A318458 -id:A318458 - cp's OEIS frontend

A324398 a(1) = 0; for n > 1, a(n) = A318458(A156552(n)).

0, 0, 0, 1, 0, 1, 0, 1, 6, 0, 0, 1, 0, 1, 8, 9, 0, 1, 0, 1, 16, 1, 0, 1, 0, 1, 10, 1, 0, 1, 0, 1, 0, 1, 20, 9, 0, 1, 66, 1, 0, 1, 0, 1, 6, 1, 0, 1, 0, 0, 2, 1, 0, 1, 36, 1, 258, 1, 0, 1, 0, 1, 6, 41, 0, 1, 0, 1, 0, 1, 0, 17, 0, 1, 16, 1, 32, 1, 0, 1, 10, 1, 0, 1, 132, 1, 1026, 1, 0, 33, 72, 1, 0, 1, 256, 25, 0, 0, 66, 17, 0, 1, 0, 1, 34

Offset: 1

Antti Karttunen, Mar 05 2019

nonn

Cf. A000203, A156552, A318458, A323243, A323244, A324396, A324397.

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))));
A318458(n) = bitand(n, sigma(n)-n);
A324398(n) = if(1==n,0,A318458(A156552(n)));

a(1) = 0; for n > 1, a(n) = A318458(A156552(n)).

a(n) = A156552(n) AND (A323243(n) - A156552(n)).

A324649 Numbers k such that A318458(k) (bitwise-AND of k and sigma(k)-k) is equal to k.

Original entry on oeis.org

6, 20, 28, 36, 66, 72, 88, 100, 104, 114, 132, 150, 240, 258, 264, 272, 280, 304, 354, 368, 392, 402, 464, 496, 498, 516, 550, 552, 642, 644, 680, 708, 748, 770, 774, 784, 786, 834, 836, 840, 860, 978, 1026, 1032, 1040, 1044, 1056, 1062, 1064, 1068, 1074, 1092, 1104, 1120, 1184, 1232, 1266, 1312, 1362, 1376, 1410, 1504

Offset: 1

Antti Karttunen, Mar 14 2019

nonn

Positions of zeros in A324648. Fixed points of A318458, also positions of the records in the latter.
Intersection with A324652 gives A324643.
The odd terms are: 7425, 76545, 92565, ... (A324897).

Cf. A318458, A324648, A324652.

Cf. A000396, A324643, A324897, A324898 (subsequences).

Select[Range@ 1600, BitAnd[#, DivisorSigma[1, #] - #] == # &] (* Michael De Vlieger, Apr 21 2019, after Vincenzo Librandi at A318458 *)

for(n=1,oo,if(bitand(n,sigma(n)-n)==n, print1(n, ", ")));

A324389 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A009194(n), A318458(n)] for all other numbers, except f(1) = -1.

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 3, 2, 2, 5, 3, 6, 3, 7, 8, 2, 3, 9, 3, 10, 3, 11, 3, 12, 2, 13, 14, 15, 3, 16, 3, 2, 17, 18, 3, 19, 3, 11, 3, 20, 3, 21, 3, 22, 23, 7, 3, 6, 2, 24, 25, 26, 3, 27, 28, 29, 28, 30, 3, 31, 3, 32, 33, 2, 3, 34, 3, 18, 17, 35, 3, 36, 3, 5, 3, 37, 3, 38, 3, 39, 2, 18, 3, 40, 41, 11, 17, 42, 3, 43, 44, 45, 3, 46, 47, 12, 3, 48, 23, 49, 3, 50, 3

Offset: 1

Antti Karttunen, Mar 05 2019

For all i, j:
  A324401(i) = A324401(j) => a(i) = a(j).
Regarding the scatter plot of this sequence, see also comments in A318310. - Antti Karttunen, Feb 04 2020

Cf. A009194, A318310, A318458, A324390, A324401, A324530, A331744, A331746, A331747.

