A002487 -id:A002487 - cp's OEIS frontend

A324287 a(n) = A002487(A005187(n)).

0, 1, 2, 1, 3, 1, 3, 5, 4, 1, 4, 7, 5, 7, 7, 5, 5, 1, 5, 9, 7, 10, 11, 8, 7, 9, 9, 7, 13, 8, 3, 10, 6, 1, 6, 11, 9, 13, 15, 11, 10, 13, 14, 11, 21, 13, 5, 17, 9, 11, 11, 9, 19, 12, 5, 18, 19, 11, 3, 13, 7, 18, 15, 4, 7, 1, 7, 13, 11, 16, 19, 14, 13, 17, 19, 15, 29, 18, 7, 24, 13, 16, 17, 14, 30, 19, 8, 29, 31, 18, 5, 22, 12, 31, 26, 7

Offset: 0

Antti Karttunen, Feb 20 2019

nonn

The motivation for this kind of sequence was a question: what kind of simply defined non-injective functions f exist such that this sequence can be defined as their function, e.g., as a(n) = g(f(n)), where g is a nontrivial integer-valued function? The same question can also be asked about A324288, A324337 and A324338. Note that A005187, A283477 and A006068 used in their definitions are all injections. Of course, A324377(n) = A000265(A005187(n)) fills the bill as A002487(n) = A002487(A000265(n)), but are there any less obvious solutions? - Antti Karttunen, Feb 28 2019

Antti Karttunen, Table of n, a(n) for n = 0..16385
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to Stern's sequences

Cf. A000265, A002487, A005187, A006068, A283477, A324286, A324288, A324293, A324337, A324338, A324377.

A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); }; \\ Modified from the one given in A002487, sign not actually needed here.
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A324287(n) = A002487(A005187(n));

from functools import reduce
def A324287(n): return sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin((n<<1)-n.bit_count())[-1:2:-1],(1,0))) if n else 0 # Chai Wah Wu, May 05 2023

a(n) = A002487(A005187(n)).
a(n) = A324286(A283477(n)).
a(n) = A002487(A324377(n)).

A324288 a(n) = A002487(1+A005187(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 4, 5, 2, 1, 5, 7, 3, 7, 2, 5, 8, 1, 6, 9, 4, 10, 3, 8, 13, 9, 2, 7, 12, 8, 11, 10, 7, 1, 7, 11, 5, 13, 4, 11, 18, 13, 3, 11, 19, 13, 18, 17, 12, 11, 2, 9, 16, 12, 17, 18, 13, 11, 14, 13, 10, 18, 11, 4, 13, 1, 8, 13, 6, 16, 5, 14, 23, 17, 4, 15, 26, 18, 25, 24, 17, 16, 3, 14, 25, 19, 27, 29, 21, 18, 23, 22, 17, 31, 19, 7, 23, 13, 2

Offset: 0

Antti Karttunen, Feb 20 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to Stern's sequences

Cf. A002487, A005187, A324287, A324116.

A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); }; \\ Modified from the one given in A002487, sign not actually needed here.
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A324288(n) = A002487(1+A005187(n));

from functools import reduce
def A324288(n): return sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(1+(n<<1)-n.bit_count())[-1:2:-1],(1,0))) if n else 1 # Chai Wah Wu, May 05 2023

a(n) = A002487(1+A005187(n)).

A284013 a(n) = n - A002487(n).

Original entry on oeis.org

0, 0, 1, 1, 3, 2, 4, 4, 7, 5, 7, 6, 10, 8, 11, 11, 15, 12, 14, 12, 17, 13, 17, 16, 22, 18, 21, 19, 25, 22, 26, 26, 31, 27, 29, 26, 32, 26, 31, 29, 37, 30, 34, 30, 39, 33, 39, 38, 46, 40, 43, 39, 47, 40, 46, 44, 53, 47, 51, 48, 56, 52, 57, 57, 63, 58, 60, 56, 63, 55, 61, 58, 68, 58, 63, 57, 69, 60, 68, 66, 77, 67, 71, 64, 76, 64

Offset: 0

Antti Karttunen, Mar 23 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A002487, A265397, A283474, A284007.

a[n_] :=If[n<2, n, If[ EvenQ[n], a[n/2], a[(n - 1)/2] + a[(n + 1)/2]]]; Table[n - a[n], {n , 0, 100}] (* Indranil Ghosh, Mar 23 2017 *)

A(n) = if(n<2, n, if(n%2, A(n\2) + A((n + 1)/2), A(n/2)));
for(n=0, 151, print1(n - A(n),", ")) \\ Indranil Ghosh, Mar 23 2017

def a(n): return n if n<2 else (a(n//2) if n%2==0 else a((n - 1)//2) + a((n + 1)//2))
print([n - a(n) for n in range(101)]) # Indranil Ghosh, Mar 23 2017

from functools import reduce
def A284013(n): return n-sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n)[-1:2:-1],(1,0))) if n else 0 # Chai Wah Wu, May 18 2023

(define (A284013 n) (- n (A002487 n))) ;; Code for A002487 given under that entry.

a(n) = n - A002487(n).

