A332458 -id:A332458 - cp's OEIS frontend

A364286 Composite numbers k for which A324644(k)/A324198(k) = 2.

33, 51, 69, 91, 99, 135, 141, 145, 153, 159, 187, 207, 213, 217, 285, 295, 303, 321, 339, 391, 411, 423, 427, 435, 445, 477, 507, 519, 573, 637, 639, 679, 681, 699, 771, 783, 799, 843, 855, 861, 885, 895, 901, 909, 933, 951, 963, 1017, 1041, 1057, 1059, 1071, 1081, 1083, 1147, 1149, 1173, 1185, 1195, 1203, 1207

Offset: 1

Antti Karttunen, Jul 17 2023

nonn

See comments in A351458.

All terms are odd. Of the 63 initial terms of A349169, only term 13923 occurs also in this sequence. The first common term with A332458 is 161257. - Antti Karttunen, Mar 10 2024

Cf. A000203, A276086, A324198, A324644, A332458, A349169, A351458, A371082 (subsequence).

Subsequence of A082686.

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[1350], CompositeQ], GCD[#2, #3]/GCD[#1, #3] == 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); };
isA364286(n) = if(isprime(n), 0, my(u=A276086(n)); (gcd(sigma(n),u)==2*gcd(n,u))); \\ Antti Karttunen, Mar 10 2024

A332445 Numbers k of the form 4m+1 for which A087808(sigma(k)) is equal to 2*A087808(k).

Original entry on oeis.org

2009, 19377, 37809, 59373, 74673, 115677, 270041, 310329, 354609, 357309, 720425, 732321, 841437, 2071737, 2612269, 3131149, 3866461, 3930929, 5172093, 5593981, 7118753, 7903961, 8224173, 9327393, 9438129, 11452321, 12708025, 18857209, 18861889, 18875313, 19110321, 20278269, 20709225, 20950061, 23963597, 24895153

Offset: 1

Antti Karttunen, Feb 14 2020

nonn

Numbers k such that A332224(k) is equal to A087808(2*k) and k == 1 mod 4.
Notably, the only square among the first 299 terms is a(248) = 5808421369 = 76213^2. sigma(5808421369) = 5808497583 == 3 (mod 4) == 7 (mod 8). Of the remaining 298 terms < 2^33, 92 are such that sigma(k) == 6 (mod 8) and 206 are such that sigma(k) == 2 (mod 8), that is, are terms of A332227.
Question: Why the terms come in clusters? Compare also the scatterplots of A087808 and A332224, and a similar sequence A332465.

Intersection of A016813 and A332446.
Cf. also A228058, A332227, A332465.
Cf. A000203, A087808, A286357, A332224, A332225, A332458.

A087808(n) = if(n<1, 0, if(n%2==0, 2*A087808(n/2), A087808((n-1)/2)+1));
isA332445(n) = ((1==(n%4))&&(2*A087808(n)==A087808(sigma(n))));

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))); };

cp's OEIS Frontend

A364286 Composite numbers k for which A324644(k)/A324198(k) = 2.

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A332445 Numbers k of the form 4m+1 for which A087808(sigma(k)) is equal to 2*A087808(k).

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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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