A286164 -id:A286164 - cp's OEIS frontend

A286161 Compound filter: a(n) = T(A001511(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

1, 5, 2, 18, 2, 23, 2, 59, 7, 23, 2, 94, 2, 23, 16, 195, 2, 80, 2, 94, 16, 23, 2, 355, 7, 23, 29, 94, 2, 467, 2, 672, 16, 23, 16, 706, 2, 23, 16, 355, 2, 467, 2, 94, 67, 23, 2, 1331, 7, 80, 16, 94, 2, 302, 16, 355, 16, 23, 2, 1894, 2, 23, 67, 2422, 16, 467, 2, 94, 16, 467, 2, 2779, 2, 23, 67, 94, 16, 467, 2, 1331, 121, 23, 2, 1894, 16, 23, 16, 355, 2, 1832

Offset: 1

Antti Karttunen, May 04 2017

nonn

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

Cf. A000027, A001511, A046523, A286160, A286162, A286163, A286164, A286251.

A001511(n) = (1+valuation(n,2));
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
A286161(n) = (2 + ((A001511(n)+A046523(n))^2) - A001511(n) - 3*A046523(n))/2;
for(n=1, 10000, write("b286161.txt", n, " ", A286161(n)));

from sympy import 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 a001511(n): return 2 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a(n): return T(a001511(n), a046523(n)) # Indranil Ghosh, May 06 2017

(define (A286161 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A046523 n)) 2) (- (A001511 n)) (- (* 3 (A046523 n))) 2)))

a(n) = (1/2)*(2 + ((A001511(n)+A046523(n))^2) - A001511(n) - 3*A046523(n)).

A286160 Compound filter: a(n) = T(A000010(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 5, 12, 14, 23, 27, 59, 42, 40, 65, 109, 90, 61, 86, 261, 152, 142, 189, 179, 148, 115, 275, 473, 273, 148, 318, 265, 434, 674, 495, 1097, 320, 226, 430, 1093, 702, 271, 430, 757, 860, 832, 945, 485, 619, 373, 1127, 1969, 1032, 485, 698, 619, 1430, 838, 1030, 1105, 856, 556, 1769, 2791, 1890, 625, 1117, 4497, 1426, 1196, 2277, 935, 1220

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, A046523, A286161, A286162, A286163, A286164.

Cf. for example A061468 (one of the sequences this matches with).

A000010(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
A286160(n) = (2 + ((A000010(n)+A046523(n))^2) - A000010(n) - 3*A046523(n))/2;
for(n=1, 10000, write("b286160.txt", n, " ", A286160(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(totient(n), a046523(n)) # Indranil Ghosh, May 06 2017

(define (A286160 n) (* (/ 1 2) (+ (expt (+ (A000010 n) (A046523 n)) 2) (- (A000010 n)) (- (* 3 (A046523 n))) 2)))

a(n) = (1/2)*(2 + ((A000010(n)+A046523(n))^2) - A000010(n) - 3*A046523(n)).

A286162 Compound filter: a(n) = T(A001511(n), A278222(n)), where T(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

2, 5, 7, 9, 16, 12, 29, 14, 16, 23, 67, 18, 67, 38, 121, 20, 16, 23, 67, 31, 436, 80, 277, 25, 67, 80, 631, 48, 277, 138, 497, 27, 16, 23, 67, 31, 436, 80, 277, 40, 436, 467, 1771, 94, 1771, 302, 1129, 33, 67, 80, 631, 94, 1771, 668, 2557, 59, 277, 302, 2557, 156, 1129, 530, 2017, 35, 16, 23, 67, 31, 436, 80, 277, 40, 436, 467, 1771, 94, 1771, 302, 1129, 50

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, A001511, A278222, A286160, A286161, A286163, A286164.

A001511(n) = (1+valuation(n,2));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
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
A278222(n) = A046523(A005940(1+n));
A286162(n) = (2 + ((A001511(n)+A278222(n))^2) - A001511(n) - 3*A278222(n))/2;
for(n=1, 10000, write("b286162.txt", n, " ", A286162(n)));

from sympy import prime, factorint
import math
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def A(n): return n - 2**int(math.floor(math.log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
def a005940(n): return b(n - 1)
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 a278222(n): return a046523(a005940(n + 1))
def a001511(n): return bin(n)[2:][::-1].index("1") + 1
def a(n): return T(a001511(n), a278222(n)) # Indranil Ghosh, May 05 2017

(define (A286162 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A278222 n)) 2) (- (A001511 n)) (- (* 3 (A278222 n))) 2)))

a(n) = (1/2)*(2 + ((A001511(n)+A278222(n))^2) - A001511(n) - 3*A278222(n)).

