a(n) = A001222(A048803(n+1)). - Flávio V. Fernandes, Jan 14 2025

More terms from Henry Bottomley, Jul 03 2001

A004613 Numbers that are divisible only by primes congruent to 1 mod 4.

Original entry on oeis.org

1, 5, 13, 17, 25, 29, 37, 41, 53, 61, 65, 73, 85, 89, 97, 101, 109, 113, 125, 137, 145, 149, 157, 169, 173, 181, 185, 193, 197, 205, 221, 229, 233, 241, 257, 265, 269, 277, 281, 289, 293, 305, 313, 317, 325, 337, 349, 353, 365, 373, 377, 389, 397, 401, 409, 421

Offset: 1

N. J. A. Sloane

Also gives solutions z to x^2+y^2=z^4 with gcd(x,y,z)=1 and x,y,z positive. - John Sillcox (johnsillcox(AT)hotmail.com), Feb 20 2004
A065338(a(n)) = 1. - Reinhard Zumkeller, Jul 10 2010
Product_{k=1..A001221(a(n))} A079260(A027748(a(n),k)) = 1. - Reinhard Zumkeller, Jan 07 2013
A062327(a(n)) = A000005(a(n))^2. (These are the only numbers that satisfy this equation.) - Benedikt Otten, May 22 2013
Numbers that are positive integer divisors of 1 + 4*x^2 where x is a positive integer. - Michael Somos, Jul 26 2013
Numbers k such that there is a "knight's move" of Euclidean distance sqrt(k) which allows the whole of the 2D lattice to be reached. For example, a knight which travels 4 units in any direction and then 1 unit at right angles to the first direction moves a distance sqrt(17) for each move. This knight can reach every square of an infinite chessboard.
Also 1/7 of the area of the n-th largest octagon with angles 3*Pi/4, along the perimeter of which there are only 8 nodes of the square lattice - at its vertices. - Alexander M. Domashenko, Feb 21 2024
Sequence closed under multiplication. Odd values of A031396 and their powers. These are the only numbers m that satisfy the Pell equation (k*x)^2 - D*(m*y)^2 = -1. - Klaus Purath, May 12 2025

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

T. D. Noe, Table of n, a(n) for n = 1..10000
David Broadhurst and Wadim Zudilin, A magnetic double integral, arXiv:1708.02381 [math.NT], 2017. See p. 7
Code Golf Stack Exchange, pseudonymous Stack Exchange user 'trichoplax', N-movers: How much of the infinite board can I reach?

Subsequence of A000404; A002144 is a subsequence. Essentially same as A008846.

Cf. A004614.

a004613 n = a004613_list !! (n-1)
a004613_list = filter (all (== 1) . map a079260 . a027748_row) [1..]
-- Reinhard Zumkeller, Jan 07 2013

[n: n in [1..500] | forall{d: d in PrimeDivisors(n) | d mod 4 eq 1}]; // Vincenzo Librandi, Aug 21 2012

isA004613 := proc(n)
    local p;
    for p in numtheory[factorset](n) do
        if modp(p,4) <> 1 then
            return false;
        end if;
    end do:
    true;
end proc:
for n from 1 to 200 do
    if isA004613(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Nov 17 2014
# second Maple program:
q:= n-> andmap(i-> irem(i[1], 4)=1, ifactors(n)[2]):
select(q, [$1..500])[];  # Alois P. Heinz, Jan 13 2024

ok[1] = True; ok[n_] := And @@ (Mod[#, 4] == 1 &) /@ FactorInteger[n][[All, 1]]; Select[Range[421], ok] (* Jean-François Alcover, May 05 2011 *)
Select[Range[500],Union[Mod[#,4]&/@(FactorInteger[#][[All,1]])]=={1}&] (* Harvey P. Dale, Mar 08 2017 *)

for(n=1,1000,if(sumdiv(n,d,isprime(d)*if((d-1)%4,1,0))==0,print1(n,",")))

is(n)=n%4==1 && factorback(factor(n)[,1]%4)==1 \\ Charles R Greathouse IV, Sep 19 2016

Numbers of the form x^2 + y^2 where x is even, y is odd and gcd(x, y) = 1.

