A324054 -id:A324054 - cp's OEIS frontend

A324058 a(n) = A324121(A005940(1+n)) = gcd(A324054(n), A005940(1+n)*A106737(n)).

1, 1, 2, 1, 2, 12, 1, 1, 2, 2, 12, 4, 1, 3, 4, 1, 2, 8, 4, 6, 4, 24, 6, 12, 3, 3, 2, 1, 4, 24, 1, 3, 2, 4, 12, 56, 4, 48, 2, 10, 4, 16, 24, 24, 2, 18, 120, 4, 1, 3, 6, 1, 6, 12, 1, 3, 4, 4, 24, 8, 1, 3, 2, 1, 2, 2, 4, 12, 4, 48, 6, 8, 28, 8, 24, 112, 6, 24, 8, 2, 4, 16, 24, 336, 8, 96, 12, 120, 6, 24, 4, 6, 8, 720, 6, 36, 3, 3, 2, 21, 6, 36, 3, 15, 14, 6

Offset: 0

Antti Karttunen, Feb 15 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

Cf. A000005, A000203, A005940, A038040, A106737, A324054, A324057, A324121.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A106737(n) = sum(k=0, n, (binomial(n+k, n-k)*binomial(n, k)) % 2);
A324054(n) = { my(p=2,mp=p*p,m=1); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, if(3==(n%4),mp *= p,m *= (mp-1)/(p-1))); n>>=1); (m); };
A324058(n) = gcd(A324054(n), A005940(1+n)*A106737(n));
\\ Alternatively as:
A324121(n) = gcd(sigma(n),n*numdiv(n));
A324058(n) = A324121(A005940(1+n));

a(n) = A324121(A005940(1+n)) = gcd(A324054(n), A005940(1+n)*A106737(n)).

A324109 Numbers n such that A324108(n) = A324054(n-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 40, 41, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 56, 58, 59, 61, 62, 64, 67, 68, 71, 72, 73, 74, 76, 79, 80, 81, 82, 83, 86, 87, 88, 89, 92, 94, 96, 97, 98, 100, 101, 103, 104, 106, 107, 108, 109, 112, 113, 116, 118, 121

Offset: 1

Antti Karttunen and David A. Corneth, Feb 15 2019

nonn

Numbers n such that A324054(n-1) is equal to A324108(n), which is a multiplicative function with A324108(p^e) = A324054((p^e)-1).
Prime powers (A000961) is a subsequence by definition.
Also A070776 is a subsequence. This follows because for every n of the form 2^i * p^j (where p is an odd prime, and i >= 0, j >= 0), we have A324108(2^i * p^j) = A324054(2^i - 1)*A324054(p^j - 1) = sigma(A005940(2^i)) * sigma(A005940(p^j)). Because A005940(1) = 1, and A005940(2n) = 2*A005940(n), the powers of two are among the fixed points of A005940 (cf. A029747), thus the left half of product is sigma(2^i), while on the other hand, we know that A005940(p^j) is odd (because A005940 also preserves parity), and thus the whole product is equal to sigma(2^i * A005940(p^j)) = sigma(A005940(2^i * p^j)) = A324054((2^i * p^j)-1).
See subsequence A324111 for less regular solutions.

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

Union of A070776 and A324111.

Cf. A000961 (a subsequence), A029747, A324054, A324107, A324108, A324110 (complement).

A324054(n) = { my(p=2,mp=p*p,m=1); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, if(3==(n%4),mp *= p,m *= (mp-1)/(p-1))); n>>=1); (m); };
A324108(n) = { my(f=factor(n)); prod(i=1, #f~, A324054((f[i,1]^f[i,2])-1)); };
isA324109(n) = (A324054(n-1)==A324108(n));
for(n=1,121,if(isA324109(n), print1(n,", ")));

A324108 Multiplicative with a(p^e) = A324054((p^e)-1).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 13, 15, 8, 18, 24, 28, 31, 39, 24, 31, 12, 24, 32, 42, 52, 72, 78, 60, 57, 93, 124, 91, 156, 72, 121, 63, 96, 36, 78, 56, 72, 96, 124, 90, 96, 156, 192, 168, 48, 234, 240, 124, 133, 171, 48, 217, 342, 372, 144, 195, 128, 468, 624, 168, 781, 363, 104, 127, 186, 288, 56, 84, 312, 234, 156, 120, 112

Offset: 1

Antti Karttunen and David A. Corneth, Feb 15 2019

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

Cf. A000203, A005940, A324054, A324106, A324109, A324110.

