A048673 -id:A048673 - cp's OEIS frontend

A246351 Numbers k such that A048673(k) < k.

5, 7, 11, 13, 17, 19, 22, 23, 29, 31, 34, 37, 38, 41, 43, 46, 47, 51, 53, 55, 58, 59, 61, 62, 65, 67, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 121, 122, 123, 127, 129, 131, 133, 134, 137, 139, 141, 142, 143, 145, 146, 149, 151, 155, 157, 158, 159

Offset: 1

Antti Karttunen, Aug 24 2014

nonn

The growth rate of the sequence seems to converge:
       a(100) = 217
      a(1000) = 2231
     a(10000) = 21535
    a(100000) = 214647
   a(1000000) = 2155903
  a(10000000) = 21553153
Please see comments in A246282.

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

Complement: A246352.
Setwise difference of A246281 and A048674.
Cf. A048673, A246371, A246282.

default(primelimit, 2^22);
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus
A048673(n) = (A003961(n)+1)/2;
isA246351(n) = (A048673(n) < n);
n = 0; i = 0; while(i < 10000, n++; if(isA246351(n), i++; write("b246351.txt", i, " ", n)));

;; With Antti Karttunen's IntSeq-library.
(define A246351 (MATCHING-POS 1 1 (lambda (n) (< (A048673 n) n))))

A304759 Binary encoding of 1-digits in ternary representation of A048673(n).

Original entry on oeis.org

1, 0, 2, 2, 3, 0, 0, 6, 7, 4, 1, 2, 4, 4, 0, 14, 5, 12, 6, 10, 9, 0, 4, 6, 1, 0, 4, 10, 5, 8, 1, 30, 8, 8, 14, 26, 2, 8, 13, 22, 3, 16, 0, 2, 17, 12, 8, 14, 1, 0, 10, 2, 10, 0, 9, 22, 3, 8, 11, 18, 9, 0, 18, 62, 0, 20, 12, 18, 1, 24, 13, 54, 15, 0, 28, 18, 0, 24, 12, 46, 37, 4, 8, 34, 7, 4, 0, 6, 11, 32, 23, 26, 22, 0

Offset: 1

Antti Karttunen, May 30 2018

Compare the logarithmic scatterplot to those of A291759, A292250 and A304760.

Antti Karttunen, Table of n, a(n) for n = 1..16383
Index entries for sequences related to binary expansion of n

Cf. A048673, A289813, A304758 (rgs-transform), A340381.
Cf. also A291759, A292250, A304760, A305295.
Cf. A340376 (positions of zeros), A340378 (binary weight).

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2;
A289813(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From A289813
A304759(n) = A289813(A048673(n));

a(n) = A289813(A048673(n)).

A243501 Permutation of even numbers: a(n) = 2*A048673(n).

Original entry on oeis.org

2, 4, 6, 10, 8, 16, 12, 28, 26, 22, 14, 46, 18, 34, 36, 82, 20, 76, 24, 64, 56, 40, 30, 136, 50, 52, 126, 100, 32, 106, 38, 244, 66, 58, 78, 226, 42, 70, 86, 190, 44, 166, 48, 118, 176, 88, 54, 406, 122, 148, 96, 154, 60, 376, 92, 298, 116, 94, 62, 316, 68, 112, 276, 730, 120

Offset: 1

Antti Karttunen, Jul 21 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A243502, A245447, A245448, A244319, A245605-A245606, A245607-A245608.

a(n) = 2*A048673(n).
a(n) = A003961(n) + 1.
a(n) = A243502(A245447(n)).

A246352 Numbers n such that A048673(n) >= n.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 26, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 49, 50, 52, 54, 56, 57, 60, 63, 64, 66, 68, 69, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 91, 92, 93, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 125, 126, 128, 130, 132

Offset: 1

Antti Karttunen, Aug 24 2014

nonn

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

Complement: A246351
Union of A246282 and A048674.
Subsequence: A029744 (gives the positions of records in A048673).
Cf. A246372.

default(primelimit, 2^22);
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus
A048673(n) = (A003961(n)+1)/2;
isA246352(n) = (A048673(n) >= n);
n = 0; i = 0; while(i < 10000, n++; if(isA246352(n), i++; write("b246352.txt", i, " ", n)));
(Scheme, with Antti Karttunen's IntSeq-library)
(define A246352 (MATCHING-POS 1 1 (lambda (n) (>= (A048673 n) n))))

