A191218 -id:A191218 - cp's OEIS frontend

A324718 Odd numbers n for which bitand(2n,sigma(n)) = 2*bitand(n,sigma(n)-n), where bitand is bitwise-AND, A004198.

1, 5, 9, 17, 37, 41, 73, 137, 149, 153, 257, 261, 277, 293, 337, 405, 521, 529, 549, 577, 593, 641, 661, 673, 677, 1025, 1033, 1061, 1093, 1097, 1109, 1153, 1193, 1289, 1297, 1301, 1321, 1361, 2053, 2069, 2081, 2089, 2097, 2113, 2129, 2209, 2213, 2225, 2309, 2341, 2377, 2389, 2593, 2633, 2689, 2693, 2729, 2825, 4129, 4133, 4177, 4229

Offset: 1

Antti Karttunen, Mar 14 2019

nonn

Odd numbers n for which 2*A318458(n) = A318468(n). If there are no common terms with A324719, then there are no odd perfect numbers.

This is not a subsequence of A191218, because terms 1, 9, 529, 2209, 10609, 77841, 83521, 263169, 279841, 330625, 528529, ... are not present in A191218.

Subsequence of A324639.

Cf. A004198, A191218, A318458, A318468, A324647, A324719, A324727.

Select[Range[1, 10^4, 2], Block[{s = DivisorSigma[1, #]}, BitAnd[2*#, s] == 2* BitAnd[#, s-#]] &] (* Paolo Xausa, Mar 11 2024 *)

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

A351538 Numbers k such that both k and sigma(k) are congruent to 2 modulo 4 and the 3-adic valuation of sigma(k) is exactly 1.

Original entry on oeis.org

26, 74, 122, 146, 194, 218, 234, 314, 362, 386, 458, 482, 554, 626, 650, 666, 674, 698, 746, 794, 818, 842, 866, 914, 1082, 1098, 1154, 1202, 1226, 1314, 1322, 1346, 1418, 1466, 1514, 1538, 1658, 1706, 1746, 1754, 1850, 1874, 1962, 1994, 2018, 2042, 2066, 2106, 2138, 2186, 2234, 2258, 2306, 2402, 2426, 2474, 2498

Offset: 1

Antti Karttunen, Feb 14 2022

nonn

All the terms of the form 4u+2 in A349745 (if they exist) are found in this sequence. As A351537 is the intersection of A191218 and A329963, and the latter has asymptotic density zero, so has this sequence also. It is conjectured that A351555(a(n)) is nonzero for all n, which would imply that the intersection with A349745 is empty. - Antti Karttunen, Feb 19 2022

Antti Karttunen, Table of n, a(n) for n = 1..25000
Index entries for sequences related to sigma(n)

Cf. A191218, A329963, A349745, A351537.

Probably a subsequence of A351543. (See also A351550, A351555).

isA351538(n) = if(!(2==(n%4)),0, my(s=sigma(n)); (2 == (s%4)) && (1==valuation(s,3)));

a(n) = 2 * A351537(n).

A324719 Odd numbers n for which bitor(2n,sigma(n)) = 2*bitor(n,sigma(n)-n), where bitor is bitwise-OR, A003986.

Original entry on oeis.org

3, 7, 15, 27, 31, 51, 55, 63, 111, 119, 123, 125, 127, 219, 255, 411, 447, 485, 493, 495, 505, 511, 735, 765, 771, 831, 879, 927, 959, 965, 985, 1011, 1023, 1563, 1587, 1611, 1731, 1779, 1791, 1799, 1887, 1921, 1923, 1945, 1975, 1983, 1991, 2019, 2031, 2041, 2043, 2045, 2047, 3099, 3183, 3231, 3279, 3291, 3327, 3459, 3535, 3579

Offset: 1

Antti Karttunen, Mar 14 2019

Odd numbers n for which 2*A318456(n) = A318466(n).
If there are no common terms with A324718, then there are no odd perfect numbers.
The following subsequence of terms k are those with sigma(k) == 2 (mod 4): 3725, 7281, 15325, 24525, 25713, 32481, 51633, 52209, 59121, 63553, 114417, 117009, 120753, 121725, 122725, 123245, 130833, 208881, 236925, 241325, 245725, 253325, 261297, 384993, 411633, 457713, 468081, 482481, 482525, 482725, 483325, ..., and are thus present in A191218.

Cf. A003986, A191218, A318456, A318466, A324718, A324727.

Select[Range[1, 10^4, 2], Block[{s = DivisorSigma[1, #]}, BitOr[2*#, s] == 2* BitOr[#, s-#]] &] (* Paolo Xausa, Mar 11 2024 *)

for(n=1,oo,if((n%2) && (2*(bitor(n, sigma(n)-n))==bitor(n+n, sigma(n))),print1(n,", ")));

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,", ")));

A332227 Odd numbers k such that sigma(k) is congruent to 2 modulo 8.

