A087808 -id:A087808 - cp's OEIS frontend

A332445 Numbers k of the form 4m+1 for which A087808(sigma(k)) is equal to 2*A087808(k).

2009, 19377, 37809, 59373, 74673, 115677, 270041, 310329, 354609, 357309, 720425, 732321, 841437, 2071737, 2612269, 3131149, 3866461, 3930929, 5172093, 5593981, 7118753, 7903961, 8224173, 9327393, 9438129, 11452321, 12708025, 18857209, 18861889, 18875313, 19110321, 20278269, 20709225, 20950061, 23963597, 24895153

Offset: 1

Antti Karttunen, Feb 14 2020

nonn

Numbers k such that A332224(k) is equal to A087808(2*k) and k == 1 mod 4.
Notably, the only square among the first 299 terms is a(248) = 5808421369 = 76213^2. sigma(5808421369) = 5808497583 == 3 (mod 4) == 7 (mod 8). Of the remaining 298 terms < 2^33, 92 are such that sigma(k) == 6 (mod 8) and 206 are such that sigma(k) == 2 (mod 8), that is, are terms of A332227.
Question: Why the terms come in clusters? Compare also the scatterplots of A087808 and A332224, and a similar sequence A332465.

Intersection of A016813 and A332446.
Cf. also A228058, A332227, A332465.
Cf. A000203, A087808, A286357, A332224, A332225, A332458.

A087808(n) = if(n<1, 0, if(n%2==0, 2*A087808(n/2), A087808((n-1)/2)+1));
isA332445(n) = ((1==(n%4))&&(2*A087808(n)==A087808(sigma(n))));

A332446 Numbers k for which A087808(sigma(k)) is equal to A087808(2*k).

Original entry on oeis.org

3, 6, 11, 19, 28, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 216, 227, 251, 267, 283, 286, 307, 331, 347, 379, 419, 443, 467, 491, 496, 499, 523, 547, 563, 571, 587, 598, 619, 643, 659, 683, 691, 726, 739, 787, 811, 827, 859, 883, 907, 947, 971, 1019, 1051, 1091, 1123, 1163, 1171, 1187, 1259, 1283, 1291, 1307, 1427, 1451

Offset: 1

Antti Karttunen, Feb 14 2020

nonn

Conjecture: includes all terms of A007520. - Bill McEachen, Dec 10 2023

Cf. A000203, A007520, A087808.
Subsequences: A000396, A332445.
Cf. A331751, A331752, A332208 for similar sequences.

A087808(n) = if(n<1, 0, if(n%2==0, 2*A087808(n/2), A087808((n-1)/2)+1));
isA332446(n) = (A087808(2*n)==A087808(sigma(n)));

A332443 Numbers k such that A332224(k) = A087808(sigma(k)) is odd.

Original entry on oeis.org

1, 4, 9, 16, 18, 25, 49, 50, 64, 81, 100, 121, 169, 200, 256, 338, 361, 392, 400, 441, 450, 529, 578, 625, 648, 676, 722, 729, 784, 800, 841, 900, 961, 1024, 1089, 1156, 1225, 1250, 1296, 1352, 1369, 1568, 1600, 1682, 1800, 1849, 2025, 2116, 2209, 2312, 2401, 2450, 2592, 2601, 2704, 2738, 3042, 3136, 3200, 3249, 3362, 3364, 3481, 3600

Offset: 1

Antti Karttunen, Feb 13 2020

nonn

Numbers k such that A332224(k) is odd, or equally, that A332448(k) is zero.

Antti Karttunen, Table of n, a(n) for n = 1..26213
Index entries for sequences related to sigma(n)

Cf. A000203, A087808, A332224, A332225, A332444.

Subsequence of A028982. Positions of zeros in A332448.

A087808(n) = if(n<1, 0, if(n%2==0, 2*A087808(n/2), A087808((n-1)/2)+1));
isA332443(n) = (A087808(sigma(n))%2);
A332443list(u) = { my(v1=vector(2*u,n,2*(n^2)), v2=vector(sqrtint(v1[#v1]),n,n^2)); select(isA332443,Vec(setunion(v1,v2))); };
v332443 = A332443list(8192);
A332443(n) = v332443[n];

A332448 a(n) = A007814(A087808(sigma(n))).

