A286160 -id:A286160 - cp's OEIS frontend

A286154 Compound filter: a(n) = T(A055396(n), A000010(n)), where T(n,k) is sequence A000027 used as a pairing function.

0, 1, 5, 2, 18, 2, 40, 7, 23, 7, 96, 7, 142, 16, 38, 29, 238, 16, 308, 29, 80, 46, 444, 29, 234, 67, 173, 67, 676, 29, 791, 121, 212, 121, 328, 67, 1093, 154, 302, 121, 1339, 67, 1499, 191, 302, 232, 1785, 121, 994, 191, 530, 277, 2227, 154, 864, 277, 668, 379, 2718, 121, 2944, 436, 668, 497, 1228, 191, 3505, 497, 992, 277, 3936, 277, 4207, 631, 822, 631

Offset: 1

Antti Karttunen, May 04 2017

nonn

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

Cf. A000010, A000027, A055396, A286142, A286143, A286144, A286152, A286160, A286161, A286162, A286163, A286164.

Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ {If[n == 1, 0, PrimePi[ FactorInteger[n][[1, 1]] ]], EulerPhi@ n}, {n, 76}] (* Michael De Vlieger, May 04 2017 *)

A000010(n) = eulerphi(n);
A055396(n) = if(n==1, 0, primepi(factor(n)[1, 1])); \\ This function from Charles R Greathouse IV, Apr 23 2015
A286154(n) = (2 + ((A055396(n)+A000010(n))^2) - A055396(n) - 3*A000010(n))/2;
for(n=1, 10000, write("b286154.txt", n, " ", A286154(n)));

from sympy import primepi, isprime, primefactors, totient
def a049084(n): return primepi(n)*(1*isprime(n))
def a055396(n): return 0 if n==1 else a049084(min(primefactors(n)))
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a(n): return T(a055396(n), totient(n)) # Indranil Ghosh, May 05 2017

(define (A286154 n) (* (/ 1 2) (+ (expt (+ (A055396 n) (A000010 n)) 2) (- (A055396 n)) (- (* 3 (A000010 n))) 2)))

a(n) = (1/2)*(2 + ((A055396(n)+A000010(n))^2) - A055396(n) - 3*A000010(n)).

A300224 Filter sequence combining A046523(n) and A296078(n), prime signature of n and prime signature of phi(n)+1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 7, 8, 2, 6, 2, 9, 4, 4, 2, 10, 11, 4, 5, 6, 2, 12, 2, 13, 14, 4, 7, 15, 2, 4, 7, 16, 2, 17, 2, 18, 9, 4, 2, 19, 3, 18, 14, 9, 2, 16, 4, 10, 4, 4, 2, 20, 2, 4, 6, 21, 7, 22, 2, 18, 23, 12, 2, 24, 2, 4, 6, 6, 4, 12, 2, 25, 26, 4, 2, 27, 14, 4, 14, 16, 2, 27, 4, 28, 4, 4, 4, 29, 2, 6, 6, 15, 2, 22, 2, 10, 12

Offset: 1

Antti Karttunen, Mar 01 2018

nonn

Restricted growth sequence transform of P(A046523(n), A296078(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.

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

Cf. A000010, A046523, A296078.

Cf. also A286160, A296085, A300223, A300225.

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[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
A296078(n) = A046523(1+eulerphi(n));
Aux300224(n) = (1/2)*(2 + ((A296078(n)+A046523(n))^2) - A296078(n) - 3*A046523(n));
write_to_bfile(1,rgs_transform(vector(up_to,n,Aux300224(n))),"b300224.txt");

A286149 Compound filter: a(n) = T(A046523(n), A109395(n)), where T(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 8, 14, 17, 34, 30, 44, 19, 51, 68, 103, 93, 72, 196, 152, 155, 103, 192, 132, 72, 126, 278, 349, 32, 159, 53, 165, 437, 976, 498, 560, 709, 237, 786, 739, 705, 282, 159, 402, 863, 660, 948, 243, 337, 384, 1130, 1273, 49, 132, 1546, 288, 1433, 349, 126, 459, 282, 567, 1772, 2761, 1893, 636, 165, 2144, 2421, 1921, 2280, 390, 2707, 2046, 2558, 2773, 2703

Offset: 1

Antti Karttunen, May 04 2017

nonn

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

Cf. A000027, A046523, A109395, A285729, A286142, A286143, A286144, A286152, A286154, A286160, A286161, A286162, A286163, A286164.

Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ {Times @@ MapIndexed[ Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]] - Boole[n == 1], Denominator[EulerPhi[n]/n]}, {n, 73}] (* Michael De Vlieger, May 04 2017 *)

A109395(n) = n/gcd(n, eulerphi(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
A286149(n) = (1/2)*(2 + ((A046523(n)+A109395(n))^2) - A046523(n) - 3*A109395(n));
for(n=1, 10000, write("b286149.txt", n, " ", A286149(n)));

from sympy import factorint, totient, gcd
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), n/gcd(n, totient(n))) # Indranil Ghosh, May 05 2017

(define (A286149 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A109395 n)) 2) (- (A046523 n)) (- (* 3 (A109395 n))) 2)))

a(n) = (1/2)*(2 + ((A046523(n)+A109395(n))^2) - A046523(n) - 3*A109395(n)).

A318893 Filter sequence combining the prime signature of n (A046523) with Euler totient function (A000010).

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, 25, 21, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 34, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 42, 48, 43, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 53, 70, 59, 71, 66, 72, 73, 74, 51, 75, 76, 77, 78, 79, 80, 81, 76, 82, 83, 71

Offset: 1

Antti Karttunen, Sep 16 2018

nonn

Restricted growth sequence transform of A286160.

