A009194 -id:A009194 - cp's OEIS frontend

A331746 Lexicographically earliest infinite sequence such that a(i) = a(j) => A009194(i) = A009194(j) and A331166(i) = A331166(j) for all i, j.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 18, 21, 24, 25, 22, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 33, 37, 38, 39, 40, 41, 42, 43, 31, 44, 45, 46, 38, 47, 48, 49, 35, 41, 48, 50, 42, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 63, 71, 72, 73, 74, 75, 76, 66, 77, 78, 79, 80, 81, 82, 83, 56

Offset: 1

Antti Karttunen, Feb 04 2020

nonn

Restricted growth sequence transform of the ordered pair [A009194(n), A331166(n)].
For all i, j:
   a(i) = a(j) => A331747(i) = A331747(j).

Cf. A009194, A331166, A331300.

Cf. also A331744, A331747.

\\ Needs also code from A331166.
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; };
A009194(n) = gcd(n, sigma(n));
Aux331746(n) = [A009194(n),A331166(n)];
v331746 = rgs_transform(vector(up_to, n, Aux331746(n)));
A331746(n) = v331746[n];

A331747 Lexicographically earliest infinite sequence such that a(i) = a(j) => A009194(i) = A009194(j) and A278222(i) = A278222(j) for all i, j.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 04 2020

nonn

Restricted growth sequence transform of the ordered pair [A009194(n), A278222(n)].
For all i, j:
   A331746(i) = A331746(j) => a(i) = a(j).

Cf. A009194, A278222.

Cf. also A324400, A286622, A318310, A318311, A324389, A331744, A331745, A331746.

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; };
A009194(n) = gcd(n, sigma(n));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1)));
t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
Aux331747(n) = [A009194(n),A278222(n)];
v331747 = rgs_transform(vector(up_to, n, Aux331747(n)));
A331747(n) = v331747[n];

a(2^n) = 1 for all n >= 0.

A286570 Compound filter (prime signature of n & gcd(n, sigma(n))): a(n) = P(A046523(n), A009194(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 3, 3, 10, 3, 61, 3, 36, 10, 27, 3, 117, 3, 27, 34, 136, 3, 103, 3, 90, 21, 27, 3, 619, 10, 27, 36, 753, 3, 625, 3, 528, 34, 27, 21, 666, 3, 27, 21, 552, 3, 625, 3, 117, 103, 27, 3, 1323, 10, 78, 34, 90, 3, 430, 21, 489, 21, 27, 3, 2545, 3, 27, 78, 2080, 21, 625, 3, 90, 34, 495, 3, 2773, 3, 27, 78, 117, 21, 625, 3, 1224, 136, 27, 3, 3801, 21, 27, 34, 375, 3

Offset: 1

Antti Karttunen, May 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000027, A009194, A046523, A286360, A286571, A286591.

A009194(n) = gcd(n, sigma(n));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A286570(n) = (1/2)*(2 + ((A046523(n)+A009194(n))^2) - A046523(n) - 3*A009194(n));

from sympy import factorint, gcd, divisor_sigma
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a(n): return T(a046523(n), gcd(n, divisor_sigma(n))) # Indranil Ghosh, May 26 2017

(define (A286570 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A009194 n)) 2) (- (A046523 n)) (- (* 3 (A009194 n))) 2)))

a(n) = (1/2)*(2 + ((A046523(n)+A009194(n))^2) - A046523(n) - 3*A009194(n)).

A286591 Compound filter: a(n) = P(A009191(n), A009194(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 3, 1, 1, 1, 23, 1, 10, 6, 5, 1, 42, 1, 5, 4, 1, 1, 34, 1, 5, 1, 5, 1, 179, 1, 5, 1, 408, 1, 23, 1, 3, 4, 5, 1, 45, 1, 5, 1, 144, 1, 23, 1, 12, 13, 5, 1, 12, 1, 3, 4, 5, 1, 23, 1, 113, 1, 5, 1, 265, 1, 5, 6, 1, 1, 23, 1, 5, 4, 5, 1, 103, 1, 5, 6, 12, 1, 23, 1, 65, 1, 5, 1, 753, 1, 5, 4, 63, 1, 259, 22, 12, 1, 5, 11, 265, 1, 3, 13, 1, 1, 23, 1, 44, 4, 5, 1

Offset: 1

Antti Karttunen, May 21 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000027, A009191, A009194, A286594.

