A326048 -id:A326048 - cp's OEIS frontend

A326140 a(n) = gcd(A318878(n), A318879(n)).

1, 1, 2, 1, 4, 2, 6, 1, 5, 2, 10, 2, 12, 2, 6, 1, 16, 1, 18, 2, 10, 2, 22, 2, 19, 2, 14, 6, 28, 6, 30, 1, 18, 2, 22, 1, 36, 2, 22, 2, 40, 2, 42, 2, 12, 2, 46, 2, 41, 1, 30, 6, 52, 2, 38, 2, 34, 2, 58, 6, 60, 2, 22, 1, 46, 6, 66, 2, 42, 2, 70, 1, 72, 2, 26, 6, 58, 2, 78, 2, 41, 2, 82, 2, 62, 2, 54, 2, 88, 6, 70, 2, 58, 2

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

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

Cf. A033879, A083254, A318878, A318879, A326141.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326129, A326130, A326144.

A326140(n) = { my(t=0, u=0); fordiv(n,d, d -= 2*eulerphi(d); if(d<0, t -= d, u += d)); gcd(t,u); };

A318878(n) = sumdiv(n,d,d=(2*eulerphi(d))-d; (d>0)*d);
A318879(n) = sumdiv(n,d,d=d-(2*eulerphi(d)); (d>0)*d);
A326140(n) = gcd(A318878(n), A318879(n));

A326049 a(n) = n - A050449(n), where A050449 is the sum of divisors of the form 4k+1.

Original entry on oeis.org

0, 1, 2, 3, -1, 5, 6, 7, -1, 4, 10, 11, -1, 13, 9, 15, -1, 8, 18, 14, -1, 21, 22, 23, -6, 12, 17, 27, -1, 24, 30, 31, -1, 16, 29, 26, -1, 37, 25, 34, -1, 20, 42, 43, -15, 45, 46, 47, -1, 19, 33, 38, -1, 44, 49, 55, -1, 28, 58, 54, -1, 61, 32, 63, -19, 32, 66, 50, -1, 64, 70, 62, -1, 36, 44, 75, -1, 64, 78, 74, -10, 40

Offset: 1

Antti Karttunen, Jun 04 2019

sign

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

Cf. A033879, A050449, A326047, A326048, A326050, A326052.

A050449(n) = sumdiv(n, d, d*((d % 4) == 1)); \\ From A050449
A326049(n) = (n-A050449(n));

a(n) = n - A050449(n).

a(n) = A326050(n) + A033879(n).

A326050 a(n) = A082052(n) - n, where A082052 is the sum of divisors of n that are not of the form 4k+1.

Original entry on oeis.org

-1, 0, 0, 2, -5, 5, 0, 6, -6, 2, 0, 15, -13, 9, 3, 14, -17, 11, 0, 16, -11, 13, 0, 35, -25, 2, 3, 27, -29, 36, 0, 30, -19, 2, 7, 45, -37, 21, 3, 44, -41, 32, 0, 39, -27, 25, 0, 75, -42, 12, 3, 32, -53, 56, 11, 63, -35, 2, 0, 102, -61, 33, 10, 62, -65, 44, 0, 40, -43, 68, 0, 113, -73, 2, 18, 63, -59, 76, 0, 100, -51, 2, 0, 118, -85, 45, 3

Offset: 1

Antti Karttunen, Jun 04 2019

sign

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

Cf. A033879, A082052, A326048, A326049.

Table[Total[Select[Divisors[n],Mod[#-1,4]!=0&]]-n,{n,90}] (* Harvey P. Dale, Jul 12 2024 *)

A082052(n) = sumdiv(n, d, if(1!=(d%4), d));
A326050(n) = (A082052(n)-n);

a(n) = A082052(n) - n.

a(n) = A326049(n) - A033879(n).

