A332226 -id:A332226 - cp's OEIS frontend

A191217 Numbers n such that sigma(n) is congruent to 2 modulo 4.

5, 10, 13, 17, 20, 26, 29, 34, 37, 40, 41, 45, 52, 53, 58, 61, 68, 73, 74, 80, 82, 89, 90, 97, 101, 104, 106, 109, 113, 116, 117, 122, 136, 137, 146, 148, 149, 153, 157, 160, 164, 173, 178, 180, 181, 193, 194, 197, 202, 208, 212, 218, 226, 229, 232, 233, 234, 241, 244, 245, 257, 261, 269

Offset: 1

Luis H. Gallardo, May 26 2011

nonn

These numbers are exactly the numbers of the form 2^a * p^(4b+1) * m^2 where p is a prime number congruent to 1 modulo 4, a is a nonnegative integer, and m is a positive integer coprime to p. In particular, they are also sums of two squares: the sequence has the first 12 terms in common with A132777.

I corrected the above comment by adding the exponent (4b+1) to p, because otherwise it would miss terms like a(614) = 3125 = 5^5, a(1140) = 6250 = 2 * 5^5, a(4421) = 28125 = 5^5 * 3^2, etc. - Antti Karttunen, May 25 2022

			For n=2, a(2) = 10 since sigma(10) = 18 = 4*4 + 2 is congruent to 2 modulo 4

Ray Chandler, Table of n, a(n) for n = 1..10000

Similar to, but different from, A230779, which is a subsequence.
Cf. A191218, A228058, A332226 for other subsequences.
Cf. A353812 (characteristic function).

with(numtheory): gen := proc(b) local n,s,d; for n from 1 to b do s := sigma(n);
if modp(s,4)=2 then print(n); fi; od; end;

for(n=1,10^3,if(2==(sigma(n)%4),print1(n,", "))) /* Joerg Arndt, May 27 2011 */

A332228 Odd numbers n, not powers of primes, such that sigma(n) is congruent to 2 modulo 8.

Original entry on oeis.org

153, 325, 369, 657, 725, 801, 833, 845, 873, 925, 1017, 1233, 1325, 1377, 1445, 1525, 1737, 2009, 2057, 2097, 2169, 2313, 2525, 2529, 2725, 2817, 2925, 3033, 3177, 3321, 3577, 3609, 3681, 3725, 3757, 3897, 3925, 4041, 4113, 4205, 4325, 4361, 4525, 4689, 4753, 4901, 4925, 4961, 5121, 5193, 5337, 5409, 5537, 5553, 5725

Offset: 1

Antti Karttunen, Feb 13 2020

nonn

Proof that any odd perfect number, if such numbers exist at all, has to reside in this sequence: As all terms in A228058 are = 1 modulo 4 (their binary expansion ends as "01"), and taking sigma of an odd perfect number would multiply it by two (shift one bit-position left), the base-2 expansion of that result would end as "010", i.e., sigma(k) modulo 8 should be 2 (not 6) for such numbers k.

Cf. A000203, A332229.

Subsequence of A228058, of A332226 and of A332227.

isA332228(n) = ((n%2)&&!isprimepower(n)&&2==(sigma(n)%8));

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));

A332456 Numbers k such that sigma(sigma(k)) is congruent to 6 modulo 8.

Original entry on oeis.org

9, 67, 163, 193, 201, 202, 271, 338, 469, 489, 505, 547, 579, 606, 670, 673, 729, 813, 982, 1087, 1111, 1123, 1139, 1141, 1153, 1201, 1223, 1351, 1354, 1407, 1414, 1423, 1474, 1515, 1543, 1601, 1630, 1641, 1642, 1791, 1873, 1897, 1930, 1966, 2010, 2017, 2019, 2077, 2083, 2088, 2323, 2326, 2371, 2447, 2455, 2467, 2503

Offset: 1

Antti Karttunen, Feb 15 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n)

Cf. A000203, A051027, A332226, A332457, A332458.

isA332456(n) = (6==(sigma(sigma(n))%8));

A332457 Numbers k such that sigma(k) == 2 modulo 8 and sigma(sigma(k)) == 6 modulo 8.

Original entry on oeis.org

193, 202, 673, 1153, 1201, 1354, 1601, 1642, 1873, 2017, 2088, 2593, 2682, 2753, 3049, 3112, 3217, 3313, 3328, 3754, 3898, 4041, 4084, 4177, 4273, 4337, 4426, 4561, 5193, 5233, 5386, 5449, 5482, 5849, 6337, 6353, 6826, 6922, 7002, 7057, 7114, 7393, 7402, 7537, 7793, 8081, 8104, 8353, 8564, 8698, 8872, 9049, 9377, 9601

Offset: 1

Antti Karttunen, Feb 15 2020

nonn

That the first part of the condition is necessary for odd perfect numbers, see A332228, that the second part of the condition is necessary, see A019283 and A326181.

Cf. A000203, A019283, A051027, A326181.
Intersection of A332226 and A332456.
Cf. A332458 (a subsequence of non-primepower odd terms).

[k:k in [1..9700]| DivisorSigma(1,k) mod 8 eq 2 and DivisorSigma(1, DivisorSigma(1,k)) mod 8 eq 6]; // Marius A. Burtea, Feb 15 2020

Select[Range[10000],With[{c=DivisorSigma[1,#]},Mod[c,8]==2&&Mod[DivisorSigma[1,c],8]==6&]]  (* Harvey P. Dale, Nov 23 2024 *)

isA332457(n) = { my(s=sigma(n)); ((2==(s%8)) && (6==(sigma(s)%8))); };

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A191217 Numbers n such that sigma(n) is congruent to 2 modulo 4.

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A332228 Odd numbers n, not powers of primes, such that sigma(n) is congruent to 2 modulo 8.

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

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A332456 Numbers k such that sigma(sigma(k)) is congruent to 6 modulo 8.

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A332457 Numbers k such that sigma(k) == 2 modulo 8 and sigma(sigma(k)) == 6 modulo 8.

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