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; };
A009194(n) = gcd(n,sigma(n));
A318458(n) = bitand(n,sigma(n)-n);
Aux324389(n) = if(1==n,-1,[A009194(n), A318458(n)]);
v324389 = rgs_transform(vector(up_to,n,Aux324389(n)));
A324389(n) = v324389[n];

A324530 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A033879(n), A318458(n)] for all other numbers, except f(1) = -1.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 2, 7, 8, 9, 10, 11, 12, 13, 2, 14, 15, 16, 17, 9, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 2, 16, 28, 19, 29, 30, 31, 19, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 2, 39, 56, 57, 58, 35, 59, 60, 61, 62, 63, 64, 65, 51, 66, 67, 68, 41, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 51, 80

Offset: 1

Antti Karttunen, Mar 05 2019

nonn

Cf. A033879, A318458, A324389, A324400, A324531.

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; };
A033879(n) = (n+n-sigma(n));
A318458(n) = bitand(n,sigma(n)-n);
Aux324530(n) = if(1==n,-1,[A033879(n), A318458(n)]);
v324530 = rgs_transform(vector(up_to,n,Aux324530(n)));
A324530(n) = v324530[n];

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

A324898 Odd numbers k such that sigma(k) is congruent to 2 modulo 4 and k = A318458(k), where A318458(k) is bitwise-AND of k and sigma(k)-k.

Original entry on oeis.org

236925, 3847725, 51122925, 69468525, 151141725, 154669725, 269748225, 344211525, 415565325, 445817925, 551569725, 1111904325, 1112565825, 1113756525, 1175717025, 1400045625, 1631666925, 1695170925, 1820873925, 1915847325, 1946981925, 2179080225, 2321121825, 2453690925, 2460041325, 2491740225, 3223500525, 3493517445, 3775103325

Offset: 1

Antti Karttunen, Apr 19 2019

nonn

If this sequence has no common terms with A324647, or no terms common with A324727, then there are no odd perfect numbers.
The first 29 terms factored:
      236925 = 3^6 * 5^2 *  13,
     3847725 = 3^2 * 5^2 *  7^2 * 349,
    51122925 = 3^2 * 5^2 *  7^2 * 4637,
    69468525 = 3^2 * 5^2 *  7^2 * 6301,
   151141725 = 3^2 * 5^2 *  7^2 * 13709,
   154669725 = 3^2 * 5^2 *  7^2 * 14029,
   269748225 = 3^6 * 5^2 * 19^2 * 41,
   344211525 = 3^4 * 5^2 *  7^2 * 3469,
   415565325 = 3^2 * 5^2 *  7^2 * 37693,
   445817925 = 3^4 * 5^2 *  7^2 * 4493,
   551569725 = 3^2 * 5^2 *  7^4 * 1021,
  1111904325 = 3^2 * 5^2 *  7^2 * 100853,
  1112565825 = 3^2 * 5^2 *  7^2 * 100913,
  1113756525 = 3^2 * 5^2 *  7^2 * 101021,
  1175717025 = 3^4 * 5^2 *  7^2 * 17^2 * 41,
  1400045625 = 3^2 * 5^4 * 11^4 * 17,
  1631666925 = 3^2 * 5^2 *  7^2 * 147997,
  1695170925 = 3^2 * 5^2 *  7^2 * 153757,
  1820873925 = 3^4 * 5^2 *   13 * 263^2, [Here the unitary prime is not the largest]
  1915847325 = 3^2 * 5^2 *  7^2 * 173773,
  1946981925 = 3^2 * 5^2 *  7^2 * 176597,
  2179080225 = 3^4 * 5^2 *  7^2 * 21961,
  2321121825 = 3^4 * 5^2 * 11^2 * 9473,
  2453690925 = 3^2 * 5^2 *  7^2 * 222557,
  2460041325 = 3^2 * 5^2 *  7^2 * 223133,
  2491740225 = 3^6 * 5^2 * 13^2 * 809,
  3223500525 = 3^2 * 5^2 *  7^2 * 292381,
  3493517445 = 3^6 * 5^1 * 11^2 * 89^2,  [Here the unitary prime is not the largest]
  3775103325 = 3^2 * 5^2 *  7^2 * 342413.
Subsequence of A228058 provided this sequence does not contain any prime powers. - Antti Karttunen, Jun 17 2019
Sequence contains no prime powers up to 10^20. I believe any prime powers must be of the form (4k+1)^(4e+1), in which case I have verified this up to 10^50. - Charles R Greathouse IV, Dec 08 2021

Intersection of A191218 and A324897, also intersection of A191218 and A324649.

Cf. A228058, A318458, A324647, A324727.