A318307 Multiplicative with a(p^e) = 2^A002487(e).

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 8, 2, 8, 4, 4, 4, 4, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 8, 4, 8, 4, 4, 2, 8, 2, 4, 4, 4, 4, 8, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4, 8, 2, 8, 4, 4, 4, 4, 4, 16, 2, 4, 4, 4, 2, 8, 2, 8, 8

Offset: 1

Antti Karttunen, Aug 29 2018

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A318306, A318316, A318470, A318667.

Differs from A037445 for the first time at n=32, where a(32) = 8, while A037445(32) = 4.

f[m_] := Module[{a = 1, b = 0, n = m}, While[n > 0, If[OddQ[n], b += a, a += b]; n = Floor[n/2]]; b]; Array[Times @@ Map[2^f@ # &, FactorInteger[#][[All, -1]] ] - Boole[# == 1] &, 105] (* after Jean-François Alcover at A002487 *)

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A318307(n) = factorback(apply(e -> 2^A002487(e),factor(n)[,2]));

from functools import reduce
from sympy import factorint
def A318307(n): return 1<Chai Wah Wu, May 18 2023

a(n) = 2^A318306(n).
a(n) = A061142(A318470(n)).
a(n^2) = a(n).
a(A003557(n^2)) = A318316(n).
Dirichlet convolution square of A318667(n)/A317934(n).

A323892 Lexicographically earliest sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A033879(i) = A033879(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 5, 1, 6, 7, 8, 9, 10, 3, 11, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 8, 21, 22, 23, 24, 25, 1, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 35, 47, 48, 49, 50, 51, 52, 53, 54, 55, 1, 56, 57, 58, 8, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 47, 72, 73, 74, 75, 76, 77, 78, 79, 80, 31, 81

Offset: 1

Antti Karttunen, Feb 09 2019

nonn

Restricted growth sequence transform of the ordered pair [A002487(n), A033879(n)].

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to Stern's sequences

Cf. A002487, A033879.

Cf. also A318310, A323889.

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; };
A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A033879(n) = (2*n-sigma(n));
A323892aux(n) = [A002487(n), A033879(n)];
v323892 = rgs_transform(vector(up_to,n,A323892aux(n)));
A323892(n) = v323892[n];

a(2^n) = 1 for all n >= 0.

A323903 a(n) = A002487(A122111(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 4, 2, 1, 3, 1, 2, 4, 3, 1, 4, 1, 3, 4, 2, 1, 3, 8, 2, 7, 3, 1, 4, 1, 5, 4, 2, 8, 8, 1, 2, 4, 3, 1, 4, 1, 3, 7, 2, 1, 5, 14, 12, 4, 3, 1, 9, 8, 3, 4, 2, 1, 8, 1, 2, 7, 5, 8, 4, 1, 3, 4, 12, 1, 6, 1, 2, 18, 3, 14, 4, 1, 5, 9, 2, 1, 8, 8, 2, 4, 3, 1, 9, 14, 3, 4, 2, 8, 5, 1, 16, 7, 6, 1, 4, 1, 3, 18

Offset: 1

Antti Karttunen, Feb 09 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to Stern's sequences

Cf. A002487, A122111, A322865.

Cf. also A323168, A323174, A323902.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A323903(n) = A002487(A122111(n));

a(n) = A002487(A122111(n)) = A002487(A322865(n)).

a(p) = 1 for all primes p.

A324115 a(n) = A002487(A323244(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 0, 3, 1, 3, 1, 1, 1, 2, 1, 2, 1, 4, -2, 4, 1, 5, -1, 7, 1, 5, 1, 3, 1, 4, 3, 11, -1, 3, 1, 1, -2, 5, 1, 4, 1, 6, 1, 13, 1, 7, -2, 7, 1, 7, 1, 3, -7, 9, -2, 25, 1, 8, 1, 76, 1, 5, 3, 8, 1, 21, 7, 3, 1, 7, 1, 31, 3, 31, -3, 13, 1, 10, -2, 199, 1, 5, -4, 101, -18, 4, 1, 2, -12, 43, 11, 266, -5, 9, 1, 11, -1, 4, 1, 6, 1, 13

Offset: 1

Antti Karttunen, Feb 20 2019

sign

If there are no odd perfect numbers then A324201 gives the positions of all zeros after the initial a(1) = 0.

Antti Karttunen, Table of n, a(n) for n = 1..4473

Cf. A002487, A033879, A156552, A323244, A323902, A324201, A324116, A324117.