Original entry on oeis.org

0, 1, 2, 0, 2, 3, 3, 2, 0, 1, 3, 2, 1, 4, 4, 0, 1, 0, 2, 1, 5, 2, 4, 4, 0, 1, 3, 3, 3, 3, 5, 1, 5, 2, 5, 1, 2, 4, 3, 1, 1, 6, 4, 2, 1, 3, 5, 2, 0, 0, 3, 1, 2, 5, 3, 5, 4, 1, 4, 3, 1, 6, 3, 0, 2, 4, 2, 2, 6, 4, 3, 1, 1, 1, 2, 3, 6, 3, 4, 1, 0, 2, 2, 5, 3, 2, 5, 2, 1, 1, 4, 3, 7, 4, 5, 3, 1, 4, 2, 0, 2, 4, 3, 1, 7

Offset: 1

Antti Karttunen, Feb 14 2020

nonn

Cf. A000203, A007814, A087808, A332224, A332447.

Cf. A332443 (positions of zeros).

A087808(n) = if(n<1, 0, if(n%2==0, 2*A087808(n/2), A087808((n-1)/2)+1));
A332448(n) = valuation(A087808(sigma(n)),2);

a(n) = A007814(A332224(n)) = A332447(A000203(n)) = A007814(A087808(A000203(n))).

A332444 Numbers k that are squares or twice-squares, but for which A087808(sigma(k)) is even.

Original entry on oeis.org

2, 8, 32, 36, 72, 98, 128, 144, 162, 196, 225, 242, 288, 289, 324, 484, 512, 576, 882, 968, 1058, 1152, 1444, 1458, 1521, 1681, 1764, 1922, 1936, 2048, 2178, 2304, 2500, 2809, 2888, 2916, 3025, 3528, 3698, 3872, 4418, 4608, 4802, 5000, 5329, 5776, 5832, 6498, 7056, 7396, 7569, 7688, 7744, 7921, 7938, 8192, 8281

Offset: 1

Antti Karttunen, Feb 13 2020

nonn

Numbers k for which A000203(k) is odd and A332224(k) is even.

Antti Karttunen, Table of n, a(n) for n = 1..19515
Index entries for sequences related to sigma(n)

Setwise difference of A028982 \ A332443.

Cf. A000203, A087808, A332224, A332225.

A087808(n) = if(n<1, 0, if(n%2==0, 2*A087808(n/2), A087808((n-1)/2)+1));
isA332444(n) = (((!(n%2)&&issquare(n/2))||issquare(n)) && !(A087808(sigma(n))%2));
A332444list(u) = { my(v1=vector(2*u,n,2*(n^2)), v2=vector(sqrtint(v1[#v1]),n,n^2)); select(isA332444,Vec(setunion(v1,v2))); };
v332444 = A332444list(12000);
A332444(n) = v332444[n];

A332447 a(n) = A007814(A087808(n)).

Original entry on oeis.org

0, 1, 1, 2, 0, 2, 0, 3, 0, 1, 2, 3, 0, 1, 2, 4, 0, 1, 1, 2, 0, 3, 0, 4, 0, 1, 1, 2, 0, 3, 0, 5, 0, 1, 1, 2, 0, 2, 0, 3, 0, 1, 3, 4, 0, 1, 1, 5, 0, 1, 1, 2, 0, 2, 0, 3, 0, 1, 3, 4, 0, 1, 1, 6, 0, 1, 1, 2, 0, 2, 0, 3, 0, 1, 2, 3, 0, 1, 3, 4, 0, 1, 1, 2, 0, 4, 0, 5, 0, 1, 1, 2, 0, 2, 0, 6, 0, 1, 1, 2, 0, 2, 0, 3, 0

Offset: 1

Antti Karttunen, Feb 14 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A007814, A087808, A332448.

Cf. A000079 (positions of records), A079523 (of zeros).

A087808(n) = if(n<1, 0, if(n%2==0, 2*A087808(n/2), A087808((n-1)/2)+1));
A332447(n) = valuation(A087808(n),2);

a(n) = A007814(A087808(n)).

For n >= 0, a(2^n) = n. [These are the first occurrences of each n]

A048675 If n = p_i^e_i * ... * p_k^e_k, p_i < ... < p_k primes (with p_i = prime(i)), then a(n) = (1/2) * (e_i * 2^i + ... + e_k * 2^k).