Select[Range[10^5, 10^8, 2], And[Mod[#2, 4] == 2, BitAnd[#1, #2 - #1] == #1] & @@ {#, DivisorSigma[1, #]} &] (* Michael De Vlieger, Jun 22 2019 *)

for(n=1, oo, if((n%2)&&2==((t=sigma(n))%4)&&(bitand(n, t-n)==n), print1(n,", ")));

A324531 Lexicographically earliest sequence such that for all i, j >= 1, a(i) = a(j) => f(i) = f(j), where f(n) = [A278222(n), A318458(n)] for all other numbers, except f(1) = 0.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 05 2019

nonn

For all i, j:

a(i) = a(j) => A324532(i) = A324532(j).

Cf. A278222, A318458, A324400, A324530, A324532.

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; };
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));
A318458(n) = bitand(n,sigma(n)-n);
Aux324531(n) = if(1==n,0,[A278222(n), A318458(n)]);
v324531 = rgs_transform(vector(up_to,n,Aux324531(n)));
A324531(n) = v324531[n];

For n >= 1, a(2^n) = 2.

A324648 a(n) = n - A318458(n), where A318458(n) is bitwise-AND of n and the sum of proper divisors of n (sigma(n)-n).

Original entry on oeis.org

1, 2, 2, 4, 4, 0, 6, 8, 9, 2, 10, 12, 12, 4, 6, 16, 16, 2, 18, 0, 20, 16, 22, 24, 25, 10, 18, 0, 28, 20, 30, 32, 32, 34, 34, 0, 36, 32, 38, 8, 40, 8, 42, 4, 12, 36, 46, 48, 49, 16, 34, 16, 52, 52, 38, 56, 40, 26, 58, 16, 60, 28, 22, 64, 64, 0, 66, 68, 68, 4, 70, 0, 72, 66, 74, 12, 76, 4, 78, 16, 81, 82, 82, 80, 64, 80, 86, 0, 88

Offset: 1

Antti Karttunen, Mar 14 2019

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to sigma(n)

Cf. A001065, A004198, A318458, A324658, A324649 (positions of zeros).

Array[# - BitAnd[#, DivisorSigma[1, #] - #] &, 100] (* Paolo Xausa, Mar 12 2024 *)

A318458(n) = bitand(n,sigma(n)-n);
A324648(n) = (n-A318458(n));

PARI
```
A324648(n) = (n-bitand(n,sigma(n)-n));
```

a(n) = n - A318458(n).

A324897 Odd numbers k such that A318458(k) (bitwise-AND of k and sigma(k)-k) is equal to k.

Original entry on oeis.org

7425, 76545, 92565, 236925, 831105, 954765, 1401345, 2011905, 2048445, 2129985, 2253825, 2445345, 2621745, 2974725, 3283245, 3847725, 5709825, 6447105, 8422785, 8503425, 8945685, 10781505, 12488385, 13470345, 14322945, 15213825, 15340545, 19470465, 19502145, 20075265, 22749825, 25740225, 25756605, 26215245, 27009045

Offset: 1

Antti Karttunen, Apr 19 2019

nonn

If this sequence has no common terms with A324647, or no terms common with A324727, then there are no odd perfect numbers.
The first 16 terms factored:
      7425 = 3^3 * 5^2 * 11,
     76545 = 3^7 * 5 * 7,
     92565 = 3^2 * 5 * 11^2 * 17,
    236925 = 3^6 * 5^2 * 13,
    831105 = 3^2 * 5 * 11 * 23 * 73,
    954765 = 3^2 * 5 * 7^2 * 433,
   1401345 = 3^2 * 5 * 11 * 19 * 149,
   2011905 = 3^3 * 5 * 7 * 2129,
   2048445 = 3^2 * 5 * 7^2 * 929,
   2129985 = 3^2 * 5 * 11 * 13 * 331,
   2253825 = 3^5 * 5^2 * 7 * 53,
   2445345 = 3^2 * 5 * 7^2 * 1109,
   2621745 = 3^2 * 5 * 7^2 * 29 * 41,
   2974725 = 3^4 * 5^2 * 13 * 113,
   3283245 = 3^2 * 5 * 7^2 * 1489,
   3847725 = 3^2 * 5^2 * 7^2 * 349.

Subsequence of A324649.

Cf. A318458, A324647, A324898 (a subsequence).

