A286161 -id:A286161 - cp's OEIS frontend

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

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)).

A286164 Compound filter: a(n) = T(A055396(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 2, 5, 7, 9, 16, 14, 29, 12, 16, 20, 67, 27, 16, 23, 121, 35, 67, 44, 67, 23, 16, 54, 277, 18, 16, 38, 67, 65, 436, 77, 497, 23, 16, 31, 631, 90, 16, 23, 277, 104, 436, 119, 67, 80, 16, 135, 1129, 25, 67, 23, 67, 152, 277, 31, 277, 23, 16, 170, 1771, 189, 16, 80, 2017, 31, 436, 209, 67, 23, 436, 230, 2557, 252, 16, 80, 67, 40, 436, 275, 1129, 138, 16, 299

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, A055396, A286160, A286161, A286162, A286163.

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
A055396(n) = if(n==1, 0, primepi(factor(n)[1, 1])); \\ This function from Charles R Greathouse IV, Apr 23 2015
A286164(n) = (2 + ((A055396(n)+A046523(n))^2) - A055396(n) - 3*A046523(n))/2;
for(n=1, 10000, write("b286164.txt", n, " ", A286164(n)));

from sympy import primepi, isprime, primefactors, factorint
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 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(a055396(n), a046523(n)) # Indranil Ghosh, May 05 2017

(define (A286164 n) (* (/ 1 2) (+ (expt (+ (A055396 n) (A046523 n)) 2) (- (A055396 n)) (- (* 3 (A046523 n))) 2)))

a(n) = (1/2)*(2 + ((A055396(n)+A046523(n))^2) - A055396(n) - 3*A046523(n)).

A305891 Filter sequence combining 2-adic valuation (A007814) and the prime signature (A046523) of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 14 2018

nonn

Restricted growth sequence transform of A286161, of the ordered pair [A007814(n), A046523(n)].

For all i, j: a(i) = a(j) => A291761(i) = A291761(j).

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

Cf. A286161, A291761, A305801.

Cf. also A305893.

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);
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
Aux305891(n) = [A007814(n), A046523(n)];
v305891 = rgs_transform(vector(up_to,n,Aux305891(n)));
A305891(n) = v305891[n];

A291761 Restricted growth sequence transform of ((-1)^n)*A046523(n); filter combining the parity and the prime signature of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 11 2017

nonn

Equally, restricted growth sequence transform of sequence b defined as b(1) = 1, and for n > 1, b(n) = A046523(n) + A000035(n), which starts as 1, 2, 3, 4, 3, 6, 3, 8, 5, 6, 3, 12, 3, 6, 7, 16, 3, 12, 3, 12, ...

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

Cf. A291767, A291768 (bisections), A147516.
Cf. A046523, A101296, A286161, A286251, A286367, A291762 (related or similar filtering sequences).
Cf. A065091 (positions of 3's), A100484 (of 4 and 5's), A001248 (of 4 and 7's), A046388 (of 9's), A030078 (of 6 and 12's).
Cf. A098108 (one of the matching sequences).

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
write_to_bfile(1,rgs_transform(vector(65537,n,((-1)^n)*A046523(n))),"b291761.txt");
\\ Or alternatively:
f(n) = if(1==n,n,A046523(n)+(n%2));
write_to_bfile(1,rgs_transform(vector(16385,n,f(n))),"b291761.txt");

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)).

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)).

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Scheme

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Scheme

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Scheme

Formula

A286164 Compound filter: a(n) = T(A055396(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Scheme

Formula

A305891 Filter sequence combining 2-adic valuation (A007814) and the prime signature (A046523) of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A291761 Restricted growth sequence transform of ((-1)^n)*A046523(n); filter combining the parity and the prime signature of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Scheme

Formula

A286142 Compound filter: a(n) = T(A257993(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

Views

Author

Keywords

Links