A324054(n) = { my(p=2,mp=p*p,m=1); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, if(3==(n%4),mp *= p,m *= (mp-1)/(p-1))); n>>=1); (m); };
A324108(n) = { my(f=factor(n)); prod(i=1, #f~, A324054((f[i,1]^f[i,2])-1)); };

A324111 Numbers n for which A324108(n) = A324054(n-1) and which are neither prime powers nor of the form 2^i * p^j, where p is an odd prime, with either exponent i or j > 0.

Original entry on oeis.org

1, 87, 174, 348, 696, 1392, 2091, 2784, 4182, 5568, 8364, 11136, 16683, 16728, 22272, 33215, 33366, 33456, 44544, 66430, 66732, 66912, 89088, 132860, 133464, 133824, 178176, 265720, 266928, 267179, 267648, 356352, 531440, 533856, 534358, 535296, 712704, 1062880, 1066877, 1067712, 1068716, 1070592, 1319235, 1425408

Offset: 1

Antti Karttunen, Feb 15 2019

nonn

Setwise difference of A324109 and A070776.
Setwise difference of A070537 and A324110.
If an odd number n > 1 is present, then all 2^k * n are present also. Odd terms > 1 are given in A324112.

			87 is a term, as 87 = 3*29, A324054(3-1) = 4, A324054(29-1) = 156, and A324108(87) = 4*156 = 624 = A324054(87-1).

Antti Karttunen, Table of n, a(n) for n = 1..173

Cf. A070537, A070776, A324109, A324110, A324112.

A000265(n) = (n/2^valuation(n, 2));
A324054(n) = { my(p=2,mp=p*p,m=1); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, if(3==(n%4),mp *= p,m *= (mp-1)/(p-1))); n>>=1); (m); };
A324108(n) = { my(f=factor(n)); prod(i=1, #f~, A324054((f[i,1]^f[i,2])-1)); };
isA324111(n) = ((1!=omega(n))&&(1!=omega(A000265(n)))&&(A324054(n-1)==A324108(n)));
for(n=1,2^20,if(isA324111(n), print1(n,", ")))

A324110 Numbers k such that A324108(k) != A324054(k-1).

Original entry on oeis.org

15, 21, 30, 33, 35, 39, 42, 45, 51, 55, 57, 60, 63, 65, 66, 69, 70, 75, 77, 78, 84, 85, 90, 91, 93, 95, 99, 102, 105, 110, 111, 114, 115, 117, 119, 120, 123, 126, 129, 130, 132, 133, 135, 138, 140, 141, 143, 145, 147, 150, 153, 154, 155, 156, 159, 161, 165, 168, 170, 171, 175, 177, 180, 182, 183, 185, 186, 187, 189, 190, 195, 198, 201

Offset: 1

Antti Karttunen and David A. Corneth, Feb 15 2019

nonn

This is a subsequence of A070537. The missing terms 1, 87, 174, 348, 696, 1392, 2091, ..., are at A324111.

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

Cf. A070537, A324054, A324108, A324109 (complement), A324111.

A324054(n) = { my(p=2,mp=p*p,m=1); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, if(3==(n%4),mp *= p,m *= (mp-1)/(p-1))); n>>=1); (m); };
A324108(n) = { my(f=factor(n)); prod(i=1, #f~, A324054((f[i,1]^f[i,2])-1)); };
isA324110(n) = (A324054(n-1)!=A324108(n));
for(n=1,201,if(isA324110(n), print1(n,", ")))

A324112 Odd numbers n for which A324108(n) = A324054(n-1), and which themselves are not powers of primes (in A000961).

