A007913 -id:A007913 - cp's OEIS frontend

A284003 a(n) = A007913(A283477(n)) = A019565(A006068(n)).

1, 2, 6, 3, 30, 15, 5, 10, 210, 105, 35, 70, 7, 14, 42, 21, 2310, 1155, 385, 770, 77, 154, 462, 231, 11, 22, 66, 33, 330, 165, 55, 110, 30030, 15015, 5005, 10010, 1001, 2002, 6006, 3003, 143, 286, 858, 429, 4290, 2145, 715, 1430, 13, 26, 78, 39, 390, 195, 65, 130, 2730, 1365, 455, 910, 91, 182, 546, 273, 510510, 255255, 85085, 170170, 17017

Offset: 0

Antti Karttunen, Mar 18 2017

A squarefree analog of A302783. Each term is either a divisor or a multiple of the next one. In contrast to A302033 at each step the previous term can be multiplied (or divided), not just by a single prime, but possibly by a product of several distinct ones, A019565(A000975(k)). E.g., a(3) = 3, a(4) = 2*5*a(3) = 30. - Antti Karttunen, Apr 17 2018

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

A permutation of A005117.
Cf. A000975, A001222, A006068, A007913, A019565, A046523, A048675, A064707, A209281, A283477, A284004.
Cf. also A277811, A283475, A302033, A302783.

Table[Apply[Times, FactorInteger[#] /. {p_, e_} /; e > 0 :> Times @@ (p^Mod[e, 2])] &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e == 1 :> {Times @@ Prime@ Range@ PrimePi@ p, e}] &[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2]]], {n, 0, 52}] (* Michael De Vlieger, Mar 18 2017 *)

A007913(n) = core(n);
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From Charles R Greathouse IV, Jun 28 2015
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A283477(n) = A108951(A019565(n));
A284003(n) = A007913(A283477(n));

A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ From A006068
A284003(n) = A019565(A006068(n)); \\ (and use A019565 from above) - Antti Karttunen, Apr 16 2018

(define (A284003 n) (A007913 (A283477 n)))

a(n) = A007913(A283477(n)).
Other identities. For all n >= 0:
A048675(a(n)) = A006068(n).
A046523(a(n)) = A284004(n).
It seems that A001222(a(n)) = A209281(n).
a(n) = A019565(A006068(n)) = A302033(A064707(n)). - Antti Karttunen, Apr 16 2018

Name amended with a second formula by Antti Karttunen, Apr 16 2018

A326126 Sum of all other divisors of n except the squarefree part of n: a(n) = sigma(n) - A007913(n).

Original entry on oeis.org

0, 1, 1, 6, 1, 6, 1, 13, 12, 8, 1, 25, 1, 10, 9, 30, 1, 37, 1, 37, 11, 14, 1, 54, 30, 16, 37, 49, 1, 42, 1, 61, 15, 20, 13, 90, 1, 22, 17, 80, 1, 54, 1, 73, 73, 26, 1, 121, 56, 91, 21, 85, 1, 114, 17, 106, 23, 32, 1, 153, 1, 34, 97, 126, 19, 78, 1, 109, 27, 74, 1, 193, 1, 40, 121, 121, 19, 90, 1, 181, 120, 44, 1, 203, 23, 46, 33, 158, 1, 224, 21

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537.
Index entries for sequences related to sigma(n).

Cf. A000203, A007913, A326127, A326128, A326129.

Cf. also A326053, A326061, A326142.

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := p^Mod[e, 2]; a[n_] := Module[{f = FactorInteger[n]}, Times @@ f1 @@@ f - Times @@ f2 @@@ f]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Mar 21 2024 *)

PARI
```
A326126(n) = (sigma(n)-core(n));
```

a(n) = A000203(n) - A007913(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/20 = 0.4934802... . - Amiram Eldar, Mar 21 2024

A326128 a(n) = n - A007913(n), where A007913 gives the squarefree part of n.

Original entry on oeis.org

0, 0, 0, 3, 0, 0, 0, 6, 8, 0, 0, 9, 0, 0, 0, 15, 0, 16, 0, 15, 0, 0, 0, 18, 24, 0, 24, 21, 0, 0, 0, 30, 0, 0, 0, 35, 0, 0, 0, 30, 0, 0, 0, 33, 40, 0, 0, 45, 48, 48, 0, 39, 0, 48, 0, 42, 0, 0, 0, 45, 0, 0, 56, 63, 0, 0, 0, 51, 0, 0, 0, 70, 0, 0, 72, 57, 0, 0, 0, 75, 80, 0, 0, 63, 0, 0, 0, 66, 0, 80, 0, 69, 0, 0, 0, 90, 0, 96, 88, 99, 0, 0, 0, 78, 0

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537.

