A293214 -id:A293214 - cp's OEIS frontend

A293215 Restricted growth sequence transform of A293214, where A293214(n) = Product_{d|n, dA019565(d).

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

Offset: 1

Antti Karttunen, Oct 03 2017

nonn

For all i, j: a(i) = a(j) => A001065(i) = A001065(j).

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

Cf. A001065, A019565, A048675, A293214, A293217 (a variant), A293223, A293224, A293226.

Differs from related A293232 for the first time at n=55, where a(55) = 28, while A293232(55) = 39.

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A293214(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(d))); m; };
write_to_bfile(1,rgs_transform(vector(16384,n,A293214(n))),"b293215.txt");

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

A087207 A binary representation of the primes that divide a number, shown in decimal.

Original entry on oeis.org

0, 1, 2, 1, 4, 3, 8, 1, 2, 5, 16, 3, 32, 9, 6, 1, 64, 3, 128, 5, 10, 17, 256, 3, 4, 33, 2, 9, 512, 7, 1024, 1, 18, 65, 12, 3, 2048, 129, 34, 5, 4096, 11, 8192, 17, 6, 257, 16384, 3, 8, 5, 66, 33, 32768, 3, 20, 9, 130, 513, 65536, 7, 131072, 1025, 10, 1, 36, 19, 262144, 65, 258

Offset: 1

Mitch Cervinka (puritan(AT)planetkc.com), Oct 26 2003

The binary representation of a(n) shows which prime numbers divide n, but not the multiplicities. a(2)=1, a(3)=10, a(4)=1, a(5)=100, a(6)=11, a(10)=101, a(30)=111, etc.
For n > 1, a(n) gives the (one-based) index of the column where n is located in array A285321. A008479 gives the other index. - Antti Karttunen, Apr 17 2017
From Antti Karttunen, Jun 18 & 20 2017: (Start)
A268335 gives all n such that a(n) = A248663(n); the squarefree numbers (A005117) are all the n such that a(n) = A285330(n) = A048675(n).
For all n > 1 for which the value of A285331(n) is well-defined, we have A285331(a(n)) <= floor(A285331(n)/2), because then n is included in the binary tree A285332 and a(n) is one of its ancestors (in that tree), and thus must be at least one step nearer to its root than n itself.
Conjecture: Starting at any n and iterating the map n -> a(n), we will always reach 0 (see A288569). This conjecture is equivalent to the conjecture that at any n that is neither a prime nor a power of two, we will eventually hit a prime number (which then becomes a power of two in the next iteration). If this conjecture is false then sequence A285332 cannot be a permutation of natural numbers. On the other hand, if the conjecture is true, then A285332 must be a permutation of natural numbers, because all primes and powers of 2 occur in definite positions in that tree. This conjecture also implies the conjectures made in A019565 and A285320 that essentially claim that there are neither finite nor infinite cycles in A019565.
If there are any 2-cycles in this sequence, then both terms of the cycle should be present in A286611 and the larger one should be present in A286612.
(End)
Binary rank of the distinct prime indices of n, where the binary rank of an integer partition y is given by Sum_i 2^(y_i-1). For all prime indices (with multiplicity) we have A048675. - Gus Wiseman, May 25 2024

			a(38) = 129 because 38 = 2*19 = prime(1)*prime(8) and 129 = 2^0 + 2^7 (in binary 10000001).
a(140) = 13, binary 1101 because 140 is divisible by the first, third and fourth primes and 2^(1-1) + 2^(3-1) + 2^(4-1) = 13.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000 [First 1000 terms from _T. D. Noe_]
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization

For partial sums see A288566.
Cf. A000040, A000120, A001221, A005117, A008479, A019565, A055396, A285320, A285321, A285329, A285330, A285332.
Sequences with related definitions: A007947, A008472, A027748, A048675, A248663, A276379 (same sequence shown in base 2), A288569, A289271, A297404.
Cf. A286608 (numbers n for which a(n) < n), A286609 (n for which a(n) > n), and also A286611, A286612.
A003986, A003961, A059896 are used to express relationship between terms of this sequence.
Related to A267116 via A225546.
Positions of particular values are: A000079\{1} (1), A000244\{1} (2), A033845 (3), A000351\{1} (4), A033846 (5), A033849 (6), A143207 (7), A000420\{1} (8), A033847 (9), A033850 (10), A033851 (12), A147576 (14), A147571 (15), A001020\{1} (16), A033848 (17).
A048675 gives binary rank of prime indices.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Binary indices (listed A048793):
- length A000120, complement A023416
- min A001511, opposite A000012
- sum A029931, product A096111
- max A029837 or A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
- opposite A371572, sum A230877
Cf. A000720, A005940, A018819, A023506, A071814, A225620, A277319, A277905, A304818, A372689, A372890.