Original entry on oeis.org

87, 2091, 16683, 33215, 267179, 1066877, 1319235, 4228521, 4330579, 8668351, 9769751, 34662043, 35924003, 50892875, 68239949, 83920375, 143201615, 151730823, 311513495, 419564887, 537386921, 538253299, 539511051, 605140375

Offset: 1

Antti Karttunen, Feb 16 2019

Odd terms > 1 in A324111.

			(Here f is A324054 and g is A324108).
87 is a term, as 87 = 3 * 29, f(3-1) = 4, f(29-1) = 156, and g(87) = 4 * 156 = 624 = f(87-1).
1066877 is a term, as 1066877 = 7^2 * 21773, f(49-1) = 133, f(21773-1) = 8035200, and g(1066877) = 133 * 8035200 = 1068681600 = f(1066877-1).
537386921 is a term, as 537386921 = 2083 * 257987, f(2083-1) = 1440, f(257987-1) = 31554095246856, and g(537386921) = 1440 * 31554095246856 = 45437897155472640 = f(537386921-1).

Cf. A324111.

A005940 The Doudna sequence: write n-1 in binary; power of prime(k) in a(n) is # of 1's that are followed by k-1 0's.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 15, 12, 25, 18, 27, 16, 11, 14, 21, 20, 35, 30, 45, 24, 49, 50, 75, 36, 125, 54, 81, 32, 13, 22, 33, 28, 55, 42, 63, 40, 77, 70, 105, 60, 175, 90, 135, 48, 121, 98, 147, 100, 245, 150, 225, 72, 343, 250, 375, 108, 625, 162, 243, 64, 17, 26, 39

Offset: 1

N. J. A. Sloane and J. H. Conway

A permutation of the natural numbers. - Robert G. Wilson v, Feb 22 2005
Fixed points: A029747. - Reinhard Zumkeller, Aug 23 2006
The even bisection, when halved, gives the sequence back. - Antti Karttunen, Jun 28 2014
From Antti Karttunen, Dec 21 2014: (Start)
This irregular table can be represented as a binary tree. Each child to the left is obtained by applying A003961 to the parent, and each child to the right is obtained by doubling the parent:
                                      1
                                      |
                   ...................2...................
                  3                                       4
        5......../ \........6                   9......../ \........8
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
    7       10         15       12         25       18         27       16
  11 14   21  20     35  30   45  24     49  50   75  36    125  54   81  32
etc.
Sequence A163511 is obtained by scanning the same tree level by level, from right to left. Also in binary trees A253563 and A253565 the terms on level of the tree are some permutation of the terms present on the level n of this tree. A252464(n) gives the distance of n from 1 in all these trees.
A252737(n) gives the sum and A252738(n) the product of terms on row n (where 1 is on row 0, 2 on row 1, 3 and 4 on row 2, etc.). A252745(n) gives the number of nodes on level n whose left child is larger than the right child, A252750 the difference between left and right child for each node from node 2 onward.
(End)
-A008836(a(1+n)) gives the corresponding numerator for A323505(n). - Antti Karttunen, Jan 19 2019
(a(2n+1)-1)/2 [= A244154(n)-1, for n >= 0] is a permutation of the natural numbers. - George Beck and Antti Karttunen, Dec 08 2019
From Peter Munn, Oct 04 2020: (Start)
Each term has the same even part (equivalently, the same 2-adic valuation) as its index.
Using the tree depicted in Antti Karttunen's 2014 comment:
Numbers are on the right branch (4 and descendants) if and only if divisible by the square of their largest prime factor (cf. A070003).
Numbers on the left branch, together with 2, are listed in A102750.
(End)
According to Kutz (1981), he learned of this sequence from American mathematician Byron Leon McAllister (1929-2017) who attributed the invention of the sequence to a graduate student by the name of Doudna (first name Paul?) in the mid-1950's at the University of Wisconsin. - Amiram Eldar, Jun 17 2021
From David James Sycamore, Sep 23 2022: (Start)
Alternative (recursive) definition: If n is a power of 2 then a(n)=n. Otherwise, if 2^j is the greatest power of 2 not exceeding n, and if k = n - 2^j, then a(n) is the least m*a(k) that has not occurred previously, where m is an odd prime.
Example: Use recursion with n = 77 = 2^6 + 13. a(13) = 25 and since 11 is the smallest odd prime m such that m*a(13) has not already occurred (see a(27), a(29),a(45)), then a(77) = 11*25 = 275. (End)
The odd bisection, when transformed by replacing all prime(k)^e in a(2*n - 1) with prime(k-1)^e, returns a(n), and thus gives the sequence back. - David James Sycamore, Sep 28 2022