A019554 Smallest number whose square is divisible by n.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 4, 3, 10, 11, 6, 13, 14, 15, 4, 17, 6, 19, 10, 21, 22, 23, 12, 5, 26, 9, 14, 29, 30, 31, 8, 33, 34, 35, 6, 37, 38, 39, 20, 41, 42, 43, 22, 15, 46, 47, 12, 7, 10, 51, 26, 53, 18, 55, 28, 57, 58, 59, 30, 61, 62, 21, 8, 65, 66, 67, 34, 69, 70, 71, 12, 73, 74, 15, 38, 77

Offset: 1

R. Muller

A note on square roots of numbers: we can write sqrt(n) = b*sqrt(c) where c is squarefree. Then b = A000188(n) is the "inner square root" of n, c = A007913(n), and b*c = A019554(n) = "outer square root" of n.
Instead of the terms "inner square root" and "outer square root", we may use the terms "lower square root" and "upper square root", respectively. Upper k-th roots have been studied by Broughan (2002, 2003, 2006). - Petros Hadjicostas, Sep 15 2019
The number of times each number k appears in this sequence is A034444(k). The first time k appears is at position A102631(k). - N. J. A. Sloane, Jul 28 2021

T. D. Noe, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.
Kevin A. Broughan, Restricted divisor sums, Acta Arithmetica, 101(2) (2002), 105-114. See also here for another copy.
Kevin A. Broughan, Relationship between the integer conductor and k-th root functions, Int. J. Pure Appl. Math. 5(3) (2003), 253-275.
Kevin A. Broughan, Relaxations of the ABC Conjecture using integer k'th roots, New Zealand J. Math. 35(2) (2006), 121-136.
Florentin Smarandache, Collected Papers, Vol. II, Tempus Publ. Hse, Bucharest, 1996.
Eric Weisstein's World of Mathematics, Smarandache Ceil Function.

Cf. A000188 (inner square root), A053150 (inner 3rd root), A019555 (outer 3rd root), A053164 (inner 4th root), A053166 (outer 4th root), A015052 (outer 5th root), A015053 (outer 6th root).

Cf. also A007913, A007947, A008833, A015049, A034444, A053143, A102631, A346602, A005117, A133466, A195085.

a019554 n = product $ zipWith (^)
            (a027748_row n) (map ((`div` 2) . (+ 1)) $ a124010_row n)
-- Reinhard Zumkeller, Apr 13 2013
(Python 3.8+)
from math import prod
from sympy import factorint
def A019554(n): return n//prod(p**(q//2) for p, q in factorint(n).items()) # Chai Wah Wu, Aug 18 2021

with(numtheory):A019554 := proc(n) local i: RETURN(op(mul(i,i=map(x->x[1]^ceil(x[2]/2),ifactors(n)[2])))); end;

Flatten[Table[Select[Range[n],Divisible[#^2,n]&,1],{n,100}]] (* Harvey P. Dale, Oct 17 2011 *)
f[p_, e_] := p^Ceiling[e/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)

a(n)=n/core(n,1)[2] \\ Charles R Greathouse IV, Feb 24 2011

Replace any square factors in n by their square roots.
Multiplicative with a(p^e) = p^ceiling(e/2).
Dirichlet series:
   Sum_{n>=1} a(n)/n^s = zeta(2*s-1)*zeta(s-1)/zeta(2*s-2), (Re(s) > 2);
   Sum_{n>=1} (1/a(n))/n^s = zeta(2*s+1)*zeta(s+1)/zeta(2*s+2), (Re(s) > 0).
a(n) = n/A000188(n).
a(n) = denominator of n/n^(3/2). - Arkadiusz Wesolowski, Dec 04 2011
a(n) = Product_{k=1..A001221(n)} A027748(n,k)^ceiling(A124010(n,k)/2). - Reinhard Zumkeller, Apr 13 2013
Sum_{k=1..n} a(k) ~ 3*zeta(3)*n^2 / Pi^2. - Vaclav Kotesovec, Sep 18 2020
Sum_{k=1..n} 1/a(k) ~ 3*log(n)^2/(2*Pi^2) + (9*gamma/Pi^2 - 36*zeta'(2)/Pi^4)*log(n) + 6*gamma^2/Pi^2 - 108*gamma*zeta'(2)/Pi^4 + 432*zeta'(2)^2/Pi^6 - 36*zeta''(2)/Pi^4 - 15*sg1/Pi^2, where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Jul 27 2021
a(n) = sqrt(n*A007913(n)). - Jianing Song, May 08 2022
a(n) = sqrt(A053143(n)). - Amiram Eldar, Sep 02 2023
From Mia Boudreau, Jul 17 2025: (Start)
a(n^2) = n.
a(A005117(n)) = A005117(n).
a(A133466(n)) = A133466(n)/2.
a(A195085(n)) = A195085(n)/3. (End)

cp's OEIS Frontend

A051377 a(1)=1; for n > 1, a(n) = sum of exponential divisors (or e-divisors) of n.

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A035316 Sum of the square divisors of n.

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A008475 If n = Product (p_j^k_j) then a(n) = Sum (p_j^k_j) (a(1) = 0 by convention).

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A087207 A binary representation of the primes that divide a number, shown in decimal.

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A206369 a(p^k) = p^k - p^(k-1) + p^(k-2) - ... +- 1, and then extend by multiplicativity.

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