A286163 Compound filter: a(n) = T(A046523(n), A278222(n)), where T(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

2, 5, 12, 14, 23, 42, 38, 44, 40, 61, 80, 117, 80, 84, 216, 152, 23, 148, 80, 148, 601, 142, 302, 375, 109, 142, 911, 183, 302, 1020, 530, 560, 61, 61, 142, 856, 467, 142, 412, 430, 467, 1741, 1832, 265, 2497, 412, 1178, 1323, 109, 265, 826, 265, 1832, 1735, 2932, 489, 412, 412, 2630, 2835, 1178, 672, 2787, 2144, 61, 625, 80, 148, 601, 850, 302, 2998, 467, 601

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, A278222, A286160, A286161, A286162, A286164.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
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
A278222(n) = A046523(A005940(1+n));
A286163(n) = (2 + ((A046523(n)+A278222(n))^2) - A046523(n) - 3*A278222(n))/2;
for(n=1, 10000, write("b286163.txt", n, " ", A286163(n)));

from sympy import prime, factorint
import math
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def A(n): return n - 2**int(math.floor(math.log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
def a005940(n): return b(n - 1)
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 a278222(n): return a046523(a005940(n + 1))
def a(n): return T(a046523(n), a278222(n)) # Indranil Ghosh, May 05 2017

(define (A286163 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A278222 n)) 2) (- (A046523 n)) (- (* 3 (A278222 n))) 2)))

a(n) = (1/2)*(2 + ((A046523(n)+A278222(n))^2) - A046523(n) - 3*A278222(n)).

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

Original entry on oeis.org

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)).

A286142 Compound filter: a(n) = T(A257993(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, 328, 7, 23, 29, 80, 2, 532, 2, 530, 16, 23, 16, 706, 2, 23, 16, 302, 2, 499, 2, 80, 67, 23, 2, 1228, 7, 80, 16, 80, 2, 328, 16, 302, 16, 23, 2, 1957, 2, 23, 67, 2082, 16, 499, 2, 80, 16, 467, 2, 2704, 2, 23, 67, 80, 16, 499, 2, 1178, 121, 23, 2, 1894, 16, 23, 16, 302, 2, 1957, 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, A257993, A286144, A286160, A286161, A286162, A286163, A286164.

Differs from A286143 for the first time at n=24, where a(24) = 328, while A286143(24) = 355.

Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 - Boole[n == 1] & @@ {Module[{i = 1}, While[! CoprimeQ[Prime@ i, n], i++]; i], Times @@ MapIndexed[ Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]]}, {n, 92}] (* Michael De Vlieger, May 04 2017 *)

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
A257993(n) = { for(i=1,n,if(n%prime(i),return(i))); }
A286142(n) = (1/2)*(2 + ((A257993(n)+A046523(n))^2) - A257993(n) - 3*A046523(n));
for(n=1, 10000, write("b286142.txt", n, " ", A286142(n)));

from sympy import factorint, prime, primepi, 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 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(a257993(n), a046523(n)) # Indranil Ghosh, May 05 2017

(define (A286142 n) (* (/ 1 2) (+ (expt (+ (A257993 n) (A046523 n)) 2) (- (A257993 n)) (- (* 3 (A046523 n))) 2)))

a(n) = (1/2)*(2 + ((A257993(n)+A046523(n))^2) - A257993(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)).

A286356 Compound filter: a(n) = P(A061395(n), A046523(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 2, 5, 7, 9, 23, 14, 29, 12, 31, 20, 80, 27, 40, 31, 121, 35, 80, 44, 94, 40, 50, 54, 302, 18, 61, 38, 109, 65, 499, 77, 497, 50, 73, 40, 668, 90, 86, 61, 328, 104, 532, 119, 125, 94, 100, 135, 1178, 25, 94, 73, 142, 152, 302, 50, 355, 86, 115, 170, 1894, 189, 131, 109, 2017, 61, 566, 209, 160, 100, 532, 230, 2630, 252, 148, 94, 179, 50, 601, 275, 1228, 138

Offset: 1

Antti Karttunen, May 09 2017

nonn

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

Cf. A000027, A046523, A061395, A286164.

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
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1]))); \\ After M. F. Hasler's code for A006530.
A286356(n) = (2 + ((A061395(n)+A046523(n))^2) - A061395(n) - 3*A046523(n))/2;
for(n=1, 10000, write("b286356.txt", n, " ", A286356(n)));

from sympy import factorint
from operator import mul
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 a061395(n): return 0 if n == 1 else primepi(max(primefactors(n)))
def a(n): return T(a061395(n), a046523(n)) # Indranil Ghosh, May 09 2017

(define (A286356 n) (* (/ 1 2) (+ (expt (+ (A061395 n) (A046523 n)) 2) (- (A061395 n)) (- (* 3 (A046523 n))) 2)))

a(n) = (1/2)*(2 + ((A061395(n)+A046523(n))^2) - A061395(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)).

cp's OEIS Frontend

A286161 Compound filter: a(n) = T(A001511(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

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A286160 Compound filter: a(n) = T(A000010(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

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A286162 Compound filter: a(n) = T(A001511(n), A278222(n)), where T(n,k) is sequence A000027 used as a pairing function.

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A286163 Compound filter: a(n) = T(A046523(n), A278222(n)), where T(n,k) is sequence A000027 used as a pairing function.

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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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Mathematica

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A286142 Compound filter: a(n) = T(A257993(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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