A008833 -id:A008833 - cp's OEIS frontend

A326056 a(n) = gcd(sigma(n)-A008833(n)-n, n-A008833(n)), where sigma is the sum of divisors of n, and A008833 is the largest square dividing n.

1, 1, 2, 1, 4, 5, 6, 1, 5, 1, 10, 4, 12, 1, 2, 1, 16, 3, 18, 2, 10, 1, 22, 4, 19, 5, 2, 24, 28, 1, 30, 1, 2, 1, 2, 19, 36, 1, 2, 2, 40, 1, 42, 4, 12, 5, 46, 4, 41, 1, 10, 6, 52, 3, 2, 4, 2, 1, 58, 8, 60, 1, 2, 1, 2, 1, 66, 2, 2, 1, 70, 3, 72, 1, 2, 12, 2, 1, 78, 2, 41, 1, 82, 8, 2, 5, 2, 4, 88, 27, 10, 8, 2, 1, 2, 20, 96, 1, 6

Offset: 1

Antti Karttunen, Jun 05 2019

nonn

Composite numbers n such that a(n) = A326055(n) start as: 6, 28, 336, 496, 792, 8128, 31968, 3606912, ...

Nonsquare odd numbers n such that a(n) = abs(A326054(n)) start as: 21, 153, 301, 697, 1333, 1909, 1917, 2041, 3901, 4753, 24601, 24957, 26977, 29161, 29637, 56953, 67077, 96361, ...

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A008833, A326053, A326054, A326055.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326057, A326060, A326062, A326129, A326130, A326140, A326144.

A008833(n) = (n/core(n));
A326053(n) = (sigma(n)-A008833(n));
A326054(n) = (A326053(n)-n);
A326055(n) = (n-A008833(n));
A326056(n) = gcd(A326054(n), A326055(n));

a(n) = gcd(A326054(n), A326055(n)) = gcd((A000203(n)-A008833(n))-n, n-A008833(n)).

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

A326039 Largest square dividing the sum of divisors of n: a(n) = A008833(sigma(n)).

Original entry on oeis.org

1, 1, 4, 1, 1, 4, 4, 1, 1, 9, 4, 4, 1, 4, 4, 1, 9, 1, 4, 1, 16, 36, 4, 4, 1, 1, 4, 4, 1, 36, 16, 9, 16, 9, 16, 1, 1, 4, 4, 9, 1, 16, 4, 4, 1, 36, 16, 4, 1, 1, 36, 49, 9, 4, 36, 4, 16, 9, 4, 4, 1, 16, 4, 1, 4, 144, 4, 9, 16, 144, 36, 1, 1, 1, 4, 4, 16, 4, 16, 1, 121, 9, 4, 16, 36, 4, 4, 36, 9, 9, 16, 4, 64, 144, 4, 36

Offset: 1

Antti Karttunen, Jun 05 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A008833, A326038, A326040.

Module[{sqs=Range[100,1,-1]^2},Table[SelectFirst[sqs,Divisible[ DivisorSigma[ 1,n],#]&],{n,100}]] (* Harvey P. Dale, Jul 29 2019 *)

A008833(n) = (n/core(n));
A326039(n) = A008833(sigma(n));

a(n) = A008833(A000203(n)) = A326038(n)^2.

a(n) = A000203(n) - A326040(n).

A326040 a(n) = sigma(n) - A008833(sigma(n)).

Original entry on oeis.org

0, 2, 0, 6, 5, 8, 4, 14, 12, 9, 8, 24, 13, 20, 20, 30, 9, 38, 16, 41, 16, 0, 20, 56, 30, 41, 36, 52, 29, 36, 16, 54, 32, 45, 32, 90, 37, 56, 52, 81, 41, 80, 40, 80, 77, 36, 32, 120, 56, 92, 36, 49, 45, 116, 36, 116, 64, 81, 56, 164, 61, 80, 100, 126, 80, 0, 64, 117, 80, 0, 36, 194, 73, 113, 120, 136, 80, 164, 64, 185, 0, 117, 80

Offset: 1

Antti Karttunen, Jun 05 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A008833, A326039, A326055.

A008833(n) = (n/core(n));
A326039(n) = A008833(sigma(n));
A326040(n) = (sigma(n)-A008833(sigma(n)));

A326055(n) = (n-A008833(n));
A326040(n) = A326055(sigma(n));

a(n) = A000203(n) - A326039(n) = A000203(n) - A008833(A000203(n)).

a(n) = A326055(A000203(n)).