A326069 a(n) = gcd((sigma(n) - sigma(A032742(n))) - n, n - sigma(A032742(n))), where A032742 gives the largest proper divisor of n.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 1, 5, 2, 10, 4, 12, 2, 3, 1, 16, 1, 18, 2, 1, 2, 22, 4, 19, 2, 14, 4, 28, 6, 30, 1, 3, 2, 1, 1, 36, 2, 1, 2, 40, 2, 42, 4, 3, 2, 46, 4, 41, 1, 3, 2, 52, 2, 1, 8, 1, 2, 58, 12, 60, 2, 1, 1, 1, 6, 66, 2, 3, 2, 70, 1, 72, 2, 2, 4, 1, 2, 78, 2, 41, 2, 82, 4, 1, 2, 3, 4, 88, 6, 7, 4, 1, 2, 5, 4, 96, 1, 3, 1, 100, 6, 102, 2, 3

Offset: 1

Antti Karttunen, Jun 06 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, A032742, A326065, A326066, A326067, A326068.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A326066(n) = (sigma(n) - sigma(A032742(n)));
A326067(n) = (A326066(n) - n);
A326065(n) = sigma(A032742(n));
A326068(n) = (n - A326065(n));
A326069(n) = gcd(A326067(n), A326068(n));

a(n) = gcd(A326067(n), A326068(n)) = gcd(A326066(n) - n, n - A326065(n)).

A326147 a(n) = gcd(n-A020639(n), sigma(n)-A020639(n)-n), where A020639 gives the smallest prime factor of n, and sigma is the sum of divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 6, 1, 1, 2, 10, 2, 12, 4, 6, 1, 16, 1, 18, 2, 2, 4, 22, 2, 1, 2, 2, 26, 28, 4, 30, 1, 6, 2, 2, 1, 36, 4, 2, 2, 40, 4, 42, 2, 6, 4, 46, 2, 1, 1, 6, 2, 52, 4, 2, 2, 2, 2, 58, 2, 60, 4, 2, 1, 2, 4, 66, 2, 6, 4, 70, 1, 72, 2, 2, 2, 2, 4, 78, 26, 1, 2, 82, 2, 2, 4, 6, 2, 88, 2, 14, 2, 2, 4, 10, 2, 96, 1, 6, 1, 100, 4, 102, 2, 6

Offset: 1

Antti Karttunen, Jun 10 2019

nonn

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

Cf. A000203, A020639, A046666, A326146, A326148.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326073, A326129, A326130, A326140, A326144.

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A326147(n) = gcd(n-A020639(n), sigma(n)-A020639(n)-n);

a(n) = gcd(n-A020639(n), A000203(n)-A020639(n)-n).

For n > 1, a(n) = gcd(A046666(n), A326146(n)).

A326073 a(n) = gcd(1+n-A020639(n), 1+sigma(n)-A020639(n)-n), where A020639 gives the smallest prime factor of n (and 1 for 1), and sigma is the sum of divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 27, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 7, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 7, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Jun 10 2019

nonn

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

Cf. A000203, A020639, A326074.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326129, A326130, A326140, A326144, A326147.

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A326073(n) = gcd(1+n-A020639(n), 1+sigma(n)-A020639(n)-n);

a(n) = gcd(1+n-A020639(n), 1+A000203(n)-A020639(n)-n).

cp's OEIS Frontend

A326140 a(n) = gcd(A318878(n), A318879(n)).

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A326049 a(n) = n - A050449(n), where A050449 is the sum of divisors of the form 4k+1.

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PARI

Formula

A326050 a(n) = A082052(n) - n, where A082052 is the sum of divisors of n that are not of the form 4k+1.

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Mathematica

PARI

Formula

A326069 a(n) = gcd((sigma(n) - sigma(A032742(n))) - n, n - sigma(A032742(n))), where A032742 gives the largest proper divisor of n.

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PARI

Formula

A326147 a(n) = gcd(n-A020639(n), sigma(n)-A020639(n)-n), where A020639 gives the smallest prime factor of n, and sigma is the sum of divisors of n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A326073 a(n) = gcd(1+n-A020639(n), 1+sigma(n)-A020639(n)-n), where A020639 gives the smallest prime factor of n (and 1 for 1), and sigma is the sum of divisors of n.

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Author

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Links

Crossrefs

Programs

PARI

Formula