A286162 -id:A286162 - cp's OEIS frontend

A278222 The least number with the same prime signature as A005940(n+1).

1, 2, 2, 4, 2, 6, 4, 8, 2, 6, 6, 12, 4, 12, 8, 16, 2, 6, 6, 12, 6, 30, 12, 24, 4, 12, 12, 36, 8, 24, 16, 32, 2, 6, 6, 12, 6, 30, 12, 24, 6, 30, 30, 60, 12, 60, 24, 48, 4, 12, 12, 36, 12, 60, 36, 72, 8, 24, 24, 72, 16, 48, 32, 64, 2, 6, 6, 12, 6, 30, 12, 24, 6, 30, 30, 60, 12, 60, 24, 48, 6, 30, 30, 60, 30, 210, 60, 120, 12, 60, 60, 180, 24, 120, 48, 96, 4, 12, 12

Offset: 0

Antti Karttunen, Nov 15 2016

This sequence can be used for filtering certain base-2 related sequences, because it matches only with any such sequence b that can be computed as b(n) = f(A005940(n+1)), where f(n) is any function that depends only on the prime signature of n (some of these are listed under the index entry for "sequences computed from exponents in ...").
Matching in this context means that the sequence a matches with the sequence b iff for all i, j: a(i) = a(j) => b(i) = b(j). In other words, iff the sequence b partitions the natural numbers to the same or coarser equivalence classes (as/than the sequence a) by the distinct values it obtains.
Because the Doudna map n -> A005940(1+n) is an isomorphism from "unary-binary encoding of factorization" (see A156552) to the ordinary representation of the prime factorization of n, it follows that the equivalence classes of this sequence match with any such sequence b, where b(n) is computed from the lengths of 1-runs in the binary representation of n and the order of those 1-runs does not matter. Particularly, this holds for any sequence that is obtained as a "Run Length Transform", i.e., where b(n) = Product S(i), for some function S, where i runs through the lengths of runs of 1's in the binary expansion of n. See for example A227349.
However, this sequence itself is not a run length transform of any sequence (which can be seen for example from the fact that A046523 is not multiplicative).
Furthermore, this matches not only with sequences involving products of S(i), but with any sequence obtained with any commutative function applied cumulatively, like e.g., A000120 (binary weight, obtained in this case as Sum identity(i)), and A069010 (number of runs of 1's in binary representation of n, obtained as Sum signum(i)).

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A005940, A156552, A006068, A124859, A278159.
Similar sequences: A278217, A278219 (other base-2 related variants), A069877 (base-10 related), A278226 (primorial base), A278234-A278236 (factorial base), A278243 (Stern polynomials), A278233 (factorization in ring GF(2)[X]), A046523 (factorization in Z).
Cf. also A286622 (rgs-transform of this sequence) and A286162, A286252, A286163, A286240, A286242, A286379, A286464, A286374, A286375, A286376, A286243, A286553 (various other sequences involving this sequence).
Sequences that partition N into same or coarser equivalence classes: too many to list all here (over a hundred). At least every sequence listed under index-entry "Run Length Transforms" is included (e.g., A227349, A246660, A278159), and also sequences like A000120 and A069010, and their combinations like A136277.

f[n_, i_, x_] := Which[n == 0, x, EvenQ@ n, f[n/2, i + 1, x], True, f[(n - 1)/2, i, x Prime@ i]]; Array[If[# == 1, 1, Times @@ MapIndexed[ Prime[First[#2]]^#1 &, Sort[FactorInteger[#][[All, -1]], Greater]]] &@ f[# - 1, 1, 1] &, 99] (* Michael De Vlieger, May 09 2017 *)

A046523(n)=factorback(primes(#n=vecsort(factor(n)[, 2], , 4)), n)
a(n)=my(p=2, t=1); for(i=0,exponent(n), if(bittest(n,i), t*=p, p=nextprime(p+1))); A046523(t) \\ Charles R Greathouse IV, Nov 11 2021

from sympy import prime, factorint
import math
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 a(n): return a046523(a005940(n + 1)) # Indranil Ghosh, May 05 2017

(define (A278222 n) (A046523 (A005940 (+ 1 n))))

a(n) = A046523(A005940(1+n)).
a(n) = A124859(A278159(n)).
a(n) = A278219(A006068(n)).

Misleading part of the name removed by Antti Karttunen, Apr 07 2022

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

Original entry on oeis.org

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)).

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)).

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)).

A286260 Compound filter: a(n) = P(A001511(n), A161942(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 8, 1, 39, 4, 8, 1, 157, 79, 47, 4, 39, 22, 8, 4, 600, 37, 782, 11, 256, 1, 47, 4, 157, 466, 233, 11, 39, 106, 47, 1, 2284, 4, 380, 4, 4281, 172, 122, 22, 1132, 211, 8, 56, 256, 742, 47, 4, 600, 1597, 4373, 37, 1278, 352, 122, 37, 157, 11, 1037, 106, 256, 466, 8, 79, 8785, 211, 47, 137, 2083, 4, 47, 37, 19507, 667, 1655, 466, 669, 4, 233, 11, 4661, 7261

Offset: 1

Antti Karttunen, May 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
MathWorld, Pairing Function

Cf. A000027, A001511, A161942, A286161, A286162, A286259, A286034.

A001511(n) = (1+valuation(n,2));
A000265(n) = (n >> valuation(n, 2));
A161942(n) = A000265(sigma(n));
A286260(n) = (2 + ((A001511(n)+A161942(n))^2) - A001511(n) - 3*A161942(n))/2;
for(n=1, 16384, write("b286260.txt", n, " ", A286260(n)));

from sympy import factorint, divisors, divisor_sigma
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a000265(n): return max(list(filter(lambda i: i%2 == 1, divisors(n))))
def a161942(n): return a000265(divisor_sigma(n))
def a001511(n): return 2 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a(n): return T(a001511(n), a161942(n)) # Indranil Ghosh, May 07 2017

(define (A286260 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A161942 n)) 2) (- (A001511 n)) (- (* 3 (A161942 n))) 2)))

a(n) = (1/2)*(2 + ((A001511(n)+A161942(n))^2) - A001511(n) - 3*A161942(n)).

cp's OEIS Frontend

A278222 The least number with the same prime signature as A005940(n+1).

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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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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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A286164 Compound filter: a(n) = T(A055396(n), A046523(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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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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