For all i, j: a(i) = a(j) => A062355(i) = A062355(j).

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

Cf. A000010, A046523, A286160, A318839, A318892.

Cf. also A061468, A062355.

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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A318893aux(n) = [eulerphi(n), A046523(n)];
v318893 = rgs_transform(vector(up_to,n,A318893aux(n)));
A318893(n) = v318893[n];

A286568 Compound filter (phi(n) & 2-adic valuation of sigma(n)): a(n) = P(A000010(n), A286357(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 1, 8, 3, 14, 8, 42, 10, 21, 14, 76, 19, 90, 42, 63, 36, 152, 21, 208, 44, 148, 76, 322, 53, 210, 90, 228, 117, 434, 63, 625, 136, 296, 152, 402, 78, 702, 208, 375, 152, 860, 148, 988, 251, 324, 322, 1271, 169, 903, 210, 627, 324, 1430, 228, 943, 375, 816, 434, 1828, 187, 1890, 625, 777, 528, 1273, 296, 2344, 560, 1220, 402, 2698, 300, 2700, 702, 901

Offset: 1

Antti Karttunen, May 26 2017

nonn

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

Cf. A000010, A000027, A286154, A286160, A286357, A286451, A286572.

A000010(n) = eulerphi(n);
A001511(n) = (1+valuation(n,2));
A286357(n) = A001511(sigma(n));
A286568(n) = (1/2)*(2 + ((A000010(n)+A286357(n))^2) - A000010(n) - 3*A286357(n));

from sympy import divisor_sigma as D, totient
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a001511(n): return bin(n)[2:][::-1].index("1") + 1
def a286357(n): return a001511(D(n))
def a(n): return T(totient(n), a286357(n)) # Indranil Ghosh, May 26 2017

(define (A286568 n) (* (/ 1 2) (+ (expt (+ (A000010 n) (A286357 n)) 2) (- (A000010 n)) (- (* 3 (A286357 n))) 2)))

a(n) = (1/2)*(2 + ((A000010(n)+A286357(n))^2) - A000010(n) - 3*A286357(n)).

A289622 Compound filter (prime signature & Carmichael's lambda): a(n) = P(A046523(n), A002322(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 3, 5, 14, 12, 27, 23, 44, 40, 42, 57, 90, 80, 61, 42, 187, 138, 148, 173, 117, 61, 111, 255, 324, 257, 142, 308, 148, 408, 558, 467, 773, 111, 216, 142, 856, 668, 259, 142, 375, 822, 625, 905, 222, 265, 357, 1083, 1323, 994, 477, 216, 265, 1380, 844, 306, 430, 259, 534, 1713, 2013, 1832, 601, 148, 3145, 142, 771, 2213, 363, 357

Offset: 1

Antti Karttunen, Jul 16 2017

nonn

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

Cf. A000027, A002322, A046523, A286160, A286360, A286573.

A002322(n) = lcm(znstar(n)[2]); \\ This function from Charles R Greathouse IV, Aug 04 2012
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
A289622(n) = (1/2)*(2 + ((A046523(n)+A002322(n))^2) - A046523(n) - 3*A002322(n));

a(n) = (1/2)*(2 + ((A046523(n)+A002322(n))^2) - A046523(n) - 3*A002322(n)).

A289628 Compound filter (for the structure of the multiplicative group of integers modulo n & prime signature of n): a(n) = P(A289626(n), A101296(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 5, 8, 9, 12, 14, 41, 19, 18, 27, 50, 35, 25, 63, 99, 54, 40, 65, 86, 102, 42, 90, 203, 134, 52, 101, 131, 135, 128, 152, 342, 228, 75, 250, 221, 230, 88, 250, 399, 275, 182, 299, 271, 295, 117, 324, 517, 323, 185, 403, 295, 377, 146, 462, 623, 525, 168, 495, 549, 527, 187, 698, 728, 663, 343, 629, 460, 738, 370, 702, 889, 740, 273, 523, 590, 858, 370

Offset: 1

Antti Karttunen, Jul 19 2017

nonn

Here, instead of A046523 and A289625 we use as the components of a(n) their rgs-versions A101296 and A289626 because of the latter sequence's more moderate growth rate.
For all i, j: a(i) = a(j) => A286160(i) = A286160(j).
For all i, j: a(i) = a(j) => A289622(i) = A289622(j).

Antti Karttunen, Table of n, a(n) for n = 1..65536

Cf. A000027, A046523, A101296, A286160, A289622, A289626, A286454.

(define (A289628 n) (* (/ 1 2) (+ (expt (+ (A289626 n) (A101296 n)) 2) (- (A289626 n)) (- (* 3 (A101296 n))) 2)))

a(n) = (1/2)*(2 + ((A289626(n)+A101296(n))^2) - A289626(n) - 3*A101296(n)).

cp's OEIS Frontend

A286154 Compound filter: a(n) = T(A055396(n), A000010(n)), where T(n,k) is sequence A000027 used as a pairing function.

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A300224 Filter sequence combining A046523(n) and A296078(n), prime signature of n and prime signature of phi(n)+1.

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A286149 Compound filter: a(n) = T(A046523(n), A109395(n)), where T(n,k) is sequence A000027 used as a pairing function.

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A318893 Filter sequence combining the prime signature of n (A046523) with Euler totient function (A000010).

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A286568 Compound filter (phi(n) & 2-adic valuation of sigma(n)): a(n) = P(A000010(n), A286357(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A289622 Compound filter (prime signature & Carmichael's lambda): a(n) = P(A046523(n), A002322(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A289628 Compound filter (for the structure of the multiplicative group of integers modulo n & prime signature of n): a(n) = P(A289626(n), A101296(n)), where P(n,k) is sequence A000027 used as a pairing function.

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