a087207 = sum . map ((2 ^) . (subtract 1) . a049084) . a027748_row
-- Reinhard Zumkeller, Jul 16 2013

a[n_] := Total[ 2^(PrimePi /@ FactorInteger[n][[All, 1]] - 1)]; a[1] = 0; Table[a[n], {n, 1, 69}] (* Jean-François Alcover, Dec 12 2011 *)

a(n) = {if (n==1, 0, my(f=factor(n), v = []); forprime(p=2, vecmax(f[,1]), v = concat(v, vecsearch(f[,1], p)!=0);); fromdigits(Vecrev(v), 2));} \\ Michel Marcus, Jun 05 2017

A087207(n)=vecsum(apply(p->1<M. F. Hasler, Jun 23 2017

from sympy import factorint, primepi
def a(n):
    return sum(2**primepi(i - 1) for i in factorint(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 06 2017

(definec (A087207 n) (if (= 1 n) 0 (+ (A000079 (+ -1 (A055396 n))) (A087207 (A028234 n))))) ;; This uses memoization-macro definec
(define (A087207 n) (A048675 (A007947 n))) ;; Needs code from A007947 and A048675. - Antti Karttunen, Jun 19 2017

Additive with a(p^e) = 2^(i-1) where p is the i-th prime. - Vladeta Jovovic, Oct 29 2003
a(n) gives the m such that A019565(m) = A007947(n). - Naohiro Nomoto, Oct 30 2003
A000120(a(n)) = A001221(n); a(n) = Sum(2^(A049084(p)-1): p prime-factor of n). - Reinhard Zumkeller, Nov 30 2003
G.f.: Sum_{k>=1} 2^(k-1)*x^prime(k)/(1-x^prime(k)). - Franklin T. Adams-Watters, Sep 01 2009
From Antti Karttunen, Apr 17 2017, Jun 19 2017 & Dec 06 2018: (Start)
a(n) = A048675(A007947(n)).
a(1) = 0; for n > 1, a(n) = 2^(A055396(n)-1) + a(A028234(n)).
A000035(a(n)) = 1 - A000035(n). [a(n) and n are of opposite parity.]
A248663(n) <= a(n) <= A048675(n). [XOR-, OR- and +-variants.]
a(A293214(n)) = A218403(n).
a(A293442(n)) = A267116(n).
A069010(a(n)) = A287170(n).
A007088(a(n)) = A276379(n).
A038374(a(n)) = A300820(n) for n >= 1.
(End)
From Peter Munn, Jan 08 2020: (Start)
a(A059896(n,k)) = a(n) OR a(k) = A003986(a(n), a(k)).
a(A003961(n)) = 2*a(n).
a(n^2) = a(n).
a(n) = A267116(A225546(n)).
a(A225546(n)) = A267116(n).
(End)

More terms from Don Reble, Ray Chandler and Naohiro Nomoto, Oct 28 2003

Name clarified by Antti Karttunen, Jun 18 2017

A091954 Number of odd proper divisors of n. That is, the number of odd divisors of n that are less than n.

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Mar 12 2004

			The odd divisors of 15 that are less than 15 are 1, 3 and 5. Therefore there are three odd divisors of 15 that are less than 15.

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

Cf. A000005, A000035, A001620, A032741, A001227, A293214, A293216.
Sum of the k-th powers of the odd proper divisors of n for k=0..10: this sequence (k=0), A091570 (k=1), A351647 (k=2), A352031 (k=3), A352032 (k=4), A352033 (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).
Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: this sequence (k=0), A352047 (k=1), A352048 (k=2), A352049 (k=3), A352050 (k=4), A352051 (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

Count[Most[Divisors[#]],?OddQ]&/@Range[100] (* _Harvey P. Dale, Sep 28 2012 *)
a[n_] := DivisorSigma[0, n/2^IntegerExponent[n, 2]] - Boole[OddQ[n]]; Array[a, 100] (* Amiram Eldar, Jun 11 2022 *)

A091954(n) = sumdiv(n,d,(dAntti Karttunen, Oct 04 2017

my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=2, N, x^k/(1-x^(2*k))))) \\ Seiichi Manyama, Jan 23 2021

