A163511 -id:A163511 - cp's OEIS frontend

A364258 a(n) = A163511(n) - n.

1, 1, 2, 0, 4, 4, 0, -2, 8, 18, 8, 14, 0, 2, -4, -8, 16, 64, 36, 106, 16, 54, 28, 26, 0, 20, 4, 8, -8, -8, -16, -20, 32, 210, 128, 590, 72, 338, 212, 304, 32, 184, 108, 202, 56, 102, 52, 74, 0, 86, 40, 124, 8, 52, 16, 22, -16, 6, -16, -4, -32, -28, -40, -50, 64, 664, 420, 3058, 256, 1806, 1180, 2330, 144, 1052, 676

Offset: 0

Antti Karttunen, Jul 25 2023

Compare also to the scatter plot of A364294.

Antti Karttunen, Table of n, a(n) for n = 0..12288

Cf. A007283, A163511, A364255 [= gcd(n,a(n))], A364287 (positions of negative terms), A364292 (of terms <= 0), A364288, A364294 [= -a(A364293(n))].

f[n_] := Reverse@ Map[Ceiling[(Length@ # - 1)/2] &, DeleteCases[Split@ Join[Riffle[IntegerDigits[n, 2], 0], {0}], {k__} /; k == 1]]; {1}~Join~Table[-n + Function[t, Prime[t] Product[Prime[m]^(f[n][[m]]), {m, t}] ][DigitCount[n, 2, 1]], {n, 120}] (* Michael De Vlieger, Jul 25 2023 *)

from sympy import nextprime
def A364258(n):
    c, p, k = 1, 1, n
    while k:
        c *= (p:=nextprime(p))**(s:=(~k&k-1).bit_length())
        k >>= s+1
    return c*p-n # Chai Wah Wu, Jul 25 2023

a(n) = A364288(A163511(n)).
For n >= 1, a(2*n) = 2*a(n).
For n >= 0, a(A007283(n)) = 0.

A365805 a(n) = largest exponent m for which a representation of the form A163511(n) = k^m exists (for some k). a(0) = 0 by convention.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 01 2023

nonn

Equivalently, the largest exponent m for which a representation of the form A332214(n) = k^m exists (for some k), or similarly, for any other such variant of A163511, like A332817.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A052409, A163511, A332214, A332817, A365807, A366370.

Cf. A365808 (positions of even terms), A365801 (multiples of 3), A365802 (multiples of 5), A366287 (multiples of 7), A366391 (multiples of 11).

A052409(n) = { my(k=ispower(n)); if(k, k, n>1); };
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A365805(n) = A052409(A163511(n));

a(n) = A052409(A163511(n)).

If a(n) > 1 (or A052409(n) > 1), then a(n) <> A052409(n). [Consider A366370]

A323901 a(n) = A002487(A163511(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 2, 3, 1, 8, 4, 7, 2, 4, 3, 3, 1, 14, 8, 11, 4, 18, 7, 9, 2, 12, 4, 9, 3, 8, 3, 5, 1, 22, 14, 43, 8, 34, 11, 47, 4, 16, 18, 23, 7, 26, 9, 13, 2, 16, 12, 23, 4, 18, 9, 17, 3, 6, 8, 11, 3, 6, 5, 5, 1, 64, 22, 127, 14, 112, 43, 97, 8, 84, 34, 121, 11, 26, 47, 111, 4, 66, 16, 89, 18, 40, 23, 57, 7, 36, 26, 57, 9, 50, 13, 29, 2, 50

Offset: 0

Antti Karttunen, Feb 09 2019

nonn

Cf. A002487, A163511.

Cf. also A323902, A323903.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A054429(n) = ((3<<#binary(n\2))-n-1); \\ From A054429
A163511(n) = if(!n,1,A005940(1+A054429(n)));
A323901(n) = A002487(A163511(n));

a(n) = A002487(A163511(n)).

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

A365808 Numbers k such that A163511(k) is a square.