A326053 Sum of all other divisors of n except the largest square divisor: a(n) = sigma(n) - A008833(n).

Original entry on oeis.org

0, 2, 3, 3, 5, 11, 7, 11, 4, 17, 11, 24, 13, 23, 23, 15, 17, 30, 19, 38, 31, 35, 23, 56, 6, 41, 31, 52, 29, 71, 31, 47, 47, 53, 47, 55, 37, 59, 55, 86, 41, 95, 43, 80, 69, 71, 47, 108, 8, 68, 71, 94, 53, 111, 71, 116, 79, 89, 59, 164, 61, 95, 95, 63, 83, 143, 67, 122, 95, 143, 71, 159, 73, 113, 99, 136, 95, 167, 79, 170, 40, 125, 83

Offset: 1

Antti Karttunen, Jun 05 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A008833, A285309, A326054, A326055.

A008833(n) = (n/core(n));
A326053(n) = (sigma(n)-A008833(n));

a(n) = A000203(n) - A008833(n).
a(n) = n + A326054(n).
a(n) >= A285309(n).

A326044 a(n) = n - {the largest square dividing its sum of divisors}: a(n) = n - A008833(sigma(n)).

Original entry on oeis.org

0, 1, -1, 3, 4, 2, 3, 7, 8, 1, 7, 8, 12, 10, 11, 15, 8, 17, 15, 19, 5, -14, 19, 20, 24, 25, 23, 24, 28, -6, 15, 23, 17, 25, 19, 35, 36, 34, 35, 31, 40, 26, 39, 40, 44, 10, 31, 44, 48, 49, 15, 3, 44, 50, 19, 52, 41, 49, 55, 56, 60, 46, 59, 63, 61, -78, 63, 59, 53, -74, 35, 71, 72, 73, 71, 72, 61, 74, 63, 79, -40

Offset: 1

Antti Karttunen, Jun 05 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A008833, A033879, A326039, A326045, A326046.

A008833(n) = (n/core(n));
A326039(n) = A008833(sigma(n));
A326044(n) = (n-A326039(n));

a(n) = n - A326039(n) = n - A008833(A000203(n)).

a(n) = A033879(n) + A326045(n).

A326045 a(n) is the sum of divisors of n, minus the largest square dividing that sum, minus n: a(n) = sigma(n) - A008833(sigma(n)) - n.

Original entry on oeis.org

-1, 0, -3, 2, 0, 2, -3, 6, 3, -1, -3, 12, 0, 6, 5, 14, -8, 20, -3, 21, -5, -22, -3, 32, 5, 15, 9, 24, 0, 6, -15, 22, -1, 11, -3, 54, 0, 18, 13, 41, 0, 38, -3, 36, 32, -10, -15, 72, 7, 42, -15, -3, -8, 62, -19, 60, 7, 23, -3, 104, 0, 18, 37, 62, 15, -66, -3, 49, 11, -70, -35, 122, 0, 39, 45, 60, 3, 86, -15, 105, -81, 35, -3

Offset: 1

Antti Karttunen, Jun 05 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A008833, A326039, A326040, A326044, A326046.

A008833(n) = (n/core(n));
A326039(n) = A008833(sigma(n));
A326040(n) = (sigma(n)-A008833(sigma(n)));
A326045(n) = (A326040(n)-n);

a(n) = A326040(n) - n = sigma(n) - A008833(sigma(n)) - n.

A336642 One less than the largest square dividing n: a(n) = A008833(n)-1.

Original entry on oeis.org

0, 0, 0, 3, 0, 0, 0, 3, 8, 0, 0, 3, 0, 0, 0, 15, 0, 8, 0, 3, 0, 0, 0, 3, 24, 0, 8, 3, 0, 0, 0, 15, 0, 0, 0, 35, 0, 0, 0, 3, 0, 0, 0, 3, 8, 0, 0, 15, 48, 24, 0, 3, 0, 8, 0, 3, 0, 0, 0, 3, 0, 0, 8, 63, 0, 0, 0, 3, 0, 0, 0, 35, 0, 0, 24, 3, 0, 0, 0, 15, 80, 0, 0, 3, 0, 0, 0, 3, 0, 8, 0, 3, 0, 0, 0, 15, 0, 48, 8, 99, 0, 0, 0, 3, 0

Offset: 1

Antti Karttunen, Jul 28 2020

nonn

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

Cf. A008833.

Cf. also A336551, A336641, A336644.