A009191(n) = gcd(n, numdiv(n));
A009194(n) = gcd(n, sigma(n));
A286591(n) = (1/2)*(2 + ((A009191(n)+A009194(n))^2) - A009191(n) - 3*A009194(n));

from sympy import divisor_sigma, divisor_count, gcd
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a(n): return T(gcd(n, divisor_count(n)), gcd(n, divisor_sigma(n))) # Indranil Ghosh, May 22 2017

(define (A286591 n) (* (/ 1 2) (+ (expt (+ (A009191 n) (A009194 n)) 2) (- (A009191 n)) (- (* 3 (A009194 n))) 2)))

a(n) = (1/2)*(2 + ((A009191(n)+A009194(n))^2) - A009191(n) - 3*A009194(n)).

A319338 Filter sequence combining the 2-adic valuation of n (A007814) with gcd(n,sigma(n)) (A009194).

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 6, 8, 9, 1, 10, 1, 11, 1, 6, 1, 12, 1, 6, 1, 13, 1, 4, 1, 14, 8, 6, 1, 3, 1, 6, 1, 15, 1, 4, 1, 7, 8, 6, 1, 16, 1, 2, 8, 11, 1, 4, 1, 17, 1, 6, 1, 18, 1, 6, 1, 19, 1, 4, 1, 11, 8, 6, 1, 20, 1, 6, 1, 7, 1, 4, 1, 21, 1, 6, 1, 13, 1, 6, 8, 22, 1, 23, 24, 7, 1, 6, 25, 26, 1, 2, 8, 3, 1, 4, 1, 27, 8

Offset: 1

Antti Karttunen, Sep 24 2018

nonn

Restricted growth sequence transform of ordered pair [A007814(n), A009194(n)].

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

Cf. A007814, A009194, A305801, A319346.

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; };
A007814(n) = valuation(n,2);
A009194(n) = gcd(n, sigma(n));
v319338 = rgs_transform(vector(up_to,n,[A007814(n),A009194(n)]));
A319338(n) = v319338[n];

A325633 a(n) = gcd(A009194(n), A325634(n)) = gcd(A009194(n), A091255(n, sigma(n))).

Original entry on oeis.org

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, 28, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 10, 1, 2, 1, 4, 1, 2, 1, 4, 1, 1, 3, 2, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 1, 1, 6, 1, 2, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 28, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 5, 12, 1, 1, 1, 1, 1, 2, 1, 2, 3

Offset: 1

Antti Karttunen, May 21 2019

nonn

Cf. A000203, A091255, A325634.
Differs from A009194 for the first time at n=42, where a(42) = 2, while A009194(42) = 6.
Differs from A325632 and A325640 for the first time at n=45, where a(45) = 1, while A325632(45) = 5 and A325640(45) = 3.

A009194(n) = gcd(n,sigma(n));
A091255sq(a,b) = fromdigits(Vec(lift(gcd(Pol(binary(a))*Mod(1, 2),Pol(binary(b))*Mod(1, 2)))),2);
A325634(n) = A091255sq(n, sigma(n));
A325633(n) = gcd(A009194(n),A325634(n));

a(n) = gcd(A009194(n), A325634(n)) = gcd(A009194(n), A091255(n, A000203(n))).

A331735 a(n) = A009194(A225546(n)) = gcd(A225546(n), sigma(A225546(n))).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 1, 1, 3, 1, 4, 1, 1, 1, 12, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 4, 1, 3, 1, 12, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 12, 1, 9, 1, 4, 1, 1, 1, 10, 1, 3, 1, 1, 1, 1, 1, 12, 1

Offset: 1

Antti Karttunen, Feb 04 2020

nonn

Cf. A000203, A009194, A225546, A331733, A331734.

Array[GCD[#, DivisorSigma[1, #]] &@ If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &, 105] (* Michael De Vlieger, Feb 12 2020 *)

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A331735(n) = if(issquarefree(n),1,my(f=factor(n),u=#binary(vecmax(f[, 2])),prods=vector(u,x,1),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),prods[i] *= f[k,1])); m<<=1); gcd(prod(i=1,u,prime(i)^A048675(prods[i])), prod(i=1,u,(prime(i)^(1+A048675(prods[i]))-1)/(prime(i)-1))));

a(n) = A009194(A225546(n)) = gcd(A225546(n), A331733(n)).