Cf. A007913, A033879, A326126, A326127, A326129, A336642.

Cf. also A066503.

f[p_, e_] := p^Mod[e, 2]; a[n_] := n - Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Mar 21 2024 *)

PARI
```
A326128(n) = (n-core(n));
```

a(n) = n - A007913(n).
a(n) = A326127(n) + A033879(n).
a(n) >= A066503(n).
a(n) = A007913(n) * A336642(n). - Antti Karttunen, Jul 28 2020
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1/2 - Pi^2/30 = 0.171013... . - Amiram Eldar, Mar 21 2024

A056191 Characteristic cube divisor of n: cube of g = gcd(K,F), where K is the largest square root divisor of n (A000188) and F = n/(KK) = A007913(n) is its squarefree part; g^2 divides K^2 = A008833(n) = g^2L^2 and g divides F = gf.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 27, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Labos Elemer, Aug 02 2000

This is not the largest cube which divides n. It is canonical, since the decomposition n = KKgggf is unique (factors are defined above and dependent on n).

			If n=24, largest square divisor is 4, squarefree part is 6, g=2, a(24)=8; n=81, largest square divisor is 81, both F and g is 1, a(81)=1.

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A055229, A000188, A008833, A007913, A055231, A056192.

a[n_]:=With[{sf=Times@@Power@@@({#[[1]], Mod[#[[2]], 2]}&/@FactorInteger[n])}, GCD[sf, n/sf]]; Table[a[n]^3, {n, 1, 100}] (* Vincenzo Librandi, Oct 08 2017 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1 || !(f[i,2]%2), 1,  f[i,1]^3));} \\ Amiram Eldar, Sep 05 2023

a(n) = A055229(n)^3 = g^3 = ggg; n = (KK)*(ggg)*f = K^2*g^3*f = KK*a(n)^3*f.

Multiplicative with a(p^e)=1 for even e, a(p)=1, a(p^e)=p^3 for odd e > 1. - Vladeta Jovovic, May 01 2002

A069891 a(n) = Sum_{k=1..n} A007913(k), the squarefree part of k.

Original entry on oeis.org

0, 1, 3, 6, 7, 12, 18, 25, 27, 28, 38, 49, 52, 65, 79, 94, 95, 112, 114, 133, 138, 159, 181, 204, 210, 211, 237, 240, 247, 276, 306, 337, 339, 372, 406, 441, 442, 479, 517, 556, 566, 607, 649, 692, 703, 708, 754, 801, 804, 805, 807, 858, 871, 924, 930, 985, 999

Offset: 0

Dean Hickerson, Apr 09 2002

nonn

Sum_{k=1..n, k squarefree} (1/k) = Sum{k=1..n} (mu(k)^2/k) = (1/zeta(2))*(log(n) + gamma - 2*zeta'(2)/zeta(2)) + O(1/sqrt(n)). (Suryanarayana)

D. Suryanarayana, Asymptotic formula for sum_{n <= x} mu^{2}(n)/n, Indian J. Math. 9 (1967/1968) 543-545.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Rafael Jakimczuk, Two Topics in Number Theory: Sum of Divisors of the Primorial and Sum of Squarefree Parts, International Mathematical Forum, Vol. 12, 2017, no. 7, pp. 331-338.

Cf. A007913, A046970, A069087.

[0] cat [&+[Squarefree(k):k in [1..n]]:n in [1..60]]; // Marius A. Burtea, Dec 19 2019

a[n_] := Sum[If[d == 1, 1, Times@@(1-#1[[1]]^2&) /@ FactorInteger[d]] * Binomial[Floor[n/d^2]+1, 2], {d, 1, Floor[Sqrt[n]]}]; Array[a, 100, 0] (* corrected by Amiram Eldar, Apr 02 2020 *)

a(n) = sum(k=1, n, core(k)); \\ Michel Marcus, Dec 19 2019

a(n) = Sum_{d=1..floor(sqrt(n))} f(d)*binomial(floor(n/d^2)+1, 2) where f(d)=A046970(d) is the product of 1-p^2 over all prime divisors p of d.

a(n) is asymptotic to r*n^2, where r = Pi^2/30 = 0.3289868...