A253889 a(n) = A048673(floor(A064216(n)/2)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 3, 8, 14, 4, 13, 5, 5, 7, 17, 6, 6, 18, 7, 38, 32, 8, 28, 23, 9, 15, 11, 10, 26, 16, 11, 41, 53, 12, 33, 39, 13, 10, 113, 14, 43, 12, 15, 22, 63, 16, 25, 59, 17, 203, 74, 18, 48, 30, 19, 188, 50, 20, 122, 68, 21, 9, 149, 22, 138, 83, 23, 60, 86, 24, 35, 29, 25, 73, 62, 26, 24, 123, 27, 27, 128, 28, 313

Offset: 1

Antti Karttunen, Jan 22 2015

nonn

When A048673 is represented as a binary tree, then any non-root node k (>= 2) which contains value n = A048673(k) has as its parent a(n) = A048673(floor(k/2)).

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A048673, A064216, A253888, A253893.

f[n_] := Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1]; g[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n; Array[Floor@ g[Floor[f[#]/2]] &, 84] (* Michael De Vlieger, Sep 16 2017 *)

(define (A253889 n) (A048673 (floor->exact (/ (A064216 n) 2))))

a(n) = A048673(floor(A064216(n)/2)).
Other identities. For all n >= 0:
a(3n+2) = n+1.

A286585 a(n) = A053735(A048673(n)).

Original entry on oeis.org

1, 2, 1, 3, 2, 4, 2, 4, 3, 3, 3, 5, 1, 5, 2, 5, 2, 4, 2, 4, 2, 4, 3, 6, 5, 6, 3, 6, 4, 7, 3, 6, 3, 3, 3, 5, 3, 5, 5, 5, 4, 3, 4, 5, 4, 6, 1, 7, 5, 6, 4, 7, 2, 8, 4, 7, 4, 5, 3, 8, 4, 4, 4, 7, 4, 6, 2, 4, 5, 6, 3, 6, 4, 6, 5, 6, 4, 6, 4, 6, 7, 5, 3, 4, 5, 7, 6, 6, 5, 5, 4, 7, 3, 8, 1, 8, 5, 6, 3, 7, 6, 7, 2, 8

Offset: 1

Antti Karttunen, May 31 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000079, A048673, A053735, A286582, A286583, A286584, A286586.

from sympy.ntheory.factor_ import digits
from sympy import factorint, nextprime
from operator import mul
def a053735(n): return sum(digits(n, 3)[1:])
def a048673(n):
    f = factorint(n)
    return 1 if n==1 else (1 + reduce(mul, [nextprime(i)**f[i] for i in f]))//2
def a(n): return a053735(a048673(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 12 2017

(define (A286585 n) (A053735 (A048673 n)))

a(n) = A053735(A048673(n)).

For all n >= 0, a(A000079(n)) = n+1.

A323893 Dirichlet inverse of A048673, where A048673(n) = (A003961(n)+1) / 2, and A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

Original entry on oeis.org

1, -2, -3, -1, -4, 4, -6, -2, -4, 5, -7, 3, -9, 7, 6, -4, -10, 8, -12, 4, 8, 8, -15, 8, -9, 10, -12, 6, -16, 5, -19, -8, 9, 11, 9, 8, -21, 13, 11, 11, -22, 11, -24, 7, 16, 16, -27, 20, -25, 18, 12, 9, -30, 32, 10, 17, 14, 17, -31, 6, -34, 20, 24, -16, 12, 14, -36, 10, 17, 20, -37, 16, -40, 22, 27, 12, 12, 20, -42, 28, -36, 23, -45, 12, 13

Offset: 1

Antti Karttunen, Feb 08 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..16383
Index entries for sequences related to prime indices in the factorization of n.

Cf. A003961, A048673, A323894, A349134, A378520 (Möbius transform).

up_to = 20000;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2;
v323893 = DirInverse(vector(up_to,n,A048673(n)));
A323893(n) = v323893[n];

memoA323893 = Map();
A323893(n) = if(1==n,1,my(v); if(mapisdefined(memoA323893,n,&v), v, v = -sumdiv(n,d,if(dA048673(n/d)*A323893(d),0)); mapput(memoA323893,n,v); (v))); \\ Antti Karttunen, Nov 30 2024

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA048673(n/d) * a(d).

a(n) = A349134(A003961(n)). - Antti Karttunen, Nov 30 2024

A278224 a(n) = A046523(A048673(n)).