			From _N. J. A. Sloane_, Aug 22 2022: (Start)
Let c_i = number of 1's in binary expansion of n-1 that have i 0's to their right, and let p(j) = j-th prime.  Then a(n) = Product_i p(i+1)^c_i.
If n=9, n-1 is 1000, c_3 = 1, a(9) = p(4)^1 = 7.
If n=10, n-1 = 1001, c_0 = 1, c_2 = 1, a(10) = p(1)*p(3) = 2*5 = 10.
If n=11, n-1 = 1010, c_1 = 1, c_2 = 1, a(11) = p(2)*p(3) = 15. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Antti Karttunen, Table of n, a(n) for n = 1..8192 (terms 1..1024 from Reinhard Zumkeller)
Michael De Vlieger, 6 row Doudna Tree diagram as mentioned in Comments.
Michael De Vlieger, Annotated fan-style binary tree showing 10 levels, with a color function where 2^m is shown in medium blue in row m, k < 2^m is darker blue, and k > 2^m is brighter green, with records in each row shown in red.
Ronald E. Kutz, Two unusual sequences, Two-Year College Mathematics Journal, Vol. 12, No. 5 (1981), pp. 316-319.
Index entries for sequences that are permutations of the natural numbers

Cf. A103969. Inverse is A005941 (A156552).
Cf. A125106. [From Franklin T. Adams-Watters, Mar 06 2010]
Cf. A252737 (gives row sums), A252738 (row products), A332979 (largest on row).
Related permutations of positive integers: A163511 (via A054429), A243353 (via A006068), A244154, A253563 (via A122111), A253565, A332977, A334866 (via A225546).
A000120, A003602, A003961, A006519, A053645, A070939, A246278, A250246, A252753, A253552 are used in a formula defining this sequence.
Formulas for f(a(n)) are given for f = A000265, A003963, A007949, A055396, A056239.
Numbers that occur at notable sets of positions in the binary tree representation of the sequence: A000040, A000079, A002110, A070003, A070826, A102750.
Cf. also A000142, A001511, A002450, A112798, A252463, A252464, A252745, A252750, A324054, A324106, A323505, A323508.
Cf. A106737, A290077, A323915, A324052, A324054, A324055, A324056, A324057, A324058, A324114, A324335, A324340, A324348, A324349 for various number-theoretical sequences applied to (i.e., permuted by) this sequence.
k-adic valuation: A007814 (k=2), A337821 (k=3).
Positions of multiples of 3: A091067.
Primorial deflation: A337376 / A337377.
Sum of prime indices of a(n) is A161511, reverse version A359043.
A048793 lists binary indices, ranked by A019565.
A066099 lists standard comps, partial sums A358134 (ranked by A358170).
Cf. A001222, A029837, A059893, A241916, A242628, A253566, A359042.

