A286142 -id:A286142 - cp's OEIS frontend

A285729 Compound filter: a(n) = T(A032742(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

1, 2, 2, 12, 2, 31, 2, 59, 18, 50, 2, 142, 2, 73, 50, 261, 2, 199, 2, 220, 73, 131, 2, 607, 33, 166, 129, 314, 2, 961, 2, 1097, 131, 248, 73, 1396, 2, 295, 166, 923, 2, 1246, 2, 550, 340, 401, 2, 2509, 52, 655, 248, 692, 2, 1252, 131, 1303, 295, 590, 2, 3946, 2, 661, 517, 4497, 166, 1924, 2, 1024, 401, 2051, 2, 5707, 2, 898, 655, 1214, 131, 2317, 2, 3781, 888

Offset: 1

Antti Karttunen, May 04 2017

nonn

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

Cf. A000027, A032742, A046523, A286142, A286143, A286144, A286149, A286152, A286154, A286160, A286161, A286162, A286163, A286164.

Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ {Sort[Flatten@ Apply[ TensorProduct, # /. {p_, e_} /; p > 1 :> p^Range[0, e]]][[-2]], Times @@ MapIndexed[Prime[First@ #2]^#1 &, Sort[#[[All, -1]], Greater]] - Boole[n == 1]} &@ FactorInteger@ n, {n, 81}] (* Michael De Vlieger, May 04 2017 *)

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
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
A285729(n) = (1/2)*(2 + ((A032742(n)+A046523(n))^2) - A032742(n) - 3*A046523(n));
for(n=1, 10000, write("b285729.txt", n, " ", A285729(n)));

from sympy import divisors, factorint
def a032742(n): return 1 if n==1 else max(divisors(n)[:-1])
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(a032742(n), a046523(n)) # Indranil Ghosh, May 05 2017

(define (A285729 n) (* (/ 1 2) (+ (expt (+ (A032742 n) (A046523 n)) 2) (- (A032742 n)) (- (* 3 (A046523 n))) 2)))

a(n) = (1/2)*(2 + ((A032742(n)+A046523(n))^2) - A032742(n) - 3*A046523(n)).

A286143 Compound filter: a(n) = T(A055881(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 2, 12, 2, 31, 2, 38, 7, 23, 2, 94, 2, 23, 16, 138, 2, 94, 2, 80, 16, 23, 2, 355, 7, 23, 29, 80, 2, 499, 2, 530, 16, 23, 16, 706, 2, 23, 16, 302, 2, 499, 2, 80, 67, 23, 2, 1279, 7, 80, 16, 80, 2, 328, 16, 302, 16, 23, 2, 1894, 2, 23, 67, 2082, 16, 499, 2, 80, 16, 467, 2, 2779, 2, 23, 67, 80, 16, 499, 2, 1178, 121, 23, 2, 1894, 16, 23, 16, 302, 2, 1894, 16

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, A055881, A286144, A286160, A286161, A286162, A286163, A286164.

Differs from A286142 for the first time at n=24, where a(24) = 355, while A286142(24) = 328.

Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 - Boole[n == 1] & @@ {Module[{m = 1}, While[Mod[n, m!] == 0, m++]; m - 1], Times @@ MapIndexed[ Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]]}, {n, 92}] (* Michael De Vlieger, May 04 2017, after Robert G. Wilson v at A055881 *)

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
A055881(n) = { my(i); i=2; while((0 == (n%i)), n = n/i; i++); return(i-1); }
A286143(n) = (1/2)*(2 + ((A055881(n)+A046523(n))^2) - A055881(n) - 3*A046523(n));
for(n=1, 10000, write("b286143.txt", n, " ", A286143(n)));

from sympy import factorial, factorint
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 a055881(n):
    m = 1
    while n%factorial(m)==0:
        m+=1
    return m - 1
def a(n): return T(a055881(n), a046523(n)) # Indranil Ghosh, May 05 2017

(define (A286143 n) (* (/ 1 2) (+ (expt (+ (A055881 n) (A046523 n)) 2) (- (A055881 n)) (- (* 3 (A046523 n))) 2)))

a(n) = (1/2)*(2 + ((A055881(n)+A046523(n))^2) - A055881(n) - 3*A046523(n)).

A286144 Compound filter: a(n) = T(A000010(n), A257993(n)), where T(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 3, 5, 10, 8, 21, 14, 21, 14, 55, 19, 78, 27, 36, 44, 136, 34, 171, 44, 78, 65, 253, 53, 210, 90, 171, 90, 406, 63, 465, 152, 210, 152, 300, 103, 666, 189, 300, 152, 820, 103, 903, 230, 300, 275, 1081, 169, 903, 230, 528, 324, 1378, 208, 820, 324, 666, 434, 1711, 187, 1830, 495, 666, 560, 1176, 251, 2211, 560, 990, 324, 2485, 349, 2628, 702, 820, 702

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, A257993, A286142, A286143, A286152, A286160, A286161, A286162, A286163, A286164.

Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ {EulerPhi@ n, Module[{i = 1}, While[! CoprimeQ[Prime@ i, n], i++]; i]}, {n, 74}] (* Michael De Vlieger, May 04 2017 *)

A000010(n) = eulerphi(n);
A257993(n) = { for(i=1,n,if(n%prime(i),return(i))); }
A286144(n) = (2 + ((A000010(n)+A257993(n))^2) - A000010(n) - 3*A257993(n))/2;
for(n=1, 10000, write("b286144.txt", n, " ", A286144(n)));

from sympy import prime, primepi, gcd, totient
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a053669(n):
    x=1
    while True:
        if gcd(prime(x), n) == 1: return prime(x)
        else: x+=1
def a257993(n): return primepi(a053669(n))
def a(n): return T(totient(n), a257993(n)) # Indranil Ghosh, May 05 2017

(define (A286144 n) (* (/ 1 2) (+ (expt (+ (A000010 n) (A257993 n)) 2) (- (A000010 n)) (- (* 3 (A257993 n))) 2)))

a(n) = (1/2)*(2 + ((A000010(n)+A257993(n))^2) - A000010(n) - 3*A257993(n)).

A286152 Compound filter: a(n) = T(A051953(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 2, 2, 12, 2, 40, 2, 59, 18, 61, 2, 179, 2, 86, 73, 261, 2, 265, 2, 265, 100, 148, 2, 757, 33, 185, 129, 367, 2, 1297, 2, 1097, 166, 271, 131, 1735, 2, 320, 205, 1105, 2, 1741, 2, 619, 517, 430, 2, 3113, 52, 850, 295, 769, 2, 1747, 205, 1517, 346, 625, 2, 5297, 2, 698, 730, 4497, 248, 2821, 2, 1117, 460, 2821, 2, 7069, 2, 941, 1070, 1315, 248, 3457, 2, 4513

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, A051953, A285729, A286142, A286143, A286144, A286149, A286154, A286160, A286161, A286162, A286163, A286164.

Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ {n - EulerPhi@ n, Times @@ MapIndexed[Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]] - Boole[n == 1]}, {n, 80}] (* Michael De Vlieger, May 04 2017 *)

A051953(n) = (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
A286152(n) = (2 + ((A051953(n)+A046523(n))^2) - A051953(n) - 3*A046523(n))/2;
for(n=1, 10000, write("b286152.txt", n, " ", A286152(n)));

from sympy import factorint, totient
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(n - totient(n), a046523(n)) # Indranil Ghosh, May 05 2017

(define (A286152 n) (* (/ 1 2) (+ (expt (+ (A051953 n) (A046523 n)) 2) (- (A051953 n)) (- (* 3 (A046523 n))) 2)))

a(n) = (1/2)*(2 + ((A051953(n)+A046523(n))^2) - A051953(n) - 3*A046523(n)).

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

Original entry on oeis.org

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)).

A328470 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A053669(i) = A053669(j) for all i, j.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 8, 3, 9, 3, 8, 10, 11, 3, 9, 3, 12, 10, 8, 3, 13, 7, 8, 14, 12, 3, 15, 3, 16, 10, 8, 10, 17, 3, 8, 10, 18, 3, 19, 3, 12, 20, 8, 3, 21, 7, 12, 10, 12, 3, 13, 10, 18, 10, 8, 3, 22, 3, 8, 20, 23, 10, 19, 3, 12, 10, 24, 3, 25, 3, 8, 20, 12, 10, 19, 3, 26, 27, 8, 3, 28, 10, 8, 10, 18, 3, 22, 10, 12, 10, 8, 10, 29, 3, 12, 20, 30, 3, 19, 3, 18, 31

Offset: 1

Antti Karttunen, Oct 19 2019

nonn

Restricted growth sequence transform of A286142, or equally, of the ordered pair [A046523(n), A053669(n)], where A053669(n) gives the smallest prime not dividing n, while A046523(n) gives the prime signature of n.
For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j) => A291761(i) = A291761(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A046523, A053669, A286142, A291761, A305801, A328469.

up_to = 100000;
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
A053669(n) = forprime(p=2, , if(n%p, return(p))); \\ From A053669
Aux328470(n) = [A046523(n), A053669(n)];
v328470 = rgs_transform(vector(up_to, n, Aux328470(n)));
A328470(n) = v328470[n];

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)).

A286382 Compound filter: a(n) = P(A257993(n), A278226(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

2, 5, 16, 12, 67, 9, 16, 23, 436, 80, 1771, 18, 67, 80, 1771, 668, 16111, 48, 277, 302, 7141, 2630, 64621, 156, 1129, 1178, 28681, 10442, 258841, 14, 16, 23, 436, 80, 1771, 31, 436, 467, 21946, 1832, 87991, 94, 1771, 1832, 87991, 16292, 793171, 328, 7141, 7262, 352381, 64982, 3173941, 1228, 28681, 28922, 1410361, 259562, 12698281, 25, 67, 80, 1771, 668, 16111

Offset: 1

Antti Karttunen, May 08 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..11550
Eric Weisstein's World of Mathematics, Pairing Function
Index entries for sequences related to primorial base

Cf. A000027, A257993, A278226, A286142.

Differs from A286381 for the first time at n=24, where a(24) = 156 while A286381(24) = 14.

(define (A286382 n) (* (/ 1 2) (+ (expt (+ (A257993 n) (A278226 n)) 2) (- (A257993 n)) (- (* 3 (A278226 n))) 2)))

a(n) = (1/2)*(2 + ((A257993(n)+A278226(n))^2) - A257993(n) - 3*A278226(n)).

cp's OEIS Frontend

A285729 Compound filter: a(n) = T(A032742(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

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A286143 Compound filter: a(n) = T(A055881(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

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A286144 Compound filter: a(n) = T(A000010(n), A257993(n)), where T(n,k) is sequence A000027 used as a pairing function.

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A286152 Compound filter: a(n) = T(A051953(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

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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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A328470 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A053669(i) = A053669(j) for all i, j.

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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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