Original entry on oeis.org

1, 2, 2, 2, 4, 8, 6, 6, 2, 2, 2, 2, 4, 2, 12, 2, 6, 6, 12, 32, 12, 12, 6, 12, 4, 6, 12, 12, 16, 2, 2, 6, 6, 2, 6, 2, 6, 6, 2, 6, 6, 2, 24, 2, 24, 12, 8, 6, 2, 6, 48, 6, 30, 12, 6, 2, 6, 2, 2, 6, 6, 24, 30, 6, 60, 12, 36, 6, 2, 12, 2, 12, 24, 6, 6, 24, 72, 128, 30, 12, 2, 6, 12, 24, 2, 2, 30, 48, 4, 2, 6, 2, 6, 48, 16, 96, 6, 30, 2, 6, 12, 6, 24, 30, 2, 2, 6

Offset: 1

Antti Karttunen, Nov 16 2016

nonn

This sequence works as a "sentinel" for sequence A048673 by matching to any other sequence that is obtained as f(A048673(n)), where f(n) is any function that depends only on the prime signature of n (see the index entry for "sequences computed from exponents in ..."). As of Nov 11 2016 no such sequences were present in the database.

Antti Karttunen, Table of n, a(n) for n = 1..10500
Index entries for sequences computed from exponents in factorization of n

Cf. A046523, A048673, A278223.

from sympy import factorint, nextprime
from operator import mul
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 a048673(n):
    f = factorint(n)
    return 1 if n==1 else (1 + reduce(mul, [nextprime(i)**f[i] for i in f]))//2
def a(n): return a046523(a048673(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 12 2017

(define (A278224 n) (A046523 (A048673 n)))

a(n) = A046523(A048673(n)).

A292247 a(n) = A292244(A048673(n)).

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 3, 0, 0, 0, 0, 4, 5, 6, 7, 0, 0, 0, 1, 0, 0, 0, 1, 8, 8, 10, 11, 12, 12, 14, 14, 0, 1, 0, 1, 0, 1, 2, 2, 0, 0, 0, 1, 0, 0, 2, 3, 16, 16, 16, 17, 20, 21, 22, 22, 24, 24, 24, 24, 28, 28, 28, 29, 0, 1, 2, 3, 0, 0, 2, 2, 0, 0, 2, 3, 4, 5, 4, 5, 0, 0, 0, 1, 0, 0, 2, 3, 0, 0, 0, 0, 4, 5, 6, 7, 32, 33, 32, 32, 32, 32, 34, 35, 40, 40

Offset: 1

Antti Karttunen, Sep 16 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A048673, A292244, A292248.

a(n) = A292244(A048673(n)).

a(n) + A292248(n) = n.

A292248 a(n) = A292245(A048673(n)).

Original entry on oeis.org

1, 2, 2, 4, 5, 4, 4, 8, 9, 10, 11, 8, 8, 8, 8, 16, 17, 18, 18, 20, 21, 22, 22, 16, 17, 16, 16, 16, 17, 16, 17, 32, 32, 34, 34, 36, 36, 36, 37, 40, 41, 42, 42, 44, 45, 44, 44, 32, 33, 34, 34, 32, 32, 32, 33, 32, 33, 34, 35, 32, 33, 34, 34, 64, 64, 64, 64, 68, 69, 68, 69, 72, 73, 72, 72, 72, 72, 74, 74, 80, 81, 82, 82, 84, 85, 84, 84, 88, 89, 90, 91

Offset: 1

Antti Karttunen, Sep 16 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A048673, A292244, A292247.

f[n_] := Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1]; g[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n; Map[FromDigits[#, 2] &[IntegerDigits[#, 3] /. 2 -> 0] &, Map[a, Array[g, 91]]] (* Michael De Vlieger, Sep 16 2017 *)

a(n) = A292245(A048673(n)).

a(n) + A292247(n) = n.

cp's OEIS Frontend

A246351 Numbers k such that A048673(k) < k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Scheme

A304759 Binary encoding of 1-digits in ternary representation of A048673(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A243501 Permutation of even numbers: a(n) = 2*A048673(n).

Views

Author

Keywords

Links

Crossrefs

Formula

A246352 Numbers n such that A048673(n) >= n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A253889 a(n) = A048673(floor(A064216(n)/2)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Scheme

Formula

A286585 a(n) = A053735(A048673(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Python

Scheme

Formula

A323893 Dirichlet inverse of A048673, where A048673(n) = (A003961(n)+1) / 2, and A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A278224 a(n) = A046523(A048673(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Scheme

Formula

A292247 a(n) = A292244(A048673(n)).

Views