A326060 -id:A326060 - cp's OEIS frontend

A325975 a(n) = gcd(A325977(n), A325978(n)).

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

Offset: 1

Antti Karttunen, Jun 02 2019

nonn

See comments in A325979 and A325981.

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

Cf. A034460, A048107, A325313, A325314, A325814, A325973, A325974, A325977, A325978, A325979, A325981.

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

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325313(n) = (A048250(n) - n);
A325314(n) = (n - A162296(n));
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
A325977(n) = ((A034460(n)+A325313(n))/2);
A325978(n) = ((A325314(n)+A325814(n))/2);
A325975(n) = gcd(A325977(n), A325978(n));
\\ Or alternatively, as:
A325975(n) = (1/2)*gcd((A034460(n)+A325313(n)),(A325814(n)+A325314(n)));

a(n) = gcd(A325977(n), A325978(n)).

a(n) = (1/2)*gcd(A034460(n)+A325313(n), A325814(n)+A325314(n)).

A326057 a(n) = gcd(A003961(n)-2n, A003961(n)-sigma(n)), where A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 1, 1, 1, 43, 1, 3, 5, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 3, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 19, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 3, 3, 5, 7, 1, 1, 3, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 3, 1, 1

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

Terms a(n) larger than 1 and equal to A252748(n) occur at n = 6, 28, 69, 91, 496, ..., see A326134. See also A349753.

Records 1, 3, 43, 45, 2005, 79243, ... occur at n = 1, 6, 28, 360, 496, 8128, ...

Cf. A000203, A000360, A003961, A033879, A033880, A252748, A286385, A326134.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326060, A326062, A349753.

Array[GCD[#3 - #1, #3 - #2] & @@ {2 #, DivisorSigma[1, #], Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1]} &, 78] (* Michael De Vlieger, Feb 22 2021 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A252748(n) = (A003961(n) - (2*n));
A286385(n) = (A003961(n) - sigma(n));
A326057(n) = gcd(A252748(n), A286385(n));

a(n) = gcd(A252748(n), A286385(n)) = gcd(A003961(n) - 2n, A003961(n) - A000203(n)).

a(n) = gcd(A252748(n), A033879(n)) = gcd(A286385(n), A033879(n)). [Also A033880 can be used] - Antti Karttunen, May 06 2024

A326046 a(n) = gcd(n-A326039(n), A326040(n)-n).

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 3, 1, 1, 1, 1, 4, 12, 2, 1, 1, 8, 1, 3, 1, 5, 2, 1, 4, 1, 5, 1, 24, 28, 6, 15, 1, 1, 1, 1, 1, 36, 2, 1, 1, 40, 2, 3, 4, 4, 10, 1, 4, 1, 7, 15, 3, 4, 2, 19, 4, 1, 1, 1, 8, 60, 2, 1, 1, 1, 6, 3, 1, 1, 2, 35, 1, 72, 1, 1, 12, 1, 2, 3, 1, 1, 1, 1, 4, 1, 2, 1, 4, 8, 27, 5, 8, 29, 2, 7, 60, 48, 1, 1, 1, 100, 6, 3, 1, 1

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

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

Cf. A000203, A008833, A326039, A326040, A326044, A326045.

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

A008833(n) = (n/core(n));
A326039(n) = A008833(sigma(n));
A326040(n) = (sigma(n)-A008833(sigma(n)));
A326046(n) = gcd(n-A326039(n), A326040(n)-n);

a(n) = gcd(A326044(n), A326045(n)) = gcd(n-A326039(n), A326040(n)-n).

A326062 a(1) = gcd((sigma(n)-A032742(n))-n, n-A032742(n)), where A032742 gives the largest proper divisor of n.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 6, 1, 1, 1, 10, 2, 12, 1, 2, 1, 16, 3, 18, 2, 2, 1, 22, 12, 1, 1, 2, 14, 28, 3, 30, 1, 2, 1, 2, 1, 36, 1, 2, 10, 40, 3, 42, 2, 6, 1, 46, 4, 1, 1, 2, 2, 52, 3, 2, 4, 2, 1, 58, 6, 60, 1, 2, 1, 2, 3, 66, 2, 2, 1, 70, 3, 72, 1, 2, 2, 2, 3, 78, 2, 1, 1, 82, 14, 2, 1, 2, 4, 88, 9, 2, 2, 2, 1, 2, 12, 96, 1, 6, 1, 100, 3, 102, 2, 2

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

See comments in A326063 and A326064.

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

Cf. A000203, A032742, A060681, A318505, A326061, A326063, A326064.

