A285101 -id:A285101 - cp's OEIS frontend

A285102 a(n) = A007913(A285101(n)).

2, 6, 210, 72930, 620310, 278995269860970, 12849025509071310, 492608110538467706074890, 1342951001046021018427857601026746070, 37793589449865555275592120894959094883390892772270, 728982633030274864467458719371654181886452163442582606072870, 28339554655955912942523491885490197708224606885407444005070

Offset: 0

Antti Karttunen, Apr 15 2017

nonn

Cf. A003961, A007913, A048675, A068052, A242378, A285101, A285103.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A242378(k,n) = { while(k>0,n = A003961(n); k = k-1); n; };
A285102(n) = { if(0==n,2,lcm(A285102(n-1),A242378(n,A285102(n-1)))/gcd(A285102(n-1),A242378(n,A285102(n-1)))); };

# uses [A003961, A242378]
from sympy import factorint, prime, primepi
from sympy.ntheory.factor_ import core
from operator import mul
def a003961(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**f[i] for i in f])
def a242378(k, n):
    while k>0:
        n=a003961(n)
        k-=1
    return n
l=[2]
for n in range(1, 12):
    x=l[n - 1]
    l.append(x*a242378(n, x))
print([core(j) for j in l]) # Indranil Ghosh, Jun 27 2017

(definec (A285102 n) (if (zero? n) 2 (/ (lcm (A285102 (- n 1)) (A242378bi n (A285102 (- n 1)))) (gcd (A285102 (- n 1)) (A242378bi n (A285102 (- n 1)))))))

a(0) = 2, for n > 0, a(n) = lcm(a(n-1),A242378(n,a(n-1))) / gcd(a(n-1),A242378(n,a(n-1))).
a(n) = A007913(A285101(n)).
Other identities. For all n >= 0:
A001221(a(n)) = A001222(a(n)) = A285103(n).
A048675(a(n)) = A068052(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

A053632 Irregular triangle read by rows giving coefficients in expansion of Product_{k=1..n} (1 + x^k).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Mar 22 2000

Or, triangle T(n,k) read by rows, giving number of subsets of {1,2,...,n} with sum k. - Roger CUCULIERE (cuculier(AT)imaginet.fr), Nov 19 2000
Row n consists of A000124(n) terms. These are also the successive vectors (their nonzero elements) when one starts with the infinite vector (of zeros) with 1 inserted somewhere and then shifts it one step (right or left) and adds to the original, then shifts the result two steps and adds, three steps and adds, etc. - Antti Karttunen, Feb 13 2002
T(n,k) = number of partitions of k into distinct parts <= n. Triangle of distribution of Wilcoxon's signed rank statistic. - Mitch Harris, Mar 23 2006
T(n,k) = number of binary words of length n in which the sum of the positions of the 0's is k. Example: T(4,5)=2 because we have 0110 (sum of the positions of the 0's is 1+4=5) and 1001 (sum of the positions of the 0's is 2+3=5). - Emeric Deutsch, Jul 23 2006
A fair coin is flipped n times. You receive i dollars for a "success" on the i-th flip, 1<=i<=n. T(n,k)/2^n is the probability that you will receive exactly k dollars. Your expectation is n(n+1)/4 dollars. - Geoffrey Critzer, May 16 2010
From Gus Wiseman, Jan 02 2023: (Start)
With offset 1, also the number of integer compositions of n whose partial sums add up to k for k = n..n(n+1)/2. For example, row n = 6 counts the following compositions:
  6 15 24 33  42  51  141  231  321  411  1311 2211  3111  12111 21111 111111
          114 123 132 222  312  1131 1221 2121 11121 11211
                  213 1113 1122 1212 2112 1111
(End)