a005940 n = f (n - 1) 1 1 where
   f 0 y _          = y
   f x y i | m == 0 = f x' y (i + 1)
           | m == 1 = f x' (y * a000040 i) i
           where (x',m) = divMod x 2
-- Reinhard Zumkeller, Oct 03 2012
(Scheme, with memoization-macro definec from Antti Karttunen's IntSeq-library)
(define (A005940 n) (A005940off0 (- n 1))) ;; The off=1 version, utilizing any one of three different offset-0 implementations:
(definec (A005940off0 n) (cond ((< n 2) (+ 1 n)) (else (* (A000040 (- (A070939 n) (- (A000120 n) 1))) (A005940off0 (A053645 n))))))
(definec (A005940off0 n) (cond ((<= n 2) (+ 1 n)) ((even? n) (A003961 (A005940off0 (/ n 2)))) (else (* 2 (A005940off0 (/ (- n 1) 2))))))
(define (A005940off0 n) (let loop ((n n) (i 1) (x 1)) (cond ((zero? n) x) ((even? n) (loop (/ n 2) (+ i 1) x)) (else (loop (/ (- n 1) 2) i (* x (A000040 i)))))))
;; Antti Karttunen, Jun 26 2014

f := proc(n,i,x) option remember ; if n = 0 then x; elif type(n,'even') then procname(n/2,i+1,x) ; else procname((n-1)/2,i,x*ithprime(i)) ; end if; end proc:
A005940 := proc(n) f(n-1,1,1) ; end proc: # R. J. Mathar, Mar 06 2010

f[n_] := Block[{p = Partition[ Split[ Join[ IntegerDigits[n - 1, 2], {2}]], 2]}, Times @@ Flatten[ Table[q = Take[p, -i]; Prime[ Count[ Flatten[q], 0] + 1]^q[[1, 1]], {i, Length[p]}] ]]; Table[ f[n], {n, 67}] (* Robert G. Wilson v, Feb 22 2005 *)
Table[Times@@Prime/@(Join@@Position[Reverse[IntegerDigits[n,2]],1]-Range[DigitCount[n,2,1]]+1),{n,0,100}] (* Gus Wiseman, Dec 28 2022 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, n%2 && (t*=p) || p=nextprime(p+1)); t } \\ M. F. Hasler, Mar 07 2010; update Aug 29 2014

a(n)=my(p=2, t=1); for(i=0,exponent(n), if(bittest(n,i), t*=p, p=nextprime(p+1))); t \\ Charles R Greathouse IV, Nov 11 2021

from sympy import prime
import math
def A(n): return n - 2**int(math.floor(math.log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
print([b(n - 1) for n in range(1, 101)]) # Indranil Ghosh, Apr 10 2017

from math import prod
from itertools import accumulate
from collections import Counter
from sympy import prime
def A005940(n): return prod(prime(len(a)+1)**b for a, b in Counter(accumulate(bin(n-1)[2:].split('1')[:0:-1])).items()) # Chai Wah Wu, Mar 10 2023

From Reinhard Zumkeller, Aug 23 2006, R. J. Mathar, Mar 06 2010: (Start)
a(n) = f(n-1, 1, 1)
  where f(n, i, x) = x if n = 0,
                   = f(n/2, i+1, x) if n > 0 is even
                   = f((n-1)/2, i, x*prime(i)) otherwise. (End)
From Antti Karttunen, Jun 26 2014: (Start)
Define a starting-offset 0 version of this sequence as:
  b(0)=1, b(1)=2, [base cases]
and then compute the rest either with recurrence:
  b(n) = A000040(1+(A070939(n)-A000120(n))) * b(A053645(n)).
or
  b(2n) = A003961(b(n)), b(2n+1) = 2 * b(n). [Compare this to the similar recurrence given for A163511.]
Then define a(n) = b(n-1), where a(n) gives this sequence A005940 with the starting offset 1.
Can be also defined as a composition of related permutations:
a(n+1) = A243353(A006068(n)).
a(n+1) = A163511(A054429(n)). [Compare the scatter plots of this sequence and A163511 to each other.]
This permutation also maps between the partitions as enumerated in the lists A125106 and A112798, providing identities between:
A161511(n) = A056239(a(n+1)). [The corresponding sums ...]
A243499(n) = A003963(a(n+1)). [... and the products of parts of those partitions.]
(End)
From Antti Karttunen, Dec 21 2014  - Jan 04 2015: (Start)
A002110(n) =  a(1+A002450(n)). [Primorials occur at (4^n - 1)/3 in the offset-0 version of the sequence.]
a(n) = A250246(A252753(n-1)).
a(n) = A122111(A253563(n-1)).
For n >= 1, A055396(a(n+1)) = A001511(n).
For n >= 2, a(n) = A246278(1+A253552(n)).
(End)
From Peter Munn, Oct 04 2020: (Start)
A000265(a(n)) = a(A000265(n)) = A003961(a(A003602(n))).
A006519(a(n)) = a(A006519(n)) = A006519(n).
a(n) = A003961(a(A003602(n))) * A006519(n).
A007814(a(n)) = A007814(n).
A007949(a(n)) = A337821(n) = A007814(A003602(n)).
a(n) = A225546(A334866(n-1)).
(End)
a(2n) = 2*a(n), or generally a(2^k*n) = 2^k*a(n). - Amiram Eldar, Oct 03 2022
If n-1 = Sum_{i} 2^(q_i-1), then a(n) = Product_{i} prime(q_i-i+1). These are the Heinz numbers of the rows of A125106. If the offset is changed to 0, the inverse is A156552. - Gus Wiseman, Dec 28 2022

More terms from Robert G. Wilson v, Feb 22 2005
Sign in a formula switched and Maple program added by R. J. Mathar, Mar 06 2010
Binary tree illustration and keyword tabf added by Antti Karttunen, Dec 21 2014

A323243 a(1) = 0; for n > 1, a(n) = A000203(A156552(n)).

Original entry on oeis.org

0, 1, 3, 4, 7, 6, 15, 8, 12, 13, 31, 12, 63, 18, 18, 24, 127, 14, 255, 20, 39, 48, 511, 24, 28, 84, 24, 48, 1023, 32, 2047, 32, 54, 176, 42, 40, 4095, 258, 144, 56, 8191, 38, 16383, 68, 36, 800, 32767, 48, 60, 31, 252, 132, 65535, 30, 91, 72, 528, 1302, 131071, 44, 262143, 2736, 60, 104, 126, 96, 524287, 304, 774, 42, 1048575, 72, 2097151, 4356, 42

Offset: 1

Antti Karttunen, Jan 10 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000203, A156552, A323244, A323247, A323248, A324118, A324543 (Möbius transform), A324396, A324823.

Cf. A323173, A324054, A324184, A324545 for other permutations of sigma, and also A324573, A324653.

Array[If[# == 0, 0, DivisorSigma[1, #]] &@ Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]] &, 75] (* Michael De Vlieger, Apr 21 2019 *)

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))));
A323243(n) = if(1==n, 0, sigma(A156552(n)));

\\ For computing terms a(n), with n > ~4000 use Hans Havermann's factorization file https://oeis.org/A156552/a156552.txt
v156552sigs = readvec("a156552.txt"); \\ First read it in as a PARI-vector.
A323243(n) = if(n<=2,n-1,my(prsig=v156552sigs[n],ps=prsig[1],es=prsig[2]); prod(i=1,#ps,((ps[i]^(1+es[i]))-1)/(ps[i]-1))); \\ Then play sigma
\\ Antti Karttunen, Mar 15 2019

from sympy import divisor_sigma, primepi, factorint
def A323243(n): return divisor_sigma(sum((1< 1 else 0 # Chai Wah Wu, Mar 10 2023

a(1) = 0; for n > 1, a(n) = A000203(A156552(n)).
a(n) = 2*A156552(n) - A323244(n).
a(n) = A323247(n) - A323248(n).
From Antti Karttunen, Mar 12 2019: (Start)
a(A000040(n)) = A000225(n).
a(n) = Sum_{d|n} A324543(d).
For n > 1, a(2*A246277(n)) = A324118(n).
gcd(a(n), A156552(n)) = A324396(n).
A000035(a(n)) = A324823(n).
(End)

A324055 Deficiency of Doudna-sequence: a(n) = A033879(A005940(1+n)).

