A325385 -id:A325385 - cp's OEIS frontend

A326130 a(n) = gcd(A000120(n), A294898(n)) = gcd(A000120(n), sigma(n)-A005187(n)).

1, 1, 2, 1, 2, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 1, 2, 1, 3, 2, 1, 1, 2, 2, 1, 1, 2, 3, 4, 4, 5, 1, 2, 2, 1, 1, 3, 1, 2, 2, 1, 3, 2, 1, 4, 4, 1, 2, 1, 1, 2, 3, 4, 4, 1, 1, 2, 2, 1, 4, 5, 1, 2, 1, 2, 2, 3, 2, 3, 1, 2, 1, 3, 1, 2, 3, 2, 4, 1, 2, 1, 1, 2, 1, 2, 4, 1, 1, 4, 2, 5, 4, 1, 1, 2, 2, 3, 1, 2, 1, 4, 4, 1, 1, 2

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

Cf. A000120, A000203, A005187, A033879, A326131.

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

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n)-sigma(n));
A326130(n) = gcd(hammingweight(n), A294898(n));

A326130(n) = gcd(hammingweight(n), sigma(n)-A005187(n));

a(n) = gcd(A000120(n), A294898(n)) = gcd(A000120(n), A000203(n)-A005187(n)).

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

Original entry on oeis.org

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));

A126795 a(n) = gcd(n, Product_{p|n} (p+1)), where the product is over the distinct primes p that divide n.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 12, 1, 2, 3, 1, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 1, 4, 1, 6, 1, 1, 3, 2, 1, 12, 1, 2, 1, 2, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 2, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 12, 1, 2, 3, 4, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 3, 4, 1, 18, 7, 4, 1, 2, 5, 12, 1, 2, 3, 2, 1

Offset: 1

Leroy Quet, Mar 14 2007

nonn

First occurrence of k: 1, 10, 15, 28, 95, 6, 91, 56, 153, 190, 473, 12, 1339, 182, 285, 496, 1139, 90, 703, 380, ..., . - Robert G. Wilson v

			The distinct primes that divide 28 are 2 and 7. So a(28) = GCD(28, (2+1)(7+1)) = GCD(28, 24) = 4.

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

Cf. A048250, A325313, A325385.

with(numtheory): a:=proc(n) local fs: fs:=factorset(n): gcd(n,product(1+fs[i],i=1..nops(fs))) end: seq(a(n),n=1..120); # Emeric Deutsch, Mar 27 2007

f[n_] := GCD[n, Times @@ (First /@ FactorInteger[n] + 1)]; Array[f, 101] (* Robert G. Wilson v *)

A126795(n) = gcd(n,factorback(apply(p -> p+1,factor(n)[,1]))); \\ Antti Karttunen, Sep 10 2018

a(n) = gcd(n, A048250(n)).

a(n) = gcd(n, A325313(n)) = gcd(n, A048250(n)-n). - Antti Karttunen, Apr 24 2019

More terms from Emeric Deutsch, Mar 27 2007

A325636 a(n) = gcd(2n, sigma(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 12, 2, 1, 1, 2, 2, 4, 2, 4, 6, 1, 2, 3, 2, 2, 2, 4, 2, 12, 1, 2, 2, 56, 2, 12, 2, 1, 6, 2, 2, 1, 2, 4, 2, 10, 2, 12, 2, 4, 6, 4, 2, 4, 1, 1, 6, 2, 2, 12, 2, 8, 2, 2, 2, 24, 2, 4, 2, 1, 2, 12, 2, 2, 6, 4, 2, 3, 2, 2, 2, 4, 2, 12, 2, 2, 1, 2, 2, 56, 2, 4, 6, 4, 2, 18, 14, 8, 2, 4, 10, 12, 2, 1, 6, 1, 2, 12, 2, 2, 6

Offset: 1

Antti Karttunen, May 21 2019

nonn

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

Cf. A000203, A009194, A033879, A325385, A325635, A325637 (n such that a(n) = 2n).

PARI
```
A325636(n) = gcd(n+n,sigma(n));
```

a(n) = gcd(2n, A000203(n)).

a(n) = gcd(2n, A033879(n)) = gcd(sigma(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

A326130 a(n) = gcd(A000120(n), A294898(n)) = gcd(A000120(n), sigma(n)-A005187(n)).

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

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A126795 a(n) = gcd(n, Product_{p|n} (p+1)), where the product is over the distinct primes p that divide n.

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A325636 a(n) = gcd(2n, sigma(n)).

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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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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.

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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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