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 8, 3, 4, 5, 16, 4, 32, 9, 6, 4, 64, 5, 128, 6, 10, 17, 256, 5, 8, 33, 6, 10, 512, 7, 1024, 5, 18, 65, 12, 6, 2048, 129, 34, 7, 4096, 11, 8192, 18, 8, 257, 16384, 6, 16, 9, 66, 34, 32768, 7, 20, 11, 130, 513, 65536, 8, 131072, 1025, 12, 6, 36, 19

Offset: 1

Antti Karttunen, Jul 14 1999

nonn

The original motivation for this sequence was to encode the prime factorization of n in the binary representation of a(n), each such representation being unique as long as this map is restricted to A005117 (squarefree numbers, resulting a permutation of nonnegative integers A048672) or any of its subsequence, resulting an injective function like A048623 and A048639.
However, also the restriction to A260443 (not all terms of which are squarefree) results a permutation of nonnegative integers, namely A001477, the identity permutation.
When a polynomial with nonnegative integer coefficients is encoded with the prime factorization of n (e.g., as in A206296, A260443), then a(n) gives the evaluation of that polynomial at x=2.
The primitive completely additive integer sequence that satisfies a(n) = a(A225546(n)), n >= 1. By primitive, we mean that if b is another such sequence, then there is an integer k such that b(n) = k * a(n) for all n >= 1. - Peter Munn, Feb 03 2020
If the binary rank of an integer partition y is given by Sum_i 2^(y_i-1), and the Heinz number is Product_i prime(y_i), then a(n) is the binary rank of the integer partition with Heinz number n. Note the function taking a set s to Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices), and the function taking a multiset m to Product_i prime(m_i) is the inverse of A112798 (prime indices). - Gus Wiseman, May 22 2024

			From _Gus Wiseman_, May 22 2024: (Start)
The A018819(7) = 6 cases of binary rank 7 are the following, together with their prime indices:
   30: {1,2,3}
   40: {1,1,1,3}
   54: {1,2,2,2}
   72: {1,1,1,2,2}
   96: {1,1,1,1,1,2}
  128: {1,1,1,1,1,1,1}
(End)

Antti Karttunen, Table of n, a(n) for n = 1..1024
Hans Havermann, Factorization of the first 10000 terms, in format [[primes], [exponents]]

Row 2 of A104244.
Similar logarithmic functions: A001414, A056239, A090880, A289506, A293447.
Left inverse of the following sequences: A000079, A019565, A038754, A068911, A134683, A260443, A332824.
A003961, A028234, A032742, A055396, A064989, A067029, A225546, A297845 are used to express relationship between terms of this sequence.
Cf. also A048623, A048676, A099884, A277896 and tables A277905, A285325.
Cf. A297108 (Möbius transform), A332813 and A332823 [= a(n) mod 3].
Pairs of sequences (f,g) that satisfy a(f(n)) = g(n), possibly with offset change: (A000203,A331750), (A005940,A087808), (A007913,A248663), (A007947,A087207), (A097248,A048675), (A206296,A000129), (A248692,A056239), (A283477,A005187), (A284003,A006068), (A285101,A028362), (A285102,A068052), (A293214,A001065), (A318834,A051953), (A319991,A293897), (A319992,A293898), (A320017,A318674), (A329352,A069359), (A332461,A156552), (A332462,A156552), (A332825,A000010) and apparently (A163511,A135529).
See comments/formulas in A277333, A331591, A331740 giving their relationship to this sequence.
The formula section details how the sequence maps the terms of A329050, A329332.
A277892, A322812, A322869, A324573, A324575 give properties of the n-th term of this sequence.
The term k appears A018819(k) times.
The inverse transformation is A019565 (Heinz number of binary indices).
The version for distinct prime indices is A087207.
Numbers k such that a(k) is prime are A277319, counts A372688.
Grouping by image gives A277905.
A014499 lists binary indices of prime numbers.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Binary indices:
- listed A048793, sum A029931
- reversed A272020
- opposite A371572, sum A230877
- length A000120, complement A023416
- min A001511, opposite A000012
- max A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
Cf. A000720, A035100, A071814, A096111, A225620, A243055, A304818, A355536, A358136, A372890.
Cf. A087207, A097248.