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 5, 1, 6, 2, 6, -4, 19, -3, 14, 1, 10, 4, 10, -2, 22, -12, 12, -12, 41, 7, 26, -19, 94, -12, 41, 1, 12, 8, 18, 0, 38, -12, 22, -10, 58, -4, 18, -48, 102, -54, 30, -28, 109, 25, 66, -17, 148, -72, 47, -51, 286, 32, 126, -64, 469, -39, 122, 1, 16, 10, 22, 4, 46, -12, 42, -8, 70, 4, 42, -56, 178, -60, 58, -26, 118, 20

Offset: 0

Antti Karttunen, Feb 14 2019

Both here and in the mirror image sequence A324185, the lowermost (asinh) scatter plot shows on the y = 0 line the numbers that correspond to the perfect numbers. Compare also to the scatter plot of A243492.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..70327
Index entries for sequences related to binary expansion of n
Index entries for sequences related to sigma(n)

Cf. A000120, A033879, A054429.
See A106737, A290077, A323915, A324052, A324054, A324056, A324057, A324058, A324114, A324335, A324340, A324348, A324349, A324394, A324395 for other sequences as permuted by A005940, and compare their scatter plots.
Cf. also A243492, A323174, A323244, A324185, A324346, A324347, A324654.

Array[Block[{p = Partition[Split[Join[IntegerDigits[#, 2], {2}]], 2]}, 2 # - DivisorSigma[1, #] &[Times @@ Flatten@ Table[Prime[Count[Flatten@ #, 0] + 1]^#[[1, 1]] &@ Take[p, -i], {i, Length[p]}]]] &, 82, 0] (* Michael De Vlieger, Mar 11 2019, after Robert G. Wilson v at A005940 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A033879(n) = (2*n-sigma(n));
A324055(n) = A033879(A005940(1+n));

A324055(n) = { my(m1=2,m2=1,p=2,mp=p*p); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, m1 *= p; if(3==(n%4),mp *= p,m2 *= (mp-1)/(p-1))); n>>=1); (m1-m2); };

a(n) = A033879(A005940(1+n)).
a(n) = 2*A005940(1+n) - A324054(n).
For n > 0, a(n) = A324185(A054429(n)).
a(n) = A324348(n) + A000120(A005940(1+n)).

A324653 a(n) = A000203(A276086(n)).

Original entry on oeis.org

1, 3, 4, 12, 13, 39, 6, 18, 24, 72, 78, 234, 31, 93, 124, 372, 403, 1209, 156, 468, 624, 1872, 2028, 6084, 781, 2343, 3124, 9372, 10153, 30459, 8, 24, 32, 96, 104, 312, 48, 144, 192, 576, 624, 1872, 248, 744, 992, 2976, 3224, 9672, 1248, 3744, 4992, 14976, 16224, 48672, 6248, 18744, 24992, 74976, 81224, 243672, 57, 171, 228

Offset: 0

Antti Karttunen, Mar 10 2019

nonn

Cf. A000203, A002110, A276086.
Cf. A267263, A276150, A324650, A324655 for omega, bigomega, phi and tau analogs, and also A324654.
Cf. also A324054.

A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324653(n) = sigma(A276086(n));

a(n) = A000203(A276086(n)).

For n >= 1, a(A002110(n-1)) = 1+A000040(n).

cp's OEIS Frontend

A324058 a(n) = A324121(A005940(1+n)) = gcd(A324054(n), A005940(1+n)*A106737(n)).

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A324109 Numbers n such that A324108(n) = A324054(n-1).

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A324108 Multiplicative with a(p^e) = A324054((p^e)-1).

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A324111 Numbers n for which A324108(n) = A324054(n-1) and which are neither prime powers nor of the form 2^i * p^j, where p is an odd prime, with either exponent i or j > 0.

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A324110 Numbers k such that A324108(k) != A324054(k-1).

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A324112 Odd numbers n for which A324108(n) = A324054(n-1), and which themselves are not powers of primes (in A000961).

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A005940 The Doudna sequence: write n-1 in binary; power of prime(k) in a(n) is # of 1's that are followed by k-1 0's.

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A323243 a(1) = 0; for n > 1, a(n) = A000203(A156552(n)).

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