A002487(n) = if(abs(n)<=1, n, A002487(n\2) + if( n%2, A002487(n\2 + 1))); \\ This version works consistently also with negative arguments, so that a(-n) = -a(n). Except that it is very slow on large n.
A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); }; \\ So we use this one, modified from the one given in A002487
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A323244(n) = if(1==n, 0, my(k=A156552(n)); (2*k)-sigma(k));
A324115(n) = A002487(A323244(n));

a(n) = A002487(A323244(n)), with the definition of A002487 extended to the negative arguments so that A002487(-n) = -A002487(n).

a(A324201(n)) = 0.

A324116 a(n) = A002487(1+A323247(n)) = A324288(A156552(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 4, 1, 2, 5, 5, 1, 3, 1, 6, 7, 8, 1, 2, 1, 4, 9, 7, 1, 13, 7, 8, 5, 5, 1, 3, 1, 7, 11, 9, 10, 12, 1, 10, 13, 18, 1, 4, 1, 6, 8, 11, 1, 12, 9, 2, 15, 7, 1, 11, 13, 23, 17, 12, 1, 19, 1, 13, 11, 13, 16, 5, 1, 8, 19, 3, 1, 13, 1, 14, 7, 9, 13, 6, 1, 17, 10, 15, 1, 26, 19, 16, 21, 28, 1, 18, 17, 10, 23, 17, 22, 23, 1, 2

Offset: 1

Antti Karttunen, Feb 20 2019

nonn

Like A323902, this also has quite a moderate growth rate, even though a certain subset of terms of A156552 soon grow quite big.

Cf. A002487, A005187, A156552, A323247, A323902, A324115, A324288.

A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); }; \\ So we use this one, modified from the one given in A002487
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A324288(n) = A002487(1+A005187(n));
A324116(n) = A324288(A156552(n));

a(n) = A002487(1+A323247(n)) = A324288(A156552(n)).

a(p) = 1 for all primes p.

A020946 a(n) is the smallest number k such that A002487(k) = n.

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 33, 19, 21, 35, 39, 37, 45, 43, 69, 73, 93, 77, 75, 83, 189, 85, 141, 139, 153, 151, 147, 155, 267, 149, 165, 173, 279, 275, 171, 283, 315, 277, 537, 325, 297, 293, 579, 301, 309, 365, 333, 299, 567, 331, 339, 553, 549, 563, 1275, 341, 585, 565, 615, 629

Offset: 0

N. J. A. Sloane and David W. Wilson, Jun 27 2002

nonn

			A002487(33) = 6 and this is the first time 6 appears, so a(6) = 33.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Equals A020950 + 1.

Cf. A020943-A020945, A002487, A020947-A020950.

aa = {}; a[0] = 0; a[1] = 1; a[n_] := a[n] = If[EvenQ[n], a[n/2], a[(n - 1)/2] + a[(n + 1)/2]]; Do[k = 0; While[a[k] != p, k++]; AppendTo[aa, k], {p, 0, 100}]; aa (* Artur Jasinski, Dec 06 2010 *)

fusc(n)={my(a=1, b=0);while(n,if(bitand(n, 1), b+=a, a+=b);n>>=1); b};
list(N)={
    my(v=vector(N),k);
    forstep(n=1,9e99,2,
        k=fusc(n);
        if(k<=N && !v[k],
            v[k]=n;
            if(vecmin(v),return(v))
        )
    )
}; \\ Charles R Greathouse IV, Dec 20 2011

from itertools import count
from functools import reduce
def A020946(n): return next(filter(lambda k:sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(k)[-1:2:-1],(1,0)))==n,count(1))) if n else 0 # Chai Wah Wu, May 05 2023

A091945 Numbers n such that there is no k, 0 <= k < n, satisfying A002487(k) = A002487(n).

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 19, 21, 33, 35, 37, 39, 43, 45, 69, 73, 75, 77, 83, 85, 93, 139, 141, 147, 149, 151, 153, 155, 165, 171, 173, 189, 267, 275, 277, 279, 283, 293, 297, 299, 301, 309, 315, 325, 331, 333, 339, 341, 365, 537, 549, 553, 555, 563, 565, 567, 579, 585

Offset: 1

Benoit Cloitre, Mar 11 2004

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..7297
Rémy Sigrist, PARI program
Index entries for sequences related to Stern's sequences

Cf. A002487, A091926.

isok(n) = (A091926(n) == n); \\ Michel Marcus, Dec 06 2013

PARI
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See Links section.
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Leading 0 inserted and name adapted by Rémy Sigrist, Dec 07 2022

cp's OEIS Frontend

A324287 a(n) = A002487(A005187(n)).

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A324288 a(n) = A002487(1+A005187(n)).

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A284013 a(n) = n - A002487(n).

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A318307 Multiplicative with a(p^e) = 2^A002487(e).

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A323892 Lexicographically earliest sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A033879(i) = A033879(j), for all i, j >= 1.

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A323903 a(n) = A002487(A122111(n)).

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A324115 a(n) = A002487(A323244(n)).

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A324116 a(n) = A002487(1+A323247(n)) = A324288(A156552(n)).

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