A286240 -id:A286240 - 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

A286255 Compound filter: a(n) = P(A046523(n), A046523(1+n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

2, 5, 12, 14, 23, 27, 38, 63, 40, 27, 80, 90, 23, 61, 216, 152, 80, 90, 80, 148, 61, 27, 302, 375, 40, 84, 179, 90, 467, 495, 530, 698, 61, 61, 826, 702, 23, 61, 412, 324, 467, 495, 80, 265, 148, 27, 1178, 1323, 109, 148, 142, 90, 302, 430, 412, 430, 61, 27, 1832, 1890, 23, 142, 2787, 2410, 601, 495, 80, 148, 601, 495, 2630, 2700, 23, 142, 265, 148, 601, 495, 1178

Offset: 1

Antti Karttunen, May 07 2017

nonn

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

Cf. A000027, A046523, A286240, A286256, A286257, A286258.
Cf. A005383 (after its initial term 3, gives the positions of 23's in this sequence).
Cf. A051950 (one of the matches not matched by A046523 alone).

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
A286255(n) = (2 + ((A046523(n)+A046523(1+n))^2) - A046523(n) - 3*A046523(1+n))/2;
for(n=1, 10000, write("b286255.txt", n, " ", A286255(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 a(n): return T(a046523(n), a046523(n + 1)) # Indranil Ghosh, May 07 2017

(define (A286255 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A046523 (+ 1 n))) 2) (- (A046523 n)) (- (* 3 (A046523 (+ 1 n)))) 2)))

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

A286242 Compound filter: a(n) = P(A278222(n), A278219(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 12, 14, 12, 84, 40, 44, 12, 142, 216, 183, 40, 265, 86, 152, 12, 142, 826, 265, 216, 1860, 607, 489, 40, 832, 607, 1117, 86, 619, 226, 560, 12, 142, 826, 265, 826, 5080, 2497, 619, 216, 2956, 4308, 4155, 607, 8575, 1105, 1533, 40, 832, 2497, 2116, 607, 5731, 4501, 3475, 86, 1402, 1105, 3475, 226, 1759, 698, 2144, 12, 142, 826, 265, 826, 5080, 2497, 619, 826

Offset: 0

Antti Karttunen, May 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
MathWorld, Pairing Function

Cf. A000027, A003188, A278219, A278222, A286240, A286241, A286242.

A003188(n) = bitxor(n, n>>1);
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));
A278219(n) = A278222(A003188(n));
A286242(n) = (2 + ((A278222(n)+A278219(n))^2) - A278222(n) - 3*A278219(n))/2;
for(n=0, 16383, write("b286242.txt", n, " ", A286242(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 a003188(n): return n^(n>>1)
def a243353(n): return a005940(1 + a003188(n))
def a278219(n): return a046523(a243353(n))
def a278222(n): return a046523(a005940(n + 1))
def a(n): return T(a278222(n), a278219(n)) # Indranil Ghosh, May 07 2017

(define (A286242 n) (* (/ 1 2) (+ (expt (+ (A278222 n) (A278219 n)) 2) (- (A278222 n)) (- (* 3 (A278219 n))) 2)))

a(n) = (1/2)*(2 + ((A278222(n)+A278219(n))^2) - A278222(n) - 3*A278219(n)).

a(n) = (1/2)*(2 + ((A278222(n)+A278222(A003188(n)))^2) - A278222(n) - 3*A278222(A003188(n))).

A286241 Compound filter: a(n) = P(A278219(n), A278219(1+n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

2, 12, 14, 12, 59, 86, 27, 12, 109, 363, 269, 86, 142, 148, 27, 12, 109, 1093, 1117, 363, 1097, 1517, 489, 86, 601, 1408, 619, 148, 142, 148, 27, 12, 109, 1093, 1117, 1093, 5707, 8587, 2545, 363, 1969, 6153, 4529, 1517, 4489, 4537, 489, 86, 601, 3946, 3976, 1408, 2509, 5719, 2545, 148, 601, 1408, 619, 148, 142, 148, 27, 12, 109, 1093, 1117, 1093, 5707, 8587

Offset: 0

Antti Karttunen, May 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000027, A003188, A005940, A046523, A243353, A278219, A278222, A286240, A286242, A286255.

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]]; g[n_] := If[n == 1, 1, Times @@ MapIndexed[ Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]]]; h[n_] := g@ f[BitXor[n, Floor[n/2]], 1, 1]; Map[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ # & /@ # &, Table[{h[n], h[n + 1]}, {k, 12}, {n, k (k - 1)/2, k (k + 1)/2 - 1}]] // Flatten (* Michael De Vlieger, May 09 2017 *)

A003188(n) = bitxor(n, n>>1);
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));
A278219(n) = A278222(A003188(n));
A286241(n) = (2 + ((A278219(n)+A278219(1+n))^2) - A278219(n) - 3*A278219(1+n))/2;
for(n=0, 16383, write("b286241.txt", n, " ", A286241(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 a003188(n): return n^int(n/2)
def a243353(n): return a005940(1 + a003188(n))
def a278219(n): return a046523(a243353(n))
def a(n): return T(a278219(n), a278219(n + 1)) # Indranil Ghosh, May 07 2017

(define (A286241 n) (* (/ 1 2) (+ (expt (+ (A278219 n) (A278219 (+ 1 n))) 2) (- (A278219 n)) (- (* 3 (A278219 (+ 1 n)))) 2)))

a(n) = (1/2)*(2+((A278219(n)+A278219(1+n))^2) - A278219(n) - 3*A278219(1+n)).

cp's OEIS Frontend

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

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A286255 Compound filter: a(n) = P(A046523(n), A046523(1+n)), where P(n,k) is sequence A000027 used as a pairing function.

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A286242 Compound filter: a(n) = P(A278222(n), A278219(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A286241 Compound filter: a(n) = P(A278219(n), A278219(1+n)), where P(n,k) is sequence A000027 used as a pairing function.

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Programs

Mathematica

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