A285102 a(n) = A007913(A285101(n)).

Original entry on oeis.org

2, 6, 210, 72930, 620310, 278995269860970, 12849025509071310, 492608110538467706074890, 1342951001046021018427857601026746070, 37793589449865555275592120894959094883390892772270, 728982633030274864467458719371654181886452163442582606072870, 28339554655955912942523491885490197708224606885407444005070

Offset: 0

Antti Karttunen, Apr 15 2017

nonn

Cf. A003961, A007913, A048675, A068052, A242378, A285101, A285103.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A242378(k,n) = { while(k>0,n = A003961(n); k = k-1); n; };
A285102(n) = { if(0==n,2,lcm(A285102(n-1),A242378(n,A285102(n-1)))/gcd(A285102(n-1),A242378(n,A285102(n-1)))); };

# uses [A003961, A242378]
from sympy import factorint, prime, primepi
from sympy.ntheory.factor_ import core
from operator import mul
def a003961(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**f[i] for i in f])
def a242378(k, n):
    while k>0:
        n=a003961(n)
        k-=1
    return n
l=[2]
for n in range(1, 12):
    x=l[n - 1]
    l.append(x*a242378(n, x))
print([core(j) for j in l]) # Indranil Ghosh, Jun 27 2017

(definec (A285102 n) (if (zero? n) 2 (/ (lcm (A285102 (- n 1)) (A242378bi n (A285102 (- n 1)))) (gcd (A285102 (- n 1)) (A242378bi n (A285102 (- n 1)))))))

a(0) = 2, for n > 0, a(n) = lcm(a(n-1),A242378(n,a(n-1))) / gcd(a(n-1),A242378(n,a(n-1))).
a(n) = A007913(A285101(n)).
Other identities. For all n >= 0:
A001221(a(n)) = A001222(a(n)) = A285103(n).
A048675(a(n)) = A068052(n).

A322589 Lexicographically earliest such sequence a that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = 0 for odd primes, and f(n) = A007913(n) for any other number.

Original entry on oeis.org

1, 2, 3, 1, 3, 4, 3, 2, 1, 5, 3, 6, 3, 7, 8, 1, 3, 2, 3, 9, 10, 11, 3, 4, 1, 12, 6, 13, 3, 14, 3, 2, 15, 16, 17, 1, 3, 18, 19, 5, 3, 20, 3, 21, 9, 22, 3, 6, 1, 2, 23, 24, 3, 4, 25, 7, 26, 27, 3, 8, 3, 28, 13, 1, 29, 30, 3, 31, 32, 33, 3, 2, 3, 34, 6, 35, 36, 37, 3, 9, 1, 38, 3, 10, 39, 40, 41, 11, 3, 5, 42, 43, 44, 45, 46, 4, 3, 2, 21, 1, 3, 47, 3, 12, 48

Offset: 1

Antti Karttunen, Dec 18 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A007913, A322587, A322588, A322591, A322592.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
Aux322589(n) = if((n>2)&&isprime(n),0,core(n));
v322589 = rgs_transform(vector(up_to, n, Aux322589(n)));
A322589(n) = v322589[n];

A349374 Dirichlet convolution of Kimberling's paraphrases (A003602) with squarefree part of n (A007913).

Original entry on oeis.org

1, 3, 5, 4, 8, 15, 11, 6, 12, 24, 17, 20, 20, 33, 42, 7, 26, 36, 29, 32, 58, 51, 35, 30, 29, 60, 34, 44, 44, 126, 47, 9, 90, 78, 94, 48, 56, 87, 106, 48, 62, 174, 65, 68, 110, 105, 71, 35, 54, 87, 138, 80, 80, 102, 146, 66, 154, 132, 89, 168, 92, 141, 153, 10, 172, 270, 101, 104, 186, 282, 107, 72, 110, 168, 167, 116

Offset: 1

Antti Karttunen, Nov 15 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A003602, A007913.

Cf. A347954, A347955, A347956, A349136, A349370, A349371, A349372, A349374, A349375, A349390, A349431, A349444, A349447 for Dirichlet convolutions of other sequences with A003602.

f[p_, e_] := p^Mod[e, 2]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, k[#] * s[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

A003602(n) = (1+(n>>valuation(n,2)))/2;
A349374(n) = sumdiv(n,d,A003602(n/d)*core(d));

a(n) = Sum_{d|n} A003602(n/d) * A007913(d).