Original entry on oeis.org

0, 2, 5, 8, 11, 17, 20, 23, 32, 35, 41, 44, 47, 65, 68, 71, 80, 83, 89, 92, 95, 128, 131, 137, 140, 143, 161, 164, 167, 176, 179, 185, 188, 191, 257, 260, 263, 272, 275, 281, 284, 287, 320, 323, 329, 332, 335, 353, 356, 359, 368, 371, 377, 380, 383, 512, 515, 521, 524, 527, 545, 548, 551, 560, 563, 569, 572, 575

Offset: 1

Antti Karttunen, Oct 01 2023

nonn

The sequence is defined inductively as:
 (a) it contains 0 and 2,
   and
 (b) for any nonzero term a(n), (2*a(n)) + 1 and 4*a(n) are also included as terms.
Because the inductive definition guarantees that all terms after 0 are of the form 3k+2 (A016789), and because for any n >= 0, n^2 == 0 or 1 (mod 3), (i.e., squares are in A032766), it follows that there are no squares in this sequence after the initial 0.

Index entries for sequences related to binary expansion of n

Cf. A000290, A010052, A032766, A163511, A365807 (characteristic function).
Positions of even terms in A365805.
Sequence A243071(n^2), n >= 1, sorted into ascending order.
Subsequences: A004171, A055010, A365809 (odd terms).
Subsequence of A016789 (after the initial 0).
Cf. also A277335, A365801, A365802, A365806.

A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
isA365808v2(n) = issquare(A163511(n));

isA365808(n) = if(n<=2, !(n%2), if(n%2, isA365808((n-1)/2), if(n%4, 0, isA365808(n/4))));

from itertools import count, islice
def A365808_gen(): # generator of terms
    return map(lambda n:(3*(n+1)>>2)-1,filter(lambda n:n==1 or (n&3==3 and not '00' in bin(n)),count(1)))
A365808_list = list(islice(A365808_gen(),20)) # Chai Wah Wu, Feb 12 2025

A252737 Row sums of irregular tables A005940, A163511, and A332977.

Original entry on oeis.org

1, 2, 7, 28, 130, 702, 4384, 31516, 260068, 2445372, 25796360, 299286550, 3751803964, 50211590696, 712746859372, 10697637496288, 169490803535680, 2830925427778810, 49785906936838240, 921273098388684878, 17944637546960083042, 368472898102440537484, 7993616254370783660414, 183539682466936703629744

Offset: 0

Antti Karttunen, Dec 21 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450

Row sums of tables A005940, A163511, and A332977.
Cf. A252738 (row products).
Cf. A000079, A000225, A252745, A252746.

b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-`if`(
      i=0, j, 1), j)*ithprime(j), j=1..`if`(i=0, n, i)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Mar 04 2020

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n - If[i == 0, j, 1], j]* Prime[j], {j, 1, If[i == 0, n, i]}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jan 03 2022, after Alois P. Heinz *)

allocatemem(234567890);
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A252737print(up_to_n) = { my(s, i=0, n=0); for(n=0, up_to_n, if(0 == n, s = 1, if(1 == n, s = 2; lev = vector(1); lev[1] = 2, oldlev = lev; lev = vector(2*length(oldlev)); s = 0; for(i = 0, (2^(n-1))-1, lev[i+1] = if((i%2),A003961(oldlev[(i\2)+1]),2*oldlev[(i\2)+1]); s += lev[i+1]))); write("b252737.txt", n, " ", s)); };
A252737print(23); \\ Terms a(0) .. a(23) were computed with this program.

(define (A252737 n) (if (zero? n) 1 (add A163511 (A000079 (- n 1)) (A000225 n))))

(define (A252737 n) (if (zero? n) 1 (add (COMPOSE A005940 1+) (A000079 (- n 1)) (A000225 n))))
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (+ 1 i) (+ res (intfun i)))))))
(define (COMPOSE . funlist) (cond ((null? funlist) (lambda (x) x)) (else (lambda (x) ((car funlist) ((apply COMPOSE (cdr funlist)) x))))))

a(0) = 1; for n>1: a(n) = Sum_{k = A000079(n-1) .. A000225(n)} A163511(k) = Sum_{k = 2^(n-1) .. (2^n)-1} A163511(k).