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

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A326062(n) = gcd((sigma(n)-A032742(n))-n, n-A032742(n));

a(1) = 1; for n > 1, a(n) = gcd(A060681(n), A318505(n)).

a(n) = gcd((A000203(n)-A032742(n))-n, n-A032742(n)).

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.

Original entry on oeis.org

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)).

A326129 a(n) = gcd(A326127(n), A326128(n)).

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 6, 1, 1, 2, 10, 1, 12, 4, 6, 1, 16, 1, 18, 1, 10, 8, 22, 6, 1, 10, 2, 21, 28, 12, 30, 1, 18, 14, 22, 1, 36, 16, 22, 10, 40, 12, 42, 1, 4, 20, 46, 1, 1, 1, 30, 3, 52, 12, 38, 2, 34, 26, 58, 3, 60, 28, 2, 1, 46, 12, 66, 1, 42, 4, 70, 1, 72, 34, 2, 3, 58, 12, 78, 1, 1, 38, 82, 7, 62, 40, 54, 2, 88, 2, 70, 1, 58, 44, 70, 30

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

Question: Are there any other numbers than those in A000396 that satisfy a(k) = A326128(k)?

See also comments in A336641, where all such k should reside. - Antti Karttunen, Jul 29 2020

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

Cf. A000396, A007913, A326126, A326127, A326128.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326130, A326140, A326144, A336641, A336645.

A326127(n) = (sigma(n)-core(n)-n);
A326128(n) = (n-core(n));
A326129(n) = gcd(A326127(n), A326128(n));

a(n) = n - A336645(n). - Antti Karttunen, Jul 29 2020

A326144 a(n) = gcd(A066503(n), A326143(n)) = gcd(n - A007947(n), sigma(n) - A007947(n) - n).

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 6, 1, 1, 2, 10, 2, 12, 4, 6, 1, 16, 3, 18, 2, 10, 8, 22, 6, 1, 10, 2, 14, 28, 12, 30, 1, 18, 14, 22, 1, 36, 16, 22, 10, 40, 12, 42, 2, 6, 20, 46, 14, 1, 1, 30, 2, 52, 12, 38, 2, 34, 26, 58, 6, 60, 28, 2, 1, 46, 12, 66, 2, 42, 4, 70, 3, 72, 34, 2, 2, 58, 12, 78, 2, 1, 38, 82, 14, 62, 40, 54, 2, 88, 6, 70, 2

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

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

Cf. A000203, A007947, A066503, A326142, A326143, A326145.

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

rad[n_] := Times @@ (First /@ FactorInteger@n);
a[n_] := GCD[n - rad[n], DivisorSigma[1, n] - rad[n] - n];
Array[a, 100] (* Jean-François Alcover, Dec 01 2021 *)

A007947(n) = factorback(factorint(n)[, 1]);
A066503(n) = (n - A007947(n));
A326143(n) = (sigma(n)-A007947(n)-n);
A326144(n) = gcd(A066503(n), A326143(n));

a(n) = gcd(A066503(n), A326143(n)) = gcd(n-A007947(n), A000203(n)-A007947(n)-n).

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

Original entry on oeis.org

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));

A326058 a(n) = n - {the sum of square divisors of n}.

Original entry on oeis.org

0, 1, 2, -1, 4, 5, 6, 3, -1, 9, 10, 7, 12, 13, 14, -5, 16, 8, 18, 15, 20, 21, 22, 19, -1, 25, 17, 23, 28, 29, 30, 11, 32, 33, 34, -14, 36, 37, 38, 35, 40, 41, 42, 39, 35, 45, 46, 27, -1, 24, 50, 47, 52, 44, 54, 51, 56, 57, 58, 55, 60, 61, 53, -21, 64, 65, 66, 63, 68, 69, 70, 22, 72, 73, 49, 71, 76, 77, 78, 59, -10, 81, 82, 79

Offset: 1

Antti Karttunen, Jun 06 2019

sign

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

Cf. A033879, A035316, A326059, A326060.

A035316(n) = sumdiv(n,d,issquare(d)*d);
A326058(n) = (n-A035316(n));

a(n) = n - A035316(n).

a(n) = A033879(n) + A326059(n).

cp's OEIS Frontend

A325975 a(n) = gcd(A325977(n), A325978(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A326057 a(n) = gcd(A003961(n)-2n, A003961(n)-sigma(n)), where A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A326046 a(n) = gcd(n-A326039(n), A326040(n)-n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A326062 a(1) = gcd((sigma(n)-A032742(n))-n, n-A032742(n)), where A032742 gives the largest proper divisor of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A326129 a(n) = gcd(A326127(n), A326128(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A326144 a(n) = gcd(A066503(n), A326143(n)) = gcd(n - A007947(n), sigma(n) - A007947(n) - n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI