A324644 -id:A324644 - 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

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)));

A386422 Odd numbers k that are closer to being perfect than previous terms and also satisfy the condition that A324644(k)/A324198(k) = 2.

Original entry on oeis.org

3, 33, 99, 135, 855, 2295, 19575, 38745, 63855, 121485, 371925, 3870195, 8109585, 28306005, 36340395, 113215095, 463084245, 672363615, 675916395, 686574735, 1208140395

Offset: 1

Antti Karttunen, Jul 21 2025

Questions: Are there only multiples of 5 after the three initial terms? Are there any common terms with A228058?

Apart from initial 3, a subsequence of A364286.
Cf. A000203, A000396, A228058, A324198, A324644.
Cf. also A171929, A228059, A386419, A386420, A386421 for similar sequences.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
is_A364286(n) = if(isprime(n), 0, my(u=A276086(n)); (gcd(sigma(n),u)==2*gcd(n,u))); \\ Antti Karttunen, Jul 21 2025
m=-1; n=-1; k=0; while(m!=0, n+=2; if(!((n-1)%(2^25)),print1("("n")")); if(isprime(n) || is_A364286(n), if((m<0) || abs((sigma(n)/n)-2)

A369261 Lexicographically earliest infinite sequence such that a(i) = a(j) => A324644(i) = A324644(j) and A369445(i) = A369445(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 15, 23, 16, 20, 24, 25, 26, 27, 21, 28, 29, 30, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 29, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 35, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 62, 53, 51, 70, 71, 72, 58, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 63

Offset: 1

Antti Karttunen, Jan 25 2024

nonn

Restricted growth sequence transform of the ordered pair [A324644(n), A369445(n)], or equally, of the pair [A000203(n), A324644(n)], or equally, of the pair [A000203(n), A369445(n)].

For all i, j >= 1: a(i) = a(j) => A286603(i) = A286603(j).

Cf. A000203, A276086, A324644, A369445.

Cf. also A286603, A291751, A369260 and A353523, A369047, A369051, A369062.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324644(n) = gcd(sigma(n),A276086(n));
Aux369261(n) = { my(u=A324644(n)); [u, sigma(n)/u]; };
v369261 = rgs_transform(vector(up_to, n, Aux369261(n)));
A369261(n) = v369261[n];

A387163 Numbers k such that sigma(k) >= 3*k and A324644(k)/A324198(k) = 3.

Original entry on oeis.org

10065440, 12794600, 22862840, 24806600, 27399680, 30692480, 33904640, 41629280, 41851040, 46803680, 54625760, 54842480, 70384160, 81915680, 83545280, 87311840, 91571480, 93964640, 95221280, 98030240, 101978240, 103527200, 106719200, 110116160, 121983680, 122904320, 137106200, 137359040, 143195360, 143638880, 144491200

Offset: 1

Antti Karttunen, Aug 28 2025

This sequence contains all 3-perfect numbers (A005820) that are not multiples of three: 459818240 (= a(99)), 51001180160, and also any such hypothetical triperfects of the form 4u+2, when 2u+1 is not multiple of 3. See comments in A351458.

Antti Karttunen, Table of n, a(n) for n = 1..1268 (terms < 2^33).
Index entries for sequences related to primorial base
Index entries for sequences related to sigma(n)

Intersection of A023197 and A387161.
Cf. A000203, A005820, A276086, A324198, A324644, A351458.
Cf. also A387165.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
is_A387163(n) = if(sigma(n)<3*n, 0, my(u=A276086(n)); (gcd(sigma(n), u)==3*gcd(n, u)));

{k | sigma(k) >= 3*k, A324644(k) = 3*A324198(k)}.

A387161 Numbers k for which A324644(k)/A324198(k) = 3.

Original entry on oeis.org

2, 14, 26, 62, 74, 86, 122, 134, 146, 152, 176, 182, 206, 212, 224, 254, 272, 290, 302, 314, 326, 338, 368, 386, 422, 428, 434, 446, 476, 542, 554, 566, 578, 590, 626, 632, 644, 656, 662, 674, 680, 722, 734, 752, 782, 794, 812, 842, 848, 854, 866, 890, 914, 920, 926, 974, 1046, 1058, 1082, 1088, 1094, 1136, 1154

Offset: 1

Antti Karttunen, Aug 28 2025

Subsequences: A387163 (terms whose abundancy >= 3).
Cf. A000203, A276086, A324198, A324644, A351458.
Cf. also A364286.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
is_A387161(n) = { my(u=A276086(n)); (gcd(sigma(n), u)==3*gcd(n, u)); };

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)}.

A351458 Numbers k for which k * gcd(sigma(k), A276086(k)) is equal to sigma(k) * gcd(k, A276086(k)), where A276086 is the primorial base exp-function, and sigma gives the sum of divisors of its argument.

Original entry on oeis.org

1, 10, 56, 9196, 9504, 56160, 121176, 239096, 354892, 411264, 555520, 716040, 804384, 904704, 1063348, 1387386, 1444352, 1454112, 1884800, 2708640, 3317248, 3548920, 4009824, 4634784, 6179712, 6795360, 7285248, 14511744, 16328466, 28377216, 29855232, 31940280, 37444736, 42711552, 49762944, 52815744

Offset: 1

Antti Karttunen, Feb 13 2022

nonn

Numbers k such that k * A324644(k) = A000203(k) * A324198(k).
Numbers k such that gcd(A064987(k), A324580(k)) = gcd(A064987(k), A351252(k)).
Numbers k such that their abundancy index [sigma(k)/k] is equal to A324644(k)/A324198(k). See A364286.
A324644 gives odd values for even numbers and for the odd squares. A324198 is odd on all arguments, therefore on odd squares the above equation reduces to odd * odd = odd * odd, and on odd nonsquares as odd * even = even * odd. It is an open question whether there are any odd terms after the initial a(1)=1.
If k is even, but not a multiple of 3, then A276086(k) is a multiple of 3, but not even (i.e., is an odd multiple of 3). If for such k also sigma(k) = 3*k, then A007949(A324644(k)) = min(A007949(sigma(k)), A007949(A276086(k))) = 1, while A007949(A324198(k)) = min(A007949(k), A007949(A276086(k))) = 0, therefore all such k's do occur in this sequence, for example, the two known terms of A005820 (3-perfect numbers) that are not multiples of three: 459818240, 51001180160, but also any hypothetical term of A005820 of the form 4u+2, where 2u+1 is not multiple of 3, and which by necessity is then also an odd perfect number.
Similarly, of the 65 known 5-multiperfect numbers (A046060), those 20 that are not multiples of five are included in this sequence. Note that all 65 are multiples of six.
It is conjectured that the intersection of this sequence with the multiperfect numbers (A007691) gives A323653, see comments in the latter.
For all even terms k of this sequence, A007814(A000203(k)) = A007814(k), sigma preserves the 2-adic valuation, and A007949(A000203(k)) >= A007949(k), i.e., does not decrease the 3-adic valuation. The condition is equivalence (=) when k is a multiple of 6. With odd terms, any hypothetical odd perfect number x would yield a one greater 2-adic valuation for sigma(x) than for x, but would satisfy the main condition of this sequence. - Corrected Feb 17 2022
If k is a nonsquare positive odd number (in A088828), then it must be a term of A191218. - Antti Karttunen, Mar 10 2024

Cf. A000203, A005820, A007691, A007814, A007949, A046060, A088828, A191218, A276086, A323653, A324198, A324644, A327938, A364286, A351459 (intersection with a similar sequence, A349745).

Cf. also A351549.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA351458(n) = { my(s=sigma(n), z=A276086(n)); (n*gcd(s,z))==(s*gcd(n,z)); };

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]); \\ Works OK with rationals also!
isA351458(n) = { my(orgn=n, s=sigma(n), abi=s/n, p=2, q=A006530(abi), d, e1, e2); while((1!=abi)&&(p<=q), d = n%p; e1 = min(d, valuation(s, p)); e2 = min(d, valuation(orgn, p)); d = e1-e2; if(valuation(abi,p)!=d, return(0), abi /= (p^d)); n = n\p; p = nextprime(1+p)); (abi==1); }; \\ (This implementation does not require the construction of largish intermediate numbers, A276086, but might still be slower and return a few false positives on the long run, so please check the results with the above program). - Antti Karttunen, Feb 19 2022

A379486 Numbers k for which gcd(k,A003961(k))gcd(sigma(k),A276086(k)) is equal to gcd(k,A276086(k))gcd(sigma(k),A003961(k)), where A003961(n) is fully multiplicative with a(prime(i)) = prime(i+1), and A276086 is the primorial base exp-function.

Original entry on oeis.org

1, 2, 4, 6, 14, 16, 18, 24, 26, 28, 40, 54, 62, 64, 66, 74, 86, 102, 114, 122, 134, 138, 146, 152, 162, 169, 174, 176, 182, 184, 186, 206, 222, 234, 254, 270, 280, 282, 289, 290, 302, 304, 306, 308, 314, 318, 326, 338, 342, 354, 360, 361, 366, 368, 380, 384, 386, 402, 414, 422, 426, 434, 438, 441, 446, 448, 456, 474, 496

Offset: 1

Antti Karttunen, Jan 01 2025

nonn

Cf. A000203, A003961, A276086, A322361, A324198, A324644, A342671, A379485 (characteristic function), A379487, A379488.
Positions of 0's in A379489.
Cf. A379491 (subsequence, terms that are multiperfect numbers, A007691).

PARI
```
is_A379486 = A379485;
```

{Numbers k such that A379487(k) = A379488(k)}.

{Numbers k such that A322361(k)/A324198(k) = A324644(k)/A342671(k)}.

A379487 a(n) = gcd(n,A003961(n)) * gcd(sigma(n),A276086(n)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and A276086 is the primorial base exp-function.

Original entry on oeis.org

1, 3, 2, 1, 6, 3, 2, 15, 1, 9, 6, 3, 2, 3, 30, 1, 18, 3, 10, 3, 2, 9, 6, 15, 1, 3, 10, 1, 30, 15, 2, 21, 6, 9, 42, 63, 2, 15, 14, 45, 42, 3, 2, 21, 30, 9, 6, 3, 1, 3, 6, 7, 18, 15, 2, 15, 10, 45, 30, 105, 2, 3, 2, 1, 42, 3, 2, 21, 6, 63, 18, 45, 2, 3, 10, 35, 66, 21, 10, 3, 1, 63, 42, 21, 2, 3, 30, 45, 90, 15, 14, 21

Offset: 1

Antti Karttunen, Jan 02 2025

nonn

Cf. A000203, A003961, A276086, A322361, A324644, A379486, A379488, A379489.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A379487(n) = { my(s=sigma(n),x=A003961(n),y=A276086(n)); (gcd(n,x)*gcd(s,y)); };

a(n) = A322361(n) * A324644(n).

cp's OEIS Frontend

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

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

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A386422 Odd numbers k that are closer to being perfect than previous terms and also satisfy the condition that A324644(k)/A324198(k) = 2.

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A369261 Lexicographically earliest infinite sequence such that a(i) = a(j) => A324644(i) = A324644(j) and A369445(i) = A369445(j), for all i, j >= 1.

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A387163 Numbers k such that sigma(k) >= 3*k and A324644(k)/A324198(k) = 3.

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A387161 Numbers k for which A324644(k)/A324198(k) = 3.

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

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A351458 Numbers k for which k * gcd(sigma(k), A276086(k)) is equal to sigma(k) * gcd(k, A276086(k)), where A276086 is the primorial base exp-function, and sigma gives the sum of divisors of its argument.

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A379486 Numbers k for which gcd(k,A003961(k))*gcd(sigma(k),A276086(k)) is equal to gcd(k,A276086(k))*gcd(sigma(k),A003961(k)), where A003961(n) is fully multiplicative with a(prime(i)) = prime(i+1), and A276086 is the primorial base exp-function.

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A379486 Numbers k for which gcd(k,A003961(k))gcd(sigma(k),A276086(k)) is equal to gcd(k,A276086(k))gcd(sigma(k),A003961(k)), where A003961(n) is fully multiplicative with a(prime(i)) = prime(i+1), and A276086 is the primorial base exp-function.