A379488 -id:A379488 - cp's OEIS frontend

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.

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).

A379489 a(n) = gcd(n,A003961(n))gcd(sigma(n),A276086(n)) - gcd(n,A276086(n))gcd(sigma(n),A003961(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

0, 0, -1, 0, 5, 0, 1, 12, -2, -6, 5, 2, 1, 0, 15, 0, 17, 0, 9, -102, -1, 6, 5, 0, -24, 0, -5, 0, 29, 12, 1, 12, 3, 6, 35, 62, 1, 12, 11, 0, 41, -18, 1, 18, 15, 6, 5, 2, -6, -72, 3, 6, 17, 0, -3, -6, -5, 42, 29, 84, 1, 0, -19, 0, 35, 0, 1, 12, 3, -42, 17, 30, 1, 0, -65, 34, 59, 18, 9, -12, -2, 60, 41, 14, -3, 0, 15

Offset: 1

Antti Karttunen, Jan 02 2025

sign

Cf. A000203, A003961, A276086, A379486 (positions of 0's), A379487, A379488.

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); };
A379489(n) = { my(s=sigma(n),x=A003961(n),y=A276086(n)); (gcd(n,x)*gcd(s,y))-(gcd(n,y)*gcd(s,x)); };

a(n) = A379487(n) - A379488(n) = A322361(n)*A324644(n) - A324198(n)*A342671(n).

A379491 Multiperfect 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, 6, 28, 496, 8128, 30240, 32760, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, 1379454720, 8589869056, 43861478400, 51001180160, 66433720320, 137438691328, 153003540480, 403031236608, 704575228896, 181742883469056, 6088728021160320, 14942123276641920, 20158185857531904, 275502900594021408, 622286506811515392

Offset: 1

Antti Karttunen, Jan 02 2025

nonn

Intersection of A007691 and A379486.
Cf. A000203, A003961, A276086, A322361, A324198, A324644, A342671, A379485, A379487, A379488, A379492 (complement in A007691).
Subsequences: A323653 (conjectured), A336702 (apart from any hypothetical odd terms > 1).

isA379491(n) = (!(sigma(n)%n) && A379485(n));

A379492 Multiperfect numbers k for which gcd(k,A003961(k))gcd(sigma(k),A276086(k)) is not 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

120, 672, 523776, 1476304896, 14182439040, 31998395520, 518666803200, 13661860101120, 30823866178560, 740344994887680, 796928461056000, 212517062615531520, 69357059049509038080, 87934476737668055040, 154345556085770649600, 170206605192656148480, 1161492388333469337600, 1802582780370364661760, 9186050031556349952000

Offset: 1

Antti Karttunen, Jan 02 2025

nonn

Intersection of A007691 and A000027\A379486.
Cf. A000203, A003961, A276086, A322361, A324198, A324644, A342671, A379485, A379487, A379488, A379491 (complement in A007691).
Cf. A046061 (seems to be a subsequence), A323653.

isA379492(n) = (!(sigma(n)%n) && !A379485(n));

cp's OEIS Frontend

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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PARI

Formula

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.

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Keywords

Links

Crossrefs

Programs

PARI

Formula

A379489 a(n) = gcd(n,A003961(n))*gcd(sigma(n),A276086(n)) - gcd(n,A276086(n))*gcd(sigma(n),A003961(n)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and A276086 is the primorial base exp-function.

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Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A379491 Multiperfect 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.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A379492 Multiperfect numbers k for which gcd(k,A003961(k))*gcd(sigma(k),A276086(k)) is not 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.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

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.

A379489 a(n) = gcd(n,A003961(n))gcd(sigma(n),A276086(n)) - gcd(n,A276086(n))gcd(sigma(n),A003961(n)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and A276086 is the primorial base exp-function.

A379491 Multiperfect 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.

A379492 Multiperfect numbers k for which gcd(k,A003961(k))gcd(sigma(k),A276086(k)) is not 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.