			Triangle begins:
  1;
  1, 1;
  1, 1, 1, 1;
  1, 1, 1, 2, 1, 1, 1;
  1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1;
  1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 1, 1, 1;
  1, 1, 1, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 4, 4, 4, 3, 2, 2, 1, 1, 1;
  ...
Row n = 4 counts the following binary words, where k = sum of positions of zeros:
  1111  0111  1011  0011  0101  0110  0001  0010  0100  1000  0000
                    1101  1110  1001  1010  1100
Row n = 5 counts the following strict partitions of k with all parts <= n (0 is the empty partition):
  0  1  2  3  4  5  42  43  53  54  532  542  543  5431 5432 54321
           21 31 32 51  52  431 432 541  5321 5421
                 41 321 421 521 531 4321

A. V. Yurkin, New binomial and new view on light theory, (book), 2013, 78 pages, no publisher listed.

Alois P. Heinz, Rows n = 0..40, flattened
S. R. Finch, Signum equations and extremal coefficients.
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]
FindStat - Combinatorial Statistic Finder, The major index of an integer composition
Alexander Rosa and Štefan Znám, A combinatorial problem in the theory of congruences. (Russian), Mat.-Fys. Casopis Sloven. Akad. Vied 15 1965 49-59. [Annotated scanned copy] See Table 1.
F. Wilcoxon, Individual Comparisons by Ranking Methods, Biometrics Bulletin, v. 1, no. 6 (1945), pp. 80-83.
A. V. Yurkin, On similarity of systems of geometrical and arithmetic triangles, in Mathematics, Computing, Education Conference XIX, 2012.
A. V. Yurkin, New view on the diffraction discovered by Grimaldi and Gaussian beams, arXiv preprint arXiv:1302.6287 [physics.optics], 2013.
A. V. Yurkin, About the evident description of distribution of beams and "wavy geometrical trajectories" in long thin pipes, 2014 (original in Russian).
A. V. Yurkin, Symmetric triangle of Pascal and non-linear arithmetic parallelepiped, Book Manuscript, Research Gate 2015.

Cf. A053633, A068009.
Rows reduced modulo 2 and interpreted as binary numbers: A068052, A068053. Rows converge towards A000009.
Row sums give A000079.
Cf. A001788, A028362.
Cf. A285101 (multiplicative encoding of each row), A285103 (number of odd terms on row n), A285105 (number of even terms).
Row lengths are A000124.
A reciprocal version is (A033999, A219977, A291983, A291984, A291985, ...).
A negative version is A231599.
A version for partitions is A358194, reversed partitions A264034.
Cf. A025591, A029931, A063865, A152947, A318283, A359042.

with(gfun,seriestolist); map(op,[seq(seriestolist(series(mul(1+(z^i), i=1..n),z,binomial(n+1,2)+1)), n=0..10)]); # Antti Karttunen, Feb 13 2002
# second Maple program:
g:= proc(n) g(n):= `if`(n=0, 1, expand(g(n-1)*(1+x^n))) end:
T:= n-> seq(coeff(g(n), x, k), k=0..degree(g(n))):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 19 2012

Table[CoefficientList[ Series[Product[(1 + t^i), {i, 1, n}], {t, 0, 100}], t], {n, 0, 8}] // Grid (* Geoffrey Critzer, May 16 2010 *)

From Mitch Harris, Mar 23 2006: (Start)
T(n,k) = T(n-1, k) + T(n-1, k-n), T(0,0)=1, T(0,k) = 0, T(n,k) = 0 if k < 0 or k > (n+1 choose 2).
G.f.: (1+x)*(1+x^2)*...*(1+x^n). (End)
Sum_{k>=0} k * T(n,k) = A001788(n). - Alois P. Heinz, Feb 09 2017
max_{k>=0} T(n,k) = A025591(n). - Alois P. Heinz, Jan 20 2023

A028362 Total number of self-dual binary codes of length 2n. Totally isotropic spaces of index n in symplectic geometry of dimension 2n.