A293218 a(n) = A007913(A292270(n)).

Original entry on oeis.org

1, 1, 1, 1, 13, 1, 1, 1, 38, 1, 3, 26, 31, 1, 1, 1, 103, 73, 1, 42, 14, 7, 91, 3, 58, 14, 1, 170, 303, 1, 1, 1, 66, 1, 385, 91, 93, 301, 65, 563, 1093, 1, 11, 355, 38, 118, 83, 1, 1254, 763, 1, 1043, 39, 1, 249, 141, 238, 19, 71, 43, 133, 11, 781, 1, 649, 1, 554, 1081, 614, 1, 1633, 5, 317, 1398, 1, 269, 626, 10, 527, 1285, 1191, 1

Offset: 0

Antti Karttunen, Oct 06 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A002326, A007913, A292270, A292947, A293219.

Cf. A292938 (gives the positions of ones).

A000265(n) = (n >> valuation(n, 2));
A006519(n) = 2^valuation(n, 2);
A292270(n) = { my(x = n+n+1, z = ((1+x)/A006519(1+x)), m = A000265(1+x)); while(m!=1, z += ((x+m)/A006519(x+m)); m = A000265(x+m)); z; };
A293218(n) = core(A292270(n));

(define (A293218 n) (A007913 (A292270 n)))

a(n) = A007913(A292270(n)).

A248470 Put a [+] b = A(A(a) + A(b)), where A = A007913; a(n) is the [+]-sum of binomial(n,i), i=0,...,n.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 3, 2, 1, 1, 6, 38, 167, 2095, 1, 2030, 3, 15, 21, 138, 263, 2, 57, 1266, 3470, 7, 145742, 10, 4682335, 110, 38, 618, 366, 83, 3343, 3279, 206555, 215547, 489378, 52010, 21, 5127, 11, 54663, 6203, 5041187, 194, 63038411, 407039, 7602, 2, 2474

Offset: 0

Vladimir Shevelev, Oct 27 2014

nonn

By definition, all terms are squarefree (A005117).

			For n=4, we have binomials: 1,4,6,4,1.
To obtain a(4), we form the sums 1[+]4 = 1[+]1 = 2; 2[+]6 = 2; 2[+]4 = 2[+]1 = 3; 3[+]1=1. So a(4)=1.

Cf. A005117, A007318, A007913.

a7913[n_]:=a7913[n]=Times@@(#[[1]]^Mod[#[[2]],2])&[Transpose[FactorInteger[n]]];
ab[x_,y_]:=ab[x,y]=a7913[a7913[x]+a7913[y]];
Table[Fold[ab,First[#],Rest[#]]&[Binomial[n,#]&[Range[0,n]]],{n,0,50}] (* Peter J. C. Moses, Oct 27 2014 *)

a(n) = {s = 0; for (i=0, n, s = core(core(binomial(n, i)) + core(s))); s;} \\ Michel Marcus, Nov 14 2014

More terms from Peter J. C. Moses, Oct 27 2014

cp's OEIS Frontend

A284003 a(n) = A007913(A283477(n)) = A019565(A006068(n)).

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A326126 Sum of all other divisors of n except the squarefree part of n: a(n) = sigma(n) - A007913(n).

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A326128 a(n) = n - A007913(n), where A007913 gives the squarefree part of n.

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A056191 Characteristic cube divisor of n: cube of g = gcd(K,F), where K is the largest square root divisor of n (A000188) and F = n/(K*K) = A007913(n) is its squarefree part; g^2 divides K^2 = A008833(n) = g^2*L^2 and g divides F = gf.

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A069891 a(n) = Sum_{k=1..n} A007913(k), the squarefree part of k.

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A285102 a(n) = A007913(A285101(n)).

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A322589 Lexicographically earliest such sequence a that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = 0 for odd primes, and f(n) = A007913(n) for any other number.

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A349374 Dirichlet convolution of Kimberling's paraphrases (A003602) with squarefree part of n (A007913).

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A056191 Characteristic cube divisor of n: cube of g = gcd(K,F), where K is the largest square root divisor of n (A000188) and F = n/(KK) = A007913(n) is its squarefree part; g^2 divides K^2 = A008833(n) = g^2L^2 and g divides F = gf.