Select[Range[1, 10^7, 2], BitAnd[#, DivisorSigma[1, #] - #] == # &] (* Michael De Vlieger, Jun 22 2019, after Vincenzo Librandi at A318458 *)

isok(k) = (k%2) && (bitand(k, sigma(k)-k) == k); \\ Michel Marcus, Jul 18 2021

A324532 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A000120(n), A318458(n)] for all other numbers, except f(1) = 0.

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 5, 2, 6, 7, 5, 6, 5, 8, 9, 2, 3, 10, 5, 11, 5, 12, 13, 6, 14, 15, 9, 16, 13, 17, 18, 2, 3, 6, 5, 19, 5, 12, 13, 20, 5, 21, 13, 22, 23, 17, 18, 6, 14, 21, 24, 25, 13, 26, 27, 14, 24, 28, 18, 29, 18, 30, 31, 2, 3, 32, 5, 6, 5, 33, 13, 34, 5, 35, 13, 36, 13, 37, 18, 38, 14, 14, 13, 39, 40, 41, 18, 42, 13, 43, 27, 44, 18, 45, 46, 6, 5, 36, 23, 47, 13

Offset: 1

Antti Karttunen, Mar 05 2019

nonn

Cf. A000120, A318458, A324530, A324531.

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; };
A318458(n) = bitand(n,sigma(n)-n);
Aux324532(n) = if(1==n,0,[hammingweight(n), A318458(n)]);
v324532 = rgs_transform(vector(up_to,n,Aux324532(n)));
A324532(n) = v324532[n];

For n >= 1, a(2^n) = 2.

A336157 Lexicographically earliest infinite sequence such that a(i) = a(j) => A318458(i) = A318458(j) and A336158(i) = A336158(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 2, 1, 4, 5, 2, 6, 2, 7, 8, 1, 2, 9, 2, 10, 11, 3, 2, 6, 4, 12, 13, 14, 2, 15, 2, 1, 11, 6, 11, 16, 2, 3, 11, 17, 2, 18, 2, 19, 20, 7, 2, 6, 4, 21, 22, 23, 2, 24, 22, 6, 22, 17, 2, 25, 2, 26, 27, 1, 11, 28, 2, 6, 11, 28, 2, 29, 2, 5, 30, 31, 11, 32, 2, 31, 33, 6, 2, 34, 35, 3, 11, 36, 2, 37, 22, 38, 11, 39, 40, 6, 2, 41, 20, 42, 2, 43, 2, 44, 45

Offset: 1

Antti Karttunen, Jul 11 2020

nonn

Restricted growth sequence transform of the ordered pair [A318458(n), A336158(n)].
For all i, j:
  A324400(i) = A324400(j) => a(i) = a(j).
  A324401(i) = A324401(j) => a(i) = a(j).

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

Cf. A000265, A046523, A318458, A324400, A324401, A336158.

Cf. A324389, A324530, A324531, A324532 for other similar constructions (also similar by their scatter plots).

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; };
A000265(n) = (n>>valuation(n,2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A336158(n) = A046523(A000265(n));
A318458(n) = bitand(n, sigma(n)-n);
Aux336157(n) = [A318458(n), A336158(n)];
v336157 = rgs_transform(vector(up_to, n, Aux336157(n)));
A336157(n) = v336157[n];

cp's OEIS Frontend

A324398 a(1) = 0; for n > 1, a(n) = A318458(A156552(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A324649 Numbers k such that A318458(k) (bitwise-AND of k and sigma(k)-k) is equal to k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A324389 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A009194(n), A318458(n)] for all other numbers, except f(1) = -1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A324530 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A033879(n), A318458(n)] for all other numbers, except f(1) = -1.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A324898 Odd numbers k such that sigma(k) is congruent to 2 modulo 4 and k = A318458(k), where A318458(k) is bitwise-AND of k and sigma(k)-k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A324531 Lexicographically earliest sequence such that for all i, j >= 1, a(i) = a(j) => f(i) = f(j), where f(n) = [A278222(n), A318458(n)] for all other numbers, except f(1) = 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A324648 a(n) = n - A318458(n), where A318458(n) is bitwise-AND of n and the sum of proper divisors of n (sigma(n)-n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A324897 Odd numbers k such that A318458(k) (bitwise-AND of k and sigma(k)-k) is equal to k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A324532 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A000120(n), A318458(n)] for all other numbers, except f(1) = 0.