From Antti Karttunen, Oct 04 2017: (Start)
a(n) = Sum_{d|n, dA000035(n).
a(n) = A001227(n) - A000035(n).
a(n) = A007814(A293214(n)) = A007814(A293216(n)).
(End)
G.f.: Sum_{k>=2} x^k/(1 - x^(2*k)). - Seiichi Manyama, Jan 23 2021
Sum_{k=1..n} a(k) ~ n*log(n)/2 + (gamma + log(2)/2 - 1)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 26 2023

Corrected and extended by Harvey P. Dale, Sep 28 2012

A293442 Multiplicative with a(p^e) = A019565(e).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 6, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 12, 3, 4, 6, 6, 2, 8, 2, 10, 4, 4, 4, 9, 2, 4, 4, 12, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 12, 4, 12, 4, 4, 2, 12, 2, 4, 6, 15, 4, 8, 2, 6, 4, 8, 2, 18, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4, 4, 12, 2, 12, 4, 6, 4, 4, 4, 20, 2, 6, 6, 9, 2, 8, 2, 12, 8

Offset: 1

Antti Karttunen, Oct 31 2017

From Peter Munn, Apr 06 2021: (Start)
a(n) is determined by the prime signature of n.
Compare with the multiplicative, self-inverse A225546, which also maps 2^e to the squarefree number A019565(e). However, this sequence maps p^e to the same squarefree number for every prime p, whereas A225546 maps the e-th power of progressively larger primes to progressively greater powers of A019565(e).
Both sequences map powers of squarefree numbers to powers of squarefree numbers.
(End)

Sequences used in a definition of this sequence: A000188, A003961, A019565, A028234, A059895, A067029, A162642.
Sequences with related definitions: A225546, A293443, A293444.
Cf. also A293214.
Sequences used to express relationship between terms of this sequence: A006519, A007913, A008833, A064989, A334747.
Sequences related via this sequence: (A001222, A048675, A064547), (A007814, A162642), (A087207, A267116), (A248663, A268387).

f[n_] := If[n == 1, 1, Apply[Times, Prime@ Flatten@ Position[Reverse@ IntegerDigits[Last@ #, 2], 1]] * f[n/Apply[Power, #]] &@ FactorInteger[n][[1]]]; Array[f, 105] (* Michael De Vlieger, Oct 31 2017 *)

a(1) = 1; for n > 1, a(n) = A019565(A067029(n)) * a(A028234(n)).
Other identities. For all n >= 1:
a(a(n)) = A293444(n).
A048675(a(n)) = A001222(n).
A001222(a(n)) = A064547(n) = A048675(A293444(n)).
A007814(a(n)) = A162642(n).
A087207(a(n)) = A267116(n).
A248663(a(n)) = A268387(n).
From Peter Munn, Mar 14 2021: (Start)
Alternative definition: a(1) = 1; a(2) = 2; a(n^2) = A003961(a(n)); a(A003961(n)) = a(n); if A059895(n, k) = 1, a(n*k) = a(n) * a(k).
For n >= 3, a(n) < n.
a(2n) = A334747(a(A006519(n))) * a(n/A006519(n)), where A006519(n) is the largest power of 2 dividing n.
a(2n+1) = a(A064989(2n+1)).
a(n) = a(A007913(n)) * a(A008833(n)) = 2^A162642(n) * A003961(a(A000188(n))).
(End)

A292257 a(n) is the total number of 1's in binary expansion of all proper divisors of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 3, 3, 4, 1, 7, 1, 5, 5, 4, 1, 8, 1, 7, 6, 5, 1, 10, 3, 5, 5, 9, 1, 14, 1, 5, 6, 4, 6, 13, 1, 5, 6, 10, 1, 15, 1, 9, 11, 6, 1, 13, 4, 9, 5, 9, 1, 14, 6, 13, 6, 6, 1, 23, 1, 7, 11, 6, 6, 14, 1, 7, 7, 15, 1, 18, 1, 5, 12, 9, 7, 16, 1, 13, 9, 5, 1, 24, 5, 6, 7, 13, 1, 26, 7, 11, 8, 7, 6, 16, 1, 11, 10, 15, 1, 14, 1, 13, 18

Offset: 1

Antti Karttunen, Oct 04 2017

If a(n) == A000120(n), then n is in A175522, if a(n) < A000120(n), then n is in A175524, and if a(n) > A000120(n), then n is in A175526.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n.