nthprime := proc(n) local i; if(isprime(n)) then for i from 1 to 1000000 do if(ithprime(i) = n) then RETURN(i); fi; od; else RETURN(0); fi; end; # nthprime(2) = 1, nthprime(3) = 2, nthprime(5) = 3, etc. - this is also A049084.
A048675 := proc(n) local s,d; s := 0; for d in ifactors(n)[ 2 ] do s := s + d[ 2 ]*(2^(nthprime(d[ 1 ])-1)); od; RETURN(s); end;
# simpler alternative
f:= n -> add(2^(numtheory:-pi(t[1])-1)*t[2], t=ifactors(n)[2]):
map(f, [$1..100]); # Robert Israel, Oct 10 2016

a[1] = 0; a[n_] := Total[ #[[2]]*2^(PrimePi[#[[1]]]-1)& /@ FactorInteger[n] ]; Array[a, 100] (* Jean-François Alcover, Mar 15 2016 *)

a(n) = my(f = factor(n)); sum(k=1, #f~, f[k,2]*2^primepi(f[k,1]))/2; \\ Michel Marcus, Oct 10 2016

\\ The following program reconstructs terms (e.g. for checking purposes) from the factorization file prepared by Hans Havermann:
v048675sigs = readvec("a048675.txt");
A048675(n) = if(n<=2,n-1,my(prsig=v048675sigs[n],ps=prsig[1],es=prsig[2]); prod(i=1,#ps,ps[i]^es[i])); \\ Antti Karttunen, Feb 02 2020

from sympy import factorint, primepi
def a(n):
    if n==1: return 0
    f=factorint(n)
    return sum([f[i]*2**(primepi(i) - 1) for i in f])
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jun 19 2017

a(1) = 0, a(n) = 1/2 * (e1*2^i1 + e2*2^i2 + ... + ez*2^iz) if n = p_{i1}^e1*p_{i2}^e2*...*p_{iz}^ez, where p_i is the i-th prime. (e.g. p_1 = 2, p_2 = 3).
Totally additive with a(p^e) = e * 2^(PrimePi(p)-1), where PrimePi(n) = A000720(n). [Missing factor e added to the comment by Antti Karttunen, Jul 29 2015]
From Antti Karttunen, Jul 29 2015: (Start)
a(1) = 0; for n > 1, a(n) = 2^(A055396(n)-1) + a(A032742(n)). [Where A055396(n) gives the index of the smallest prime dividing n and A032742(n) gives the largest proper divisor of n.]
a(1) = 0; for n > 1, a(n) = (A067029(n) * (2^(A055396(n)-1))) + a(A028234(n)).
Other identities. For all n >= 0:
a(A019565(n)) = n.
a(A260443(n)) = n.
a(A206296(n)) = A000129(n).
a(A005940(n+1)) = A087808(n).
a(A007913(n)) = A248663(n).
a(A007947(n)) = A087207(n).
a(A283477(n)) = A005187(n).
a(A284003(n)) = A006068(n).
a(A285101(n)) = A028362(1+n).
a(A285102(n)) = A068052(n).
Also, it seems that a(A163511(n)) = A135529(n) for n >= 1. (End)
a(1) = 0, a(2n) = 1+a(n), a(2n+1) = 2*a(A064989(2n+1)). - Antti Karttunen, Oct 11 2016
From Peter Munn, Jan 31 2020: (Start)
a(n^2) = a(A003961(n)) = 2 * a(n).
a(A297845(n,k)) = a(n) * a(k).
a(n) = a(A225546(n)).
a(A329332(n,k)) = n * k.
a(A329050(n,k)) = 2^(n+k).
(End)
From Antti Karttunen, Feb 02-25 2020, Feb 01 2021: (Start)
a(n) = Sum_{d|n} A297108(d) = Sum_{d|A225546(n)} A297108(d).
a(n) = a(A097248(n)).
For n >= 2:
A001221(a(n)) = A322812(n), A001222(a(n)) = A277892(n).
A000203(a(n)) = A324573(n), A033879(a(n)) = A324575(n).
For n >= 1, A331750(n) = a(A000203(n)).
For n >= 1, the following chains hold:
A293447(n) >= a(n) >= A331740(n) >= A331591(n).
a(n) >= A087207(n) >= A248663(n).
(End)
a(n) = A087207(A097248(n)). - Flávio V. Fernandes, Jul 16 2025

Entry revised by Antti Karttunen, Jul 29 2015

More linking formulas added by Antti Karttunen, Apr 18 2017

A135416 a(n) = A036987(n)*(n+1)/2.