Original entry on oeis.org

17, 41, 73, 89, 97, 113, 137, 153, 193, 233, 241, 257, 281, 313, 325, 337, 353, 369, 401, 409, 433, 449, 457, 521, 569, 577, 593, 601, 617, 641, 657, 673, 725, 761, 769, 801, 809, 833, 845, 857, 873, 881, 925, 929, 937, 953, 977, 1009, 1017, 1033, 1049, 1097, 1129, 1153, 1193, 1201, 1217, 1233, 1249, 1289, 1297, 1321

Offset: 1

Antti Karttunen, Feb 13 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..25000
Index entries for sequences related to sigma(n)

Cf. A000203.
Subsequence of A191218 and of A332226.
Cf. A332228 (a subsequence, terms that are not prime powers).

Select[Range[1,1401,2],Mod[DivisorSigma[1,#],8]==2&] (* Harvey P. Dale, Apr 12 2021 *)

isA332227(n) = ((n%2)&&2==(sigma(n)%8));

A371082 Composite numbers for which A324644(n)/A324198(n) = 2 and sigma(n) == 2 (mod 4).

Original entry on oeis.org

153, 477, 637, 909, 1017, 1233, 1557, 2097, 3577, 4753, 9457, 10693, 10933, 12393, 13357, 14013, 15337, 17629, 20817, 21097, 21217, 22021, 26353, 29449, 30037, 30717, 31117, 31149, 31797, 32013, 32229, 32337, 32481, 32977, 35557, 35917, 38637, 38725, 41797, 42237, 50029, 53557, 56497, 56677, 56953, 58621, 59437, 60309

Offset: 1

Antti Karttunen, Mar 10 2024

Intersection of A191218 and A364286.
Apparently also the intersection of A228058 and A364286.
Cf. A000203, A276086, A351458.

f[x_] := Block[{m, i, n = x, p}, m = i = 1; While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m]; Select[Select[Range[2^16], CompositeQ], GCD[#2, #3]/GCD[#1, #3] == Mod[#2, 4] == 2 & @@ {#, DivisorSigma[1, #], f[#]} &] (* Michael De Vlieger, Mar 10 2024 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA371082(n) = if(isprime(n) || (2!=(sigma(n)%4)), 0, my(u=A276086(n)); (gcd(sigma(n),u)==2*gcd(n,u)));

A332465 Numbers n for which A269174(sigma(n)) is equal to 2*A269174(n).

Original entry on oeis.org

6, 28, 348, 496, 732, 886, 2924, 3573, 4972, 5448, 7544, 8128, 23388, 54842, 66928, 89200, 92296, 109786, 118064, 121552, 349512, 356488, 367472, 550432, 634784, 839984, 842452, 1234048, 1561408, 1797496, 2154584, 2364832, 2788808, 2927992, 3451456, 3585328, 5952364, 5991852, 6687136, 8238752, 10594336, 11210712, 11261020

Offset: 1

Antti Karttunen, Feb 16 2020

nonn

Numbers n such that A332464(n) is equal to A269174(2*n).
There are only three odd terms <= 2^32 among the first 113 terms of this sequence: 3573, 29255157, 936109557. Because A269174 preserves the 2-adic valuation of its argument, all such odd terms are of the form 4m+1, and must be present in A191218. Incidentally, these three terms are also present in A228058, but not in A332227.
See from the graph how unevenly the terms appear. Compare also the scatter plots of A269174 and A332464, also of a similar sequence A332445.

			          k   factorization        sigma(k)     A269174(sigma(k)) = A269174(2*k)
        348 = 2^2 * 3 * 29         840          2008,
       3573 = 3^2 * 397            5174         15486,
   29255157 = 3^2 * 3250573        42257462     126737534,
  936109557 = 3^2 * 104012173      1352158262   4055424126.

Cf. A000203, A269174, A332464.
Cf. A000396 (a subsequence).
Cf. also A191218, A228058, A332227, A332445, A332446.

b[n_] := BitAnd[BitOr[n, 2n], BitOr[BitXor[n, 2n], BitXor[n, 4n]]];
okQ[n_] := b[DivisorSigma[1, n]] == 2 b[n];
Reap[For[n = 1, n <= 12*10^6, n++, If[okQ[n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 23 2020 *)

A269174(n) = bitand(bitor(n,n<<1),bitor(bitxor(n,n<<1),bitxor(n,n<<2)));
isA332465(n) = (A269174(sigma(n))==2*A269174(n));

A351534 Odd numbers for which sigma(k) is congruent to 2 modulo 4 and the 3-adic valuation of k is one larger than the 3-adic valuation of sigma(k).