PARI
```
A336642(n) = ((n/core(n))-1);
```

A283989 Largest square dividing prime factorization representation of the n-th Stern polynomial: a(n) = A008833(A260443(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 9, 1, 1, 1, 9, 25, 9, 1, 225, 1, 1, 1, 9, 25, 225, 49, 2025, 25, 225, 1, 225, 1225, 225, 1, 11025, 1, 1, 1, 9, 25, 11025, 49, 50625, 1225, 275625, 121, 2480625, 30625, 1265625, 49, 2480625, 1225, 11025, 1, 11025, 1225, 275625, 5929, 2480625, 1225, 275625

Offset: 0

Antti Karttunen, Mar 25 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A003961, A008833, A260443, A277330, A283983.

Cf. A023758 (positions of ones).

A003961[p_?PrimeQ] := A003961[p] = Prime[ PrimePi[p] + 1]; A003961[1] = 1; A003961[n_] := A003961[n] = Times @@ ( A003961[First[#]] ^ Last[#] & ) /@ FactorInteger[n]  (* after Jean-François Alcover, Dec 01 2011 *); A260443[n_]:= If[n<2, n + 1, If[EvenQ[n], A003961[A260443[n/2]], A260443[(n - 1)/2] * A260443[(n + 1)/2]]]; A000188[n_]:= Sum[Boole[Mod[i^2, n] == 0], {i, n}]; Table[A000188[A260443[n]]^2, {n, 0, 20}] (* Indranil Ghosh, Mar 28 2017 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus
A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2)))); \\ Cf. Charles R Greathouse IV's code for "ps" in A186891 and A277013.
A008833(n) = n/core(n); \\ This function from Michael B. Porter, Oct 17 2009
A283989(n) = A008833(A260443(n));

(define (A283989 n) (A008833 (A260443 n)))
(define (A283989 n) (/ (A260443 n) (A277330 n)))

a(n) = A008833(A260443(n)).
a(n) = A260443(n) / A277330(n).
a(n) = A283983(n)^2.
a(2n) = A003961(a(n)).

A336641 Numbers k such that A007913(k) divides sigma(k) and A008833(k)-1 either divides A326127(k) (= sigma(k)-core(k)-k), or both are zero.

Original entry on oeis.org

6, 24, 28, 96, 120, 150, 294, 384, 496, 1014, 1536, 3276, 3750, 3780, 6144, 8128, 14406, 20328, 24576, 32760, 93750, 98304, 171366, 306180, 393216, 705894, 1241460, 1572864, 2343750, 6291456, 16380000, 24800580, 25165824, 28960854, 30387840, 33550336, 34588806, 58593750, 100663296, 165143160, 332226048, 402653184

Offset: 1

Antti Karttunen, Jul 28 2020

nonn

Numbers k such that A326128(k) = A326129(k) form a subsequence of this sequence. So far it is not known whether it contains any other terms apart from those of A000396. See comments in A326129.
Sequence is infinite because all numbers of the form 6*4^n (A002023) are present.
Question: Are there any odd terms?

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A000203, A007913, A228058, A326126, A326127, A326128, A326129, A336642.
Cf. A000396, A002023 (subsequences).
Cf. also A336550 for a similar construction.

isA336641(n) = { my(c=core(n), s=sigma(n), u=((n/c)-1)); (!(s%c) && (gcd(u,(s-c-n))==u)); };

cp's OEIS Frontend

A326056 a(n) = gcd(sigma(n)-A008833(n)-n, n-A008833(n)), where sigma is the sum of divisors of n, and A008833 is the largest square dividing 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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A326039 Largest square dividing the sum of divisors of n: a(n) = A008833(sigma(n)).

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A326040 a(n) = sigma(n) - A008833(sigma(n)).

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A326053 Sum of all other divisors of n except the largest square divisor: a(n) = sigma(n) - A008833(n).

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A326044 a(n) = n - {the largest square dividing its sum of divisors}: a(n) = n - A008833(sigma(n)).

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A326045 a(n) is the sum of divisors of n, minus the largest square dividing that sum, minus n: a(n) = sigma(n) - A008833(sigma(n)) - n.

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A336642 One less than the largest square dividing n: a(n) = A008833(n)-1.

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A283989 Largest square dividing prime factorization representation of the n-th Stern polynomial: a(n) = A008833(A260443(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/(KK) = A007913(n) is its squarefree part; g^2 divides K^2 = A008833(n) = g^2L^2 and g divides F = gf.