Original entry on oeis.org

1, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane, based on a message from Guy Steele and Don Knuth, Mar 01 2008

nonn

Guy Steele defines a family of 36 integer sequences, denoted here by GS(i,j) for 1 <= i, j <= 6, as follows. a[1]=1; a[2n] = i-th term of {0,1,a[n],a[n]+1,2a[n],2a[n]+1}; a[2n+1] = j-th term of {0,1,a[n],a[n]+1,2a[n],2a[n]+1}. The present sequence is GS(1,5).
The full list of 36 sequences:
GS(1,1) = A000007
GS(1,2) = A000035
GS(1,3) = A036987
GS(1,4) = A007814
GS(1,5) = A135416 (the present sequence)
GS(1,6) = A135481
GS(2,1) = A135528
GS(2,2) = A000012
GS(2,3) = A000012
GS(2,4) = A091090
GS(2,5) = A135517
GS(2,6) = A135521
GS(3,1) = A036987
GS(3,2) = A000012
GS(3,3) = A000012
GS(3,4) = A000120
GS(3,5) = A048896
GS(3,6) = A038573
GS(4,1) = A135523
GS(4,2) = A001511
GS(4,3) = A008687
GS(4,4) = A070939
GS(4,5) = A135529
GS(4,6) = A135533
GS(5,1) = A048298
GS(5,2) = A006519
GS(5,3) = A080100
GS(5,4) = A087808
GS(5,5) = A053644
GS(5,6) = A000027
GS(6,1) = A135534
GS(6,2) = A038712
GS(6,3) = A135540
GS(6,4) = A135542
GS(6,5) = A054429
GS(6,6) = A003817
(with a(0)=1): Moebius transform of A038712.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Equals A048298(n+1)/2. Cf. A036987, A182660.

GS:=proc(i,j,M) local a,n; a:=array(1..2*M+1); a[1]:=1;
for n from 1 to M do
a[2*n] :=[0,1,a[n],a[n]+1,2*a[n],2*a[n]+1][i];
a[2*n+1]:=[0,1,a[n],a[n]+1,2*a[n],2*a[n]+1][j];
od: a:=convert(a,list); RETURN(a); end;
GS(1,5,200):

i = 1; j = 5; Clear[a]; a[1] = 1; a[n_?EvenQ] := a[n] = {0, 1, a[n/2], a[n/2]+1, 2*a[n/2], 2*a[n/2]+1}[[i]]; a[n_?OddQ] := a[n] = {0, 1, a[(n-1)/2], a[(n-1)/2]+1, 2*a[(n-1)/2], 2*a[(n-1)/2]+1}[[j]]; Array[a, 105] (* Jean-François Alcover, Sep 12 2013 *)

A048298(n) = if(!n,0,if(!bitand(n,n-1),n,0));
A135416(n) = (A048298(n+1)/2); \\ Antti Karttunen, Jul 22 2018

def A135416(n): return int(not(n&(n+1)))*(n+1)>>1 # Chai Wah Wu, Jul 06 2022

G.f.: sum{k>=1, 2^(k-1)*x^(2^k-1) }.

Recurrence: a(2n+1) = 2a(n), a(2n) = 0, starting a(1) = 1.

Formulae and comments by Ralf Stephan, Jun 20 2014

A048679 Compressed fibbinary numbers (A003714), with rewrite 0->0, 01->1 applied to their binary expansion.

Original entry on oeis.org

0, 1, 2, 4, 3, 8, 5, 6, 16, 9, 10, 12, 7, 32, 17, 18, 20, 11, 24, 13, 14, 64, 33, 34, 36, 19, 40, 21, 22, 48, 25, 26, 28, 15, 128, 65, 66, 68, 35, 72, 37, 38, 80, 41, 42, 44, 23, 96, 49, 50, 52, 27, 56, 29, 30, 256, 129, 130, 132, 67, 136, 69, 70, 144, 73, 74, 76, 39, 160, 81

Offset: 0

Antti Karttunen

Permutation of the nonnegative integers (A001477); inverse permutation of A048680 i.e. A048679[ A048680[ n ] ] = n for all n.