Original entry on oeis.org

1, 3, 15, 135, 2295, 75735, 4922775, 635037975, 163204759575, 83724041661975, 85817142703524375, 175839325399521444375, 720413716161839357604375, 5902349576513949856852644375, 96709997811181068404530578084375

Offset: 1

N. J. A. Sloane

These numbers appear in the second column of A155103. - Mats Granvik, Jan 20 2009

a(n) = n terms in the sequence (1, 2, 4, 8, 16, ...) dot n terms in the sequence (1, 1, 3, 15, 135). Example: a(5) = 2295 = (1, 2, 4, 8, 16) dot (1, 1, 3, 15, 135) = (1 + 2 + 12 + 120 + 2160). - Gary W. Adamson, Aug 02 2010

			G.f. = x + 3*x^2 + 15*x^3 + 135*x^4 + 2295*x^5 + 75735*x^6 + 4922775*x^7 + ...

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 630.

T. D. Noe, Table of n, a(n) for n = 1..50
C. Bachoc and P. Gaborit, On extremal additive F_4 codes of length 10 to 18, J. Théorie Nombres Bordeaux, 12 (2000), 255-271.
Steven T. Dougherty and Esengül Saltürk, The neighbor graph of binary self-orthogonal codes, Adv. Math. Comm. (2024). See p. 6.
Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.

Cf. A000124, A003178, A003179, A028361, A028363, A048675, A053632, A068052 (XOR-analog), A285101.
Cf. A155103. - Mats Granvik, Jan 20 2009
Cf. A005329, A006088. - Paul D. Hanna, Sep 16 2009

[1] cat [&*[ 2^k+1: k in [1..n] ]: n in [1..16]]; // Vincenzo Librandi, Dec 24 2015

seq(mul(1 + 2^j, j = 1..n-1), n = 1..20); # G. C. Greubel, Jun 06 2020

Table[Product[2^i+1,{i,n-1}],{n,15}] (* or *) FoldList[Times,1, 2^Range[15]+1] (* Harvey P. Dale, Nov 21 2011 *)
Table[QPochhammer[-2, 2, n - 1], {n, 15}] (* Arkadiusz Wesolowski, Oct 29 2012 *)

{a(n)=polcoeff(sum(m=0,n,2^(m*(m-1)/2)*x^m/prod(k=0,m-1,1-2^k*x+x*O(x^n))),n)} \\ Paul D. Hanna, Sep 16 2009

{a(n) = if( n<1, 0 , prod(k=1, n-1, 2^k + 1))}; /* Michael Somos, Jan 28 2018 */

{a(n) = sum(k=0, n-1, 2^(k*(k+1)/2) * prod(j=1, k, (2^(n-j) - 1) / (2^j - 1)))}; /* Michael Somos, Jan 28 2018 */

for n in range(2,40,2):
  product = 1
  for i in range(1,n//2-1 + 1):
    product *= (2**i+1)
  print(product)
# Nathan J. Russell, Mar 01 2016

from math import prod
def A028362(n): return prod((1<Chai Wah Wu, Jun 20 2022

from ore_algebra import *
R. = QQ['x']
A. = OreAlgebra(R, 'Qx', q=2)
print((Qx - x - 1).to_list([0,1], 10))  # Ralf Stephan, Apr 24 2014

from sage.combinat.q_analogues import q_pochhammer
[q_pochhammer(n-1,-2,2) for n in (1..20)] # G. C. Greubel, Jun 06 2020

;; With memoization-macro definec.
(define (A028362 n) (A028362off0 (- n 1)))
(definec (A028362off0 n) (if (zero? n) 1 (+ (A028362off0 (- n 1)) (* (expt 2 n) (A028362off0 (- n 1))))))
;; Antti Karttunen, Apr 15 2017

