A324658 -id:A324658 - cp's OEIS frontend

A324722 Numbers k such that A324658(A156552(k)) is zero.

9, 21, 25, 35, 49, 55, 77, 95, 121, 125, 133, 143, 169, 185, 203, 209, 221, 265, 289, 299, 301, 319, 323, 343, 361, 371, 377, 413, 427, 437, 445, 451, 473, 481, 493, 497, 511, 527, 529, 531, 539, 553, 559, 583, 589, 605, 611, 623, 629, 667, 679, 689, 703, 707, 737, 763, 767, 779, 791, 793, 799, 805, 817, 841, 845, 847, 851, 869, 871, 899, 901

Offset: 1

Antti Karttunen, Mar 15 2019

nonn

First even term is A005940(1+A324647(1)) = A005940(1+1116225) = 1912898. - Typo corrected by Antti Karttunen, Jul 21 2021

Antti Karttunen, Table of n, a(n) for n = 1..694 (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. A005940, A156552, A324647, A324652, A324658, A324723.

Positions of zeros in A324716.

isA324652(n) = ((2*n)==bitand(2*n, sigma(n)));
isA324722(n) = if(n>1,isA324652(A156552(n)));
k=0; n=0; while(k<105, n++; if(isA324722(n), k++; print1(n,", ")));

A324652 Numbers k such that A318468(k) (bitwise-AND of 2k and sigma(k)) is equal to 2k.

Original entry on oeis.org

6, 12, 18, 20, 24, 28, 36, 40, 48, 56, 80, 88, 96, 100, 104, 112, 160, 176, 192, 196, 200, 204, 208, 220, 224, 260, 264, 272, 304, 320, 336, 352, 368, 384, 392, 416, 448, 464, 496, 544, 550, 580, 608, 640, 648, 650, 672, 704, 736, 768, 784, 832, 896, 928, 992, 1032, 1040, 1044, 1056, 1060, 1068, 1088, 1104, 1120, 1184, 1216

Offset: 1

Antti Karttunen, Mar 14 2019

nonn

Positions of zeros in A324658, fixed points of A324659.
Intersection with A324649 gives A324643.
Intersection with A324726 gives A000396.
In the range 1..2^32 there are only 22 odd terms. See A324647.

Cf. A000203, A191218, A318468, A324647, A324649, A324658, A324659, A324722, A324726.

Some subsequences: A000396, A324643, A324647 (the odd terms).

Select[Range[2000], 2*# == BitAnd[2*#, DivisorSigma[1, #]] &] (* Paolo Xausa, Mar 11 2024 *)

for(n=1,oo,if((n+n)==bitand(2*n,sigma(n)), print1(n, ", ")))

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).

A324659 a(n) = (1/2)A318468(n), where A318468(n) is bitwise-AND of 2n and sigma(n).

Original entry on oeis.org

0, 0, 2, 0, 1, 6, 4, 0, 0, 8, 2, 12, 5, 12, 12, 0, 1, 18, 2, 20, 16, 18, 4, 24, 9, 16, 16, 28, 13, 4, 16, 0, 0, 2, 0, 36, 1, 6, 4, 40, 1, 32, 2, 40, 37, 36, 8, 48, 16, 34, 32, 48, 17, 52, 36, 56, 40, 40, 26, 20, 29, 48, 52, 0, 0, 64, 2, 4, 0, 64, 4, 64, 1, 8, 10, 68, 0, 68, 8, 80, 16, 18, 2, 80, 20, 66, 20, 88, 9, 80

Offset: 1

Antti Karttunen, Mar 14 2019

Cf. A000203, A004198, A318458, A318468, A324648, A324658.

Cf. A324652 (fixed points).

Array[BitAnd[2*#, DivisorSigma[1, #]]/2 &, 100] (* Paolo Xausa, Mar 11 2024 *)

PARI
```
A324659(n) = (bitand(2*n,sigma(n))/2);
```

a(n) = A318468(n)/2.

a(n) = n - A324658(n).

cp's OEIS Frontend

A324722 Numbers k such that A324658(A156552(k)) is zero.

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A324652 Numbers k such that A318468(k) (bitwise-AND of 2*k and sigma(k)) is equal to 2*k.

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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).

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A324659 a(n) = (1/2)*A318468(n), where A318468(n) is bitwise-AND of 2*n and sigma(n).

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Formula

A324652 Numbers k such that A318468(k) (bitwise-AND of 2k and sigma(k)) is equal to 2k.

A324659 a(n) = (1/2)A318468(n), where A318468(n) is bitwise-AND of 2n and sigma(n).