A325383 Lexicographically earliest sequence such that a(i) = a(j) => A000203(i) = A000203(j) and A009194(i) = A009194(j) for all i, j.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 16, 20, 25, 26, 27, 28, 21, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 32, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 51, 67, 68, 69, 70, 71, 72, 73, 53, 74, 75, 64, 61, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 66

Offset: 1

Antti Karttunen, May 08 2019

nonn

Restricted growth sequence transform of the ordered pair [A000203(n), A009194(n)].

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

Cf. A000203, A009194, A295300, A325381, A325384.

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; };
A009194(n) = gcd(n,sigma(n));
v325383 = rgs_transform(vector(up_to,n,[sigma(n),A009194(n)]));
A325383(n) = v325383[n];

A325384 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A000203(n), A009194(n)] for all other numbers, except f(p) = 0 for odd primes.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 8, 3, 9, 3, 10, 11, 12, 3, 13, 3, 14, 15, 16, 3, 17, 12, 14, 18, 19, 3, 20, 3, 21, 22, 23, 24, 25, 3, 26, 27, 28, 3, 29, 3, 30, 31, 32, 3, 33, 34, 35, 36, 37, 3, 38, 39, 40, 41, 42, 3, 43, 3, 44, 45, 46, 47, 48, 3, 49, 50, 51, 3, 52, 3, 53, 54, 55, 56, 57, 3, 58, 59, 49, 3, 60, 61, 62, 63, 64, 3, 65, 66, 67, 68, 51, 69, 70, 3, 71

Offset: 1

Antti Karttunen, May 08 2019

nonn

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

Cf. A000203, A009194, A300230, A305801, A325382, A325383.

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; };
A009194(n) = gcd(n,sigma(n));
Aux325384(n) = if((n%2)&&isprime(n),0,[sigma(n),A009194(n)]);
v325384 = rgs_transform(vector(up_to,n,Aux325384(n)));
A325384(n) = v325384[n];

A325640 a(n) = A091255(n, A009194(n)) = A091255(n, gcd(n, sigma(n))).

Original entry on oeis.org

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, 28, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 10, 1, 2, 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, 1, 2, 1, 3, 1, 2, 1, 4, 1, 6, 1, 2, 1, 2, 1, 28, 1, 2, 1, 4, 1, 18, 1, 4, 1, 2, 5, 12, 1, 1, 3, 1, 1, 6, 1, 2, 3

Offset: 1

Antti Karttunen, May 21 2019

nonn

Cf. A000203, A007691 (fixed points), A009194, A091255, A325634.

Differs from A325632 and A325633 for the first time at n=45, where a(45) = 3, while A325632(45) = 5 and A325633(45) = 1.

A009194(n) = gcd(n,sigma(n));
A091255sq(a,b) = fromdigits(Vec(lift(gcd(Pol(binary(a))*Mod(1, 2),Pol(binary(b))*Mod(1, 2)))),2);
A325640(n) = A091255sq(n, A009194(n));

a(n) = A091255(n, A009194(n)) = A091255(n, gcd(n, sigma(n))).

cp's OEIS Frontend

A331746 Lexicographically earliest infinite sequence such that a(i) = a(j) => A009194(i) = A009194(j) and A331166(i) = A331166(j) for all i, j.

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A331747 Lexicographically earliest infinite sequence such that a(i) = a(j) => A009194(i) = A009194(j) and A278222(i) = A278222(j) for all i, j.

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A286570 Compound filter (prime signature of n & gcd(n, sigma(n))): a(n) = P(A046523(n), A009194(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A286591 Compound filter: a(n) = P(A009191(n), A009194(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A319338 Filter sequence combining the 2-adic valuation of n (A007814) with gcd(n,sigma(n)) (A009194).

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A325633 a(n) = gcd(A009194(n), A325634(n)) = gcd(A009194(n), A091255(n, sigma(n))).

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A331735 a(n) = A009194(A225546(n)) = gcd(A225546(n), sigma(A225546(n))).

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A325383 Lexicographically earliest sequence such that a(i) = a(j) => A000203(i) = A000203(j) and A009194(i) = A009194(j) for all i, j.

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A325384 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A000203(n), A009194(n)] for all other numbers, except f(p) = 0 for odd primes.