A324184 a(n) = sigma(A163511(n)).

Original entry on oeis.org

1, 3, 7, 4, 15, 13, 12, 6, 31, 40, 39, 31, 28, 24, 18, 8, 63, 121, 120, 156, 91, 124, 93, 57, 60, 78, 72, 48, 42, 32, 24, 12, 127, 364, 363, 781, 280, 624, 468, 400, 195, 403, 372, 342, 217, 228, 171, 133, 124, 240, 234, 248, 168, 192, 144, 96, 90, 104, 96, 72, 56, 48, 36, 14, 255, 1093, 1092, 3906, 847, 3124, 2343, 2801, 600

Offset: 0

Antti Karttunen, Feb 17 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A000203, A054429, A163511, A324054, A324183, A324185, A324186, A324187, A324188, A324189.

A324184(n) = if(!n,1,my(p=2,mp=p*p,m=1); while(n>1, if(n%2, p=nextprime(1+p); mp = p*p, if((2==n)||!(n%4),mp *= p,m *= (mp-1)/(p-1))); n >>= 1); (m*(mp-1)/(p-1)));

A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(p+1)); n >>= 1); (t*p));
A324184(n) = sigma(A163511(n));

from sympy import nextprime
def A324184(n):
    if n:
        c, p = 1, 1
        while n:
            c *= ((p:=nextprime(p))**(s:=(~n&n-1).bit_length()+1)-1)//(p-1)
            n >>= s
        return c*(p**(s+1)-1)//(p**s-1)
    return 1 # Chai Wah Wu, Jul 25 2023

a(n) = A000203(A163511(n)).

For n >= 1, a(n) = A324054(A054429(n)).

A324185 Deficiency of n permuted by A163511: a(n) = A033879(A163511(n)) = 2*A163511(n) - sigma(A163511(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 5, 0, 4, 1, 14, -3, 19, -4, 6, 2, 6, 1, 41, -12, 94, -19, 26, 7, 41, -12, 12, -12, 22, -2, 10, 4, 10, 1, 122, -39, 469, -64, 126, 32, 286, -51, 47, -72, 148, -17, 66, 25, 109, -28, 30, -54, 102, -48, 18, -4, 58, -10, 22, -12, 38, 0, 18, 8, 12, 1, 365, -120, 2344, -199, 626, 157, 2001, -168, 222, -372, 1030, -92, 458, 172, 1198

Offset: 0

Antti Karttunen, Feb 17 2019

If there are no odd perfect numbers, then all n for which a(n) is 0 are given by sequence A324200.

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A000203, A033879, A054429, A163511, A324055, A324182 (compare the scatter plot), A324184, A324187, A324188, A324189, A324199, A324200.

A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A033879(n) = (2*n-sigma(n));
A324185(n) = A033879(A163511(n));

A324184(n) = if(!n,1,my(p=2,mp=p*p,m=1); while(n>1, if(n%2, p=nextprime(1+p); mp = p*p, if((2==n)||!(n%4),mp *= p,m *= (mp-1)/(p-1))); n >>= 1); (m*(mp-1)/(p-1)));
A324185(n) = (2*A163511(n)) - A324184(n);

a(n) = A033879(A163511(n)) = 2*A163511(n) - A324184(n) = 2*A163511(n) - A000203(A163511(n)).

For n > 0, a(n) = A324055(A054429(n)).

A324186 Sum of odd divisors permuted by A163511: a(n) = A000593(A163511(n)).

Original entry on oeis.org

1, 1, 1, 4, 1, 13, 4, 6, 1, 40, 13, 31, 4, 24, 6, 8, 1, 121, 40, 156, 13, 124, 31, 57, 4, 78, 24, 48, 6, 32, 8, 12, 1, 364, 121, 781, 40, 624, 156, 400, 13, 403, 124, 342, 31, 228, 57, 133, 4, 240, 78, 248, 24, 192, 48, 96, 6, 104, 32, 72, 8, 48, 12, 14, 1, 1093, 364, 3906, 121, 3124, 781, 2801, 40, 2028, 624, 2400, 156, 1600, 400, 1464, 13, 1240, 403

Offset: 0

Antti Karttunen, Feb 17 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. A000593, A038712, A054429, A163511, A324056, A324184.