Original entry on oeis.org

45, 261, 369, 909, 1017, 1233, 1341, 1557, 2313, 2529, 2637, 2853, 3177, 3501, 3609, 4149, 4293, 4581, 5121, 5445, 5553, 5733, 5769, 5877, 6093, 6525, 6849, 7173, 7389, 7713, 8361, 8469, 8793, 9117, 9225, 9441, 9981, 10629, 10737, 10953, 11061, 11601, 11709, 12249, 12357, 12681, 12897, 13005, 13329, 13977, 14517

Offset: 1

Antti Karttunen, Feb 13 2022

nonn

Index entries for sequences related to sigma(n)

Subsequence of A351533, which is intersection of A008585 and A191218.

Cf. A000203, A007949, A351536.

Select[Range[1, 15000, 2], Mod[(s = DivisorSigma[1, #]), 4] == 2 && IntegerExponent[#, 3] - IntegerExponent[s, 3] == 1 &] (* Amiram Eldar, Feb 13 2022 *)

isA351534(n) = if(!(n%2),0,my(s=sigma(n)); (2 == (s%4)) && (valuation(n,3) == (1+valuation(s,3))));

A351537 Odd numbers k for which sigma(k) is congruent to 2 modulo 4 and is not a multiple of 3.

Original entry on oeis.org

13, 37, 61, 73, 97, 109, 117, 157, 181, 193, 229, 241, 277, 313, 325, 333, 337, 349, 373, 397, 409, 421, 433, 457, 541, 549, 577, 601, 613, 657, 661, 673, 709, 733, 757, 769, 829, 853, 873, 877, 925, 937, 981, 997, 1009, 1021, 1033, 1053, 1069, 1093, 1117, 1129, 1153, 1201, 1213, 1237, 1249, 1297, 1321, 1381, 1413

Offset: 1

Antti Karttunen, Feb 14 2022

nonn

Terms are of the form p^e*m^2 where e is 1 or 9 mod 12, p is a prime = 1 mod 12 and m is an odd number not divisible by p with sigma(m^2) not divisible by 3, i.e., q^e || m implies e is not 1 mod 3 or q = 2 mod 3. - Charles R Greathouse IV, Feb 14 2022

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Intersection of A191218 and A329963.
Contains A068228 as a subsequence.
Terms of A351538 halved.

Select[Range[1, 1440, 2], MemberQ[{2, 10}, Mod[DivisorSigma[1, #], 12]] &] (* Michael De Vlieger, Feb 14 2022 *)

isA351537(n) = if(!(n%2),0,my(s=sigma(n)); (2 == (s%4)) && (0 != (s%3)));

list(lim)=my(v=List()); forstep(m=1,sqrtint(lim\13),2, my(m2=m^2); if(sigma(m2)%3==0,next); forprimestep(p=13,lim\m2,12, m%p && listput(v,p*m2))); forstep(e=9,logint(lim\1,13),[4,8], forstep(m=1,sqrtint(lim\13^e),2, my(m2=m^2); if(sigma(m2)%3==0,next); forprimestep(p=13,lim\m2,12, m%p && listput(v,p^e*m2)))); Set(v) \\ Charles R Greathouse IV, Feb 14 2022

a(n) = A351538(n)/2.

A359151 Numbers of the form 4u+1 with an odd number of prime factors (counted with multiplicity).

Original entry on oeis.org

5, 13, 17, 29, 37, 41, 45, 53, 61, 73, 89, 97, 101, 105, 109, 113, 117, 125, 137, 149, 153, 157, 165, 173, 181, 193, 197, 229, 233, 241, 245, 257, 261, 269, 273, 277, 281, 285, 293, 313, 317, 325, 333, 337, 345, 349, 353, 357, 369, 373, 385, 389, 397, 401, 405, 409, 421, 425, 429, 433, 449, 457, 461

Offset: 1

Antti Karttunen, Dec 17 2022

nonn

Intersection of A016813 and A026424.
Setwise difference A067019 \ A359153.
Setwise difference A016813 \ A359161.
Cf. A002144, A191218 (subsequences).
Cf. A359150 (characteristic function).

PARI
```
isA359151(n) = A359150(n);
```

cp's OEIS Frontend

A324718 Odd numbers n for which bitand(2n,sigma(n)) = 2*bitand(n,sigma(n)-n), where bitand is bitwise-AND, A004198.

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A351538 Numbers k such that both k and sigma(k) are congruent to 2 modulo 4 and the 3-adic valuation of sigma(k) is exactly 1.

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A324719 Odd numbers n for which bitor(2n,sigma(n)) = 2*bitor(n,sigma(n)-n), where bitor is bitwise-OR, A003986.

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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.

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A332227 Odd numbers k such that sigma(k) is congruent to 2 modulo 8.

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A371082 Composite numbers for which A324644(n)/A324198(n) = 2 and sigma(n) == 2 (mod 4).

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A332465 Numbers n for which A269174(sigma(n)) is equal to 2*A269174(n).

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A351534 Odd numbers for which sigma(k) is congruent to 2 modulo 4 and the 3-adic valuation of k is one larger than the 3-adic valuation of sigma(k).

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A351537 Odd numbers k for which sigma(k) is congruent to 2 modulo 4 and is not a multiple of 3.