A336566 -id:A336566 - cp's OEIS frontend

A336555 Numbers k for which A326144(k) <> A336566(k).

45, 63, 75, 99, 117, 126, 144, 147, 150, 153, 171, 175, 198, 207, 216, 242, 245, 250, 252, 261, 275, 279, 294, 300, 315, 325, 333, 342, 350, 363, 369, 387, 396, 405, 414, 423, 425, 475, 477, 495, 504, 507, 525, 531, 539, 549, 550, 558, 567, 575, 585, 588, 600, 603, 605, 612, 630, 637, 639, 657, 675, 684, 693, 700, 711

Offset: 1

Antti Karttunen, Jul 28 2020

nonn

Question: Is A228058 a subsequence of this one?

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A228058, A326144, A336566.

\\ Needs also code from A326144 and A336566:
isA336555(n) = (A326144(n) != A336566(n));

A336647 a(n) = n - A336566(n).

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 8, 8, 1, 10, 1, 10, 9, 15, 1, 15, 1, 18, 11, 14, 1, 18, 24, 16, 25, 14, 1, 18, 1, 31, 15, 20, 13, 35, 1, 22, 17, 30, 1, 30, 1, 42, 42, 26, 1, 34, 48, 49, 21, 50, 1, 42, 17, 54, 23, 32, 1, 54, 1, 34, 62, 63, 19, 54, 1, 66, 27, 66, 1, 69, 1, 40, 74, 74, 19, 66, 1, 78, 80, 44, 1, 70, 23, 46, 33, 86

Offset: 1

Antti Karttunen, Jul 30 2020

sign

Some terms, for example a(600) and a(6552), are negative. - Georg Fischer, Jul 31 2020

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000203, A007947, A057723, A308135, A336563, A336564, A336566.
Cf. A336555 (positions where differs from A336646).
Cf. A336565 (positions where a(n) = 2*n - A057723(n) = n - A336563(n)).
Cf. also A336645.

A007947(n) = factorback(factorint(n)[, 1]);
A057723(n) = { my(r=A007947(n)); (r*sigma(n/r)); };
A308135(n) = (sigma(n)-A057723(n));
A336563(n) = (A057723(n)-n);
A336564(n) = (n - A308135(n));
A336566(n) = gcd(A336563(n), A336564(n));
A336647(n) = (n - A336566(n));

a(n) = n - A336566(n).

A336563 Sum of proper divisors of n that are divisible by every prime that divides n.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 6, 3, 0, 0, 6, 0, 0, 0, 14, 0, 6, 0, 10, 0, 0, 0, 18, 5, 0, 12, 14, 0, 0, 0, 30, 0, 0, 0, 36, 0, 0, 0, 30, 0, 0, 0, 22, 15, 0, 0, 42, 7, 10, 0, 26, 0, 24, 0, 42, 0, 0, 0, 30, 0, 0, 21, 62, 0, 0, 0, 34, 0, 0, 0, 96, 0, 0, 15, 38, 0, 0, 0, 70, 39, 0, 0, 42, 0, 0, 0, 66, 0, 30, 0, 46, 0, 0, 0, 90

Offset: 1

Antti Karttunen, Jul 27 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10800
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001065, A003557, A007947, A057723, A033879, A065487, A335341, A336564, A336565, A336566, A336567.

Cf. A005117 (positions of zeros).

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - n; Array[a, 100] (* Amiram Eldar, May 06 2023 *)

A007947(n) = factorback(factorint(n)[, 1]);
A057723(n) = { my(r=A007947(n)); (r*sigma(n/r)); };
A336563(n) = (A057723(n)-n);
\\ Or just as:
A336563(n) = { my(x=A007947(n),y = n/x); (x*(sigma(y)-y)); };

a(n) = A057723(n) - n.
a(n) = A007947(n) * A336567(n) = A007947(n) * A001065(A003557(n)).
a(n) = A336564(n) - A033879(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A065487 - 1 = 0.231291... . - Amiram Eldar, Dec 07 2023

A336564 a(n) = n - A308135(n), where A308135(n) is the sum of non-coreful divisors of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 0, 6, 7, 8, 2, 10, 2, 12, 4, 6, 15, 16, 3, 18, 8, 10, 8, 22, 6, 24, 10, 26, 14, 28, -12, 30, 31, 18, 14, 22, 17, 36, 16, 22, 20, 40, -12, 42, 26, 27, 20, 46, 14, 48, 17, 30, 32, 52, 12, 38, 34, 34, 26, 58, -18, 60, 28, 43, 63, 46, -12, 66, 44, 42, -4, 70, 45, 72, 34, 41, 50, 58, -12, 78, 44, 80, 38, 82, -14, 62

Offset: 1

Antti Karttunen, Jul 27 2020

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A033879, A057723, A308135, A336563, A336565, A336566.

Cf. A013661, A065487.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; a[1] = 1; a[n_] := n - Times @@ f @@@ (fct = FactorInteger[n]) + Times @@ fc @@@ fct; Array[a, 100] (* Amiram Eldar, Dec 08 2023 *)

A007947(n) = factorback(factorint(n)[, 1]);
A057723(n) = { my(r=A007947(n)); (r*sigma(n/r)); };
A308135(n) = (sigma(n)-A057723(n));
A336564(n) = (n - A308135(n));

a(n) = n - A308135(n) = n - (sigma(n) - A057723(n)).
a(n) = A336563(n) + A033879(n). [Corrected by Georg Fischer, Dec 13 2022]
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A065487 - A013661 + 1 = 0.586357... . - Amiram Eldar, Dec 08 2023

A336565 Numbers k for which (A057723(k)-k) is equal to gcd(k-A308135(k), A057723(k)-k).

Original entry on oeis.org

6, 28, 234, 496, 588, 600, 1521, 1638, 6552, 8128, 55860, 89376, 33550336, 168836850

Offset: 1

Antti Karttunen, Jul 26 2020

Numbers k for which A336563(k) = A336566(n) [= gcd(A336563(n), A336564(n))].
Numbers k such that either both A336563(k) and A336564(k) are zero (in which case k is squarefree), or A336563(k) divides A336564(k), in which case k is not squarefree.
Also numbers k for which A336647(n) = 2*n - A057723(n).
Question: Are there any other odd terms apart from 1521 = 39^2 ?

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A057723, A308135, A336563, A336564, A336566, A336647.
Cf. A000396 (a subsequence).
Cf. also A326145.

A007947(n) = factorback(factorint(n)[, 1]);
A057723(n) = { my(r=A007947(n)); (r*sigma(n/r)); };
isA336565(n) = { my(b=A057723(n), c=(sigma(n)-b), d=(b-n)); (gcd(d,(n-c))==d); };

cp's OEIS Frontend

A336555 Numbers k for which A326144(k) <> A336566(k).

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A336647 a(n) = n - A336566(n).

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A336563 Sum of proper divisors of n that are divisible by every prime that divides n.

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A336564 a(n) = n - A308135(n), where A308135(n) is the sum of non-coreful divisors of n.

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A336565 Numbers k for which (A057723(k)-k) is equal to gcd(k-A308135(k), A057723(k)-k).

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