A000593(n) = sigma(n>>valuation(n, 2)); \\ From A000593
A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A324186(n) = A000593(A163511(n));

a(n) = A000593(A163511(n)).

For n > 0, a(n) = A324056(A054429(n)).

A286531 Restricted growth sequence of A278531 (prime-signature of A163511).

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 5, 2, 6, 4, 7, 3, 7, 5, 5, 2, 8, 6, 9, 4, 10, 7, 7, 3, 9, 7, 11, 5, 7, 5, 5, 2, 12, 8, 13, 6, 14, 9, 9, 4, 14, 10, 15, 7, 10, 7, 7, 3, 13, 9, 15, 7, 15, 11, 11, 5, 9, 7, 11, 5, 7, 5, 5, 2, 16, 12, 17, 8, 18, 13, 13, 6, 19, 14, 20, 9, 14, 9, 9, 4, 18, 14, 21, 10, 21, 15, 15, 7, 14, 10, 15, 7, 10, 7, 7, 3, 17, 13, 20, 9, 21, 15, 15, 7, 20, 15

Offset: 0

Antti Karttunen, May 17 2017

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A046523, A163511, A286533, A286535, A286219, A286622, A305433, A305434.

rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,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])); }
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A278222(n) = A046523(A005940(1+n));
A054429(n) = ((3<<#binary(n\2))-n-1); \\ After M. F. Hasler, Aug 18 2014
A278531(n) = if(!n,1,A278222(A054429(n)));
write_to_bfile(0,rgs_transform(vector(65538,n,A278531(n-1))),"b286531.txt");

A364492 a(n) = A163511(n) / gcd(n, A163511(n)).

Original entry on oeis.org

1, 2, 2, 1, 2, 9, 1, 5, 2, 3, 9, 25, 1, 15, 5, 7, 2, 81, 3, 125, 9, 25, 25, 49, 1, 9, 15, 35, 5, 21, 7, 11, 2, 81, 81, 125, 3, 375, 125, 343, 9, 225, 25, 245, 25, 49, 49, 121, 1, 135, 9, 175, 15, 105, 35, 7, 5, 21, 21, 55, 7, 33, 11, 13, 2, 729, 81, 3125, 81, 625, 125, 2401, 3, 1125, 375, 343, 125, 147, 343, 1331

Offset: 0

Antti Karttunen, Jul 26 2023

Denominator of n / A163511(n).

Antti Karttunen, Table of n, a(n) for n = 0..16383

Cf. A163511, A364255, A364491 (numerators), A364493, A364496 (positions of 1's).

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A054429(n) = ((3<<#binary(n\2))-n-1);
A163511(n) = if(!n,1,A005940(1+A054429(n)))
A364492(n) = { my(u=A163511(n)); (u/gcd(n, u)); };

from math import gcd
from sympy import nextprime
def A364492(n):
    c, p, k = 1, 1, n
    while k:
        c *= (p:=nextprime(p))**(s:=(~k&k-1).bit_length())
        k >>= s+1
    return c*p//gcd(c*p,n) # Chai Wah Wu, Jul 26 2023

a(n) = A163511(n) / A364255(n) = A163511(n) / gcd(n, A163511(n)).

cp's OEIS Frontend

A364258 a(n) = A163511(n) - n.

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A365805 a(n) = largest exponent m for which a representation of the form A163511(n) = k^m exists (for some k). a(0) = 0 by convention.

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A323901 a(n) = A002487(A163511(n)).

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A365808 Numbers k such that A163511(k) is a square.

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A252737 Row sums of irregular tables A005940, A163511, and A332977.

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A324184 a(n) = sigma(A163511(n)).

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A324185 Deficiency of n permuted by A163511: a(n) = A033879(A163511(n)) = 2*A163511(n) - sigma(A163511(n)).

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A324186 Sum of odd divisors permuted by A163511: a(n) = A000593(A163511(n)).

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