Cf. A000120, A093653, A175522, A175524, A175526, A192895, A290090, A293214.

a[n_] := DivisorSum[n, DigitCount[#, 2, 1] &, # < n &]; Array[a, 100] (* Amiram Eldar, Jul 20 2023 *)
Table[Total[Flatten[IntegerDigits[#,2]&/@Most[Divisors[n]]]],{n,120}] (* Harvey P. Dale, Oct 11 2024 *)

PARI
```
A292257(n) = sumdiv(n,d,(d
				
```

a(n) = Sum_{d|n, dA000120(d).
a(n) = A093653(n) - A000120(n).
a(n) = A192895(n) + A000120(n).
a(n) = A001222(A293214(n)).
A000035(a(n)) = A000035(A290090(n)). [Parity-wise equivalent with A290090.]

A293221 a(n) = Product_{d|n, dA019565(A289813(d)); a product obtained from the 1-digits present in ternary expansions of proper divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 6, 2, 12, 6, 6, 2, 36, 2, 4, 18, 12, 2, 30, 2, 360, 12, 10, 2, 540, 6, 60, 30, 360, 2, 900, 2, 120, 30, 10, 12, 2700, 2, 4, 180, 360, 2, 540, 2, 360, 450, 6, 2, 5400, 4, 120, 30, 360, 2, 210, 30, 5040, 12, 14, 2, 1701000, 2, 84, 180, 2520, 180, 1260, 2, 840, 18, 12600, 2, 94500, 2, 140, 180, 840, 20, 18900, 2, 756000, 210, 210, 2, 23814000, 30

Offset: 1

Antti Karttunen, Oct 03 2017

Antti Karttunen, Table of n, a(n) for n = 1..6561

Cf. A019565, A289813, A293214, A293222, A293223 (restricted growth sequence transform), A293226.

Cf. also A290091.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A289813(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From Remy Sigrist
A293221(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289813(d)))); m; };

a(n) = Product_{d|n, dA019565(A289813(d)).

For all n >= 0, a(3^n) = A002110(n).

A300833 Filter sequence combining A300830(n), A300831(n) and A300832(n), three products formed from such proper divisors d of n for which mu(n/d) = 0, +1 or -1 respectively, where mu is Möbius mu function (A008683).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 16 2018

nonn

Restricted growth sequence transform of triple [A300830(n), A300831(n), A300832(n)].
For all i, j:
  a(i) = a(j) => A293215(i) = A293215(j) => A001065(i) = A001065(j).
  a(i) = a(j) => A051953(i) = A051953(j).
  a(i) = a(j) => A295885(i) = A295885(j).
Apparently this is also the restricted growth sequence transform of ordered pair [A300831(n), A300832(n)], which is true if it holds that whenever we have A300831(i) = A300831(j) and A300832(i) = A300832(j) for any i, j, then also A300830(i) = A300830(j). This has been checked for the first 65537 terms.

			a(39) = a(55) (= 28) as A300830(39) = A300830(55) = 1, A300831(39) = A300831(55) = 2 and A300832(39) = A300832(55) = 420.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A019565, A051953, A300831, A300832.

Cf. also A293214, A293215, A293226, A295885, A300825.

allocatemem(2^30);
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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A300830(n) = { my(m=1); fordiv(n,d,if(!moebius(n/d),m *= A019565(d))); m; };
A300831(n) = { my(m=1); fordiv(n,d,if((d < n)&&(1==moebius(n/d)), m *= A019565(d))); m; };
A300832(n) = { my(m=1); fordiv(n,d,if(-1==moebius(n/d), m *= A019565(d))); m; };
Aux300833(n) = [A300830(n), A300831(n), A300832(n)];
write_to_bfile(1,rgs_transform(vector(up_to,n,Aux300833(n))),"b300833.txt");

cp's OEIS Frontend

A293215 Restricted growth sequence transform of A293214, where A293214(n) = Product_{d|n, dA019565(d).

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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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A087207 A binary representation of the primes that divide a number, shown in decimal.

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A091954 Number of odd proper divisors of n. That is, the number of odd divisors of n that are less than n.

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A293442 Multiplicative with a(p^e) = A019565(e).

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A292257 a(n) is the total number of 1's in binary expansion of all proper divisors of n.

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A293221 a(n) = Product_{d|n, dA019565(A289813(d)); a product obtained from the 1-digits present in ternary expansions of proper divisors of n.

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