a(n) = Product_{i=1..n-1} (2^i+1).
Letting a(0)=1, we have a(n) = Sum_{k=0..n-1} 2^k*a(k) for n>0. a(n) is asymptotic to c*sqrt(2)^(n^2-n) where c=2.384231029031371724149899288.... = A079555 = Product_{k>=1} (1 + 1/2^k). - Benoit Cloitre, Jan 25 2003
G.f.: Sum_{n>=1} 2^(n*(n-1)/2) * x^n/(Product_{k=0..n-1} (1-2^k*x)). - Paul D. Hanna, Sep 16 2009
a(n) = 2^(binomial(n,2) - 1)*(-1; 1/2){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 23 2015
From Antti Karttunen, Apr 15 2017: (Start)
a(n) = A048675(A285101(n-1)).
a(n) = b(n-1), where b(0) = 1, and for n > 0, b(n) = b(n-1) + (2^n)*b(n-1).
a(n) = Sum_{i=1..A000124(n-1)} A053632(n-1,i-1)*(2^(i-1)) [where the indexing of both rows and columns of irregular table A053632(row,col) is considered to start from zero].
(End)
G.f. A(x) satisfies: A(x) = x * (1 + A(2*x)) / (1 - x). - Ilya Gutkovskiy, Jun 06 2020
Conjectural o.g.f. as a continued fraction of Stieltjes type (S-fraction):
  1/(1 - 3*x/(1 - 2*x/(1 - 10*x/(1 - 12*x/(1 - 36*x/(1 - 56*x/(1 - 136*x/(1 - 240*x/(1 - ... - 2^(n-1)*(2^n + 1)*x/(1 - 2^n*(2^n - 1)*x/(1 - ... ))))))))))). - Peter Bala, Sep 27 2023

A068052 Start from 1, shift one left and sum mod 2 (bitwise-XOR) to get 3 (11 in binary), then shift two steps left and XOR to get 15 (1111 in binary), then three steps and XOR to get 119 (1110111 in binary), then four steps and so on.

Original entry on oeis.org

1, 3, 15, 119, 1799, 59367, 3743271, 481693095, 123123509927, 62989418816679, 64491023022979239, 132015402419352060071, 540829047855347718631591, 4430403202865824763042320551, 72583450474242118015031400337575, 2378466805556971511916001231449723047

Offset: 0

Antti Karttunen, Feb 13 2002

a(n) = each row of A053632 reduced mod 2 and interpreted as a binary number.

Antti Karttunen, Table of n, a(n) for n = 0..64

Same sequence shown in binary: A068053.

Cf. A000120, A003987, A028362 (using + instead of XOR), A048675, A053632, A080791, A248663, A285101, A285102, A285103, A285105.

with(gfun,seriestolist); [seq(foo(map(`mod`,seriestolist(series(mul(1+(z^i),i=1..n),z,binomial(n+1,2)+1)),2)), n=0..20)];
foo := proc(a) local i; add(a[i]*2^(i-1),i=1..nops(a)); end;
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1,
      (t-> Bits[Xor](2^n*t, t))(a(n-1)))
    end:
seq(a(n), n=0..16);  # Alois P. Heinz, Mar 07 2024

FoldList[BitXor[#, #*#2]&, 1, 2^Range[20]] (* Paolo Xausa, Mar 07 2024 *)

a(n) = if(n<1, 1, bitxor(a(n - 1), 2^n*a(n - 1))); \\ Indranil Ghosh, Apr 15 2017, after formula by Antti Karttunen

a(0) = 1; for n > 0, a(n) = a(n-1) XOR (2^n)*a(n-1), where XOR is bitwise-XOR (A003987).
a(n) = A248663(A285101(n)) = A048675(A285102(n)).
A000120(a(n)) = A285103(n). [Number of ones in binary representation.]
A080791(a(n)) = A285105(n). [Number of nonleading zeros.]

Formulas added by Antti Karttunen, Apr 15 2017

cp's OEIS Frontend

A285102 a(n) = A007913(A285101(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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A053632 Irregular triangle read by rows giving coefficients in expansion of Product_{k=1..n} (1 + x^k).

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A028362 Total number of self-dual binary codes of length 2n. Totally isotropic spaces of index n in symplectic geometry of dimension 2n.

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A068052 Start from 1, shift one left and sum mod 2 (bitwise-XOR) to get 3 (11 in binary), then shift two steps left and XOR to get 15 (1111 in binary), then three steps and XOR to get 119 (1110111 in binary), then four steps and so on.

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A285104 a(n) = 2^n - A285103(n).

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