Alois P. Heinz, Table of n, a(n) for n = 0..17710 (terms n = 10001..10945 from Antti Karttunen)
Index entries for sequences that are permutations of the natural numbers

Cf. A000045, A003714, A005203, A048678, A048680, A072650, A087808, A106151, A200714, A227351, A232559, A277006, A304100, A304101.

a(n) = rewrite_0to0_x1to1(fibbinary(j)) (where fibbinary(j) = A003714[ n ])
rewrite_0to0_x1to1 := proc(n) option remember; if(0 = n) then RETURN(n); else RETURN((2 * rewrite_0to0_x1to1(floor(n/(2^(1+(n mod 2)))))) + (n mod 2)); fi; end;
fastfib := n -> round((((sqrt(5)+1)/2)^n)/sqrt(5)); fibinv_appr := n -> floor(log[ (sqrt(5)+1)/2 ](sqrt(5)*n)); fibinv := n -> (fibinv_appr(n) + floor(n/fastfib(1+fibinv_appr(n)))); fibbinary := proc(n) option remember; if(n <= 2) then RETURN(n); else RETURN((2^(fibinv(n)-2))+fibbinary_seq(n-fastfib(fibinv(n)))); fi; end;
# second Maple program:
b:= proc(n) is(n=0) end:
a:= proc(n) option remember; local h; h:= iquo(a(n-1), 2)+1;
      while b(h) do h:= h*2 od; b(h):=true; h
    end: a(0):=0:
seq(a(n), n=0..100);  # Alois P. Heinz, Sep 22 2014

b[n_] := n==0; a[n_] := a[n] = Module[{h}, h = Quotient[a[n-1], 2] + 1; While[b[h], h = h*2]; b[h] = True; h]; a[0]=0; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 27 2016, after Alois P. Heinz *)

A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A007814(n) = valuation(n,2);
A000265(n) = (n/2^valuation(n, 2));
A106151(n) = if(n<=1,n,if(n%2,1+(2*A106151((n-1)/2)),(2^(A007814(n)-1))*A106151(A000265(n))));
A048679(n) = if(!n,n,A106151(2*A003714(n))); \\ Antti Karttunen, May 13 2018, after Reinhard Zumkeller's May 09 2005 formula.

from itertools import count, islice
def A048679_gen(): # generator of terms
    return map(lambda n: int(bin(n)[2:].replace('01','1'),2),filter(lambda n:not (n<<1)&n,count(0)))
A048679_list = list(islice(A048679_gen(),20)) # Chai Wah Wu, Mar 18 2024

def A048679(n):
    tlist, s = [1,2], 0
    while tlist[-1]+tlist[-2] <= n: tlist.append(tlist[-1]+tlist[-2])
    for d in tlist[::-1]:
        if d <= n:
            s += 1
            n -= d
        else:
            s <<= 1
    return s # Chai Wah Wu, Apr 24 2025

a(n) = A106151(2*A003714(n)) for n > 0. - Reinhard Zumkeller, May 09 2005
a(n+1) = min{([a(n)/2]+1)*2^k} such that it is not yet in the sequence. - Gerard Orriols, Jun 07 2014
a(n) = A072650(A003714(n)) = A003188(A227351(n)). - Antti Karttunen, May 13 2018

cp's OEIS Frontend

A332224 a(n) = A087808(sigma(n)).

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A332445 Numbers k of the form 4m+1 for which A087808(sigma(k)) is equal to 2*A087808(k).

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A332446 Numbers k for which A087808(sigma(k)) is equal to A087808(2*k).

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A332443 Numbers k such that A332224(k) = A087808(sigma(k)) is odd.

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A332448 a(n) = A007814(A087808(sigma(n))).

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A332444 Numbers k that are squares or twice-squares, but for which A087808(sigma(k)) is even.

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A332447 a(n) = A007814(A087808(n)).

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A048675 If n = p_i^e_i * ... * p_k^e_k, p_i < ... < p_k primes (with p_i = prime(i)), then a(n) = (1/2) * (e_i * 2^i + ... + e_k * 2^k).

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