A371082 -id:A371082 - 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

A387164 Numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)), and that satisfy Euler's condition for odd perfect numbers (A228058).

Original entry on oeis.org

117, 153, 333, 369, 425, 477, 549, 637, 657, 845, 873, 909, 925, 1017, 1053, 1233, 1325, 1377, 1413, 1421, 1445, 1525, 1557, 1629, 1737, 1773, 1805, 1813, 1825, 2009, 2097, 2169, 2225, 2313, 2493, 2525, 2529, 2597, 2637, 2725, 2817, 2825, 2853, 2989, 2997, 3033, 3177, 3321, 3357, 3425, 3509, 3573, 3577, 3609, 3725

Offset: 1

Antti Karttunen, Aug 28 2025

Terms k of A228058 for which A322361(k) = A342671(k), or equally, such that A319626(k) = A349164(k).

Intersection of A228058 and A349174.
Union of A387166 and A387167.
Differs from its subsequence A387167 for the first time at n=201, where a(201) = 14157, while A387167(201) = 14225.
Cf. A000203, A003961, A319626, A322361, A342671, A349164.
Cf. also A371082.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA228058(n) = if(!(n%2)||(omega(n)<2), 0, my(f=factor(n), y=0); for(i=1, #f~, if(1==(f[i, 2]%4), if((1==y)||(1!=(f[i, 1]%4)), return(0), y=1), if(f[i, 2]%2, return(0)))); (y));
isA349174(n) = if(!(n%2), 0, my(u=A003961(n)); gcd(u, sigma(n))==gcd(u, n));
isA387164(n) = (isA228058(n) && isA349174(n));

A387165 Nondeficient numbers k for which A324644(k)/A324198(k) = 2.

Original entry on oeis.org

38745, 77805, 78435, 118755, 141075, 157815, 210735, 237195, 241605, 294975, 300105, 323505, 364455, 371925, 390195, 409185, 455715, 475335, 499905, 567945, 607635, 660825, 701415, 733005, 766395, 806085, 809325, 872235, 885465, 891135, 937755, 964845, 978705, 1101555, 1150065, 1201095, 1229445, 1265355, 1293705

Offset: 1

Antti Karttunen, Aug 28 2025

First three nonmultiples of 5 occur at a(138), a(276), a(356) = 4446981, 8909901, 11234223. (Cf. A005231, A064001).

Intersection of A023196 and A364286.
Cf. A000203, A276086.
Cf. also A371082, A386422, A387163 and A047802, A064001, A115414.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
is_A387165(n) = if(sigma(n)<2*n, 0, my(u=A276086(n)); (gcd(sigma(n),u)==2*gcd(n,u)));

{k | sigma(k) >= 2*k, A324644(k) = 2*A324198(k)}.

cp's OEIS Frontend

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

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A387164 Numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)), and that satisfy Euler's condition for odd perfect numbers (A228058).

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A387165 Nondeficient numbers k for which A324644(k)/A324198(k) = 2.

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