A378980 -id:A378980 - cp's OEIS frontend

A379477 Numbers k such that A003961(k)-2k and A003961(k)-sigma(k) have a common divisor > 1, where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

6, 7, 13, 18, 19, 24, 28, 30, 31, 37, 42, 43, 46, 54, 55, 60, 61, 66, 67, 68, 69, 72, 78, 79, 90, 91, 96, 97, 102, 103, 106, 109, 114, 120, 126, 127, 131, 132, 135, 138, 139, 140, 146, 150, 151, 162, 163, 166, 168, 174, 175, 180, 181, 186, 193, 198, 199, 200, 204, 210, 216, 222, 223, 229, 234, 240, 241, 246, 251

Offset: 1

Antti Karttunen, Dec 23 2024

nonn

Cf. A000203, A003961, A252748, A286385, A379476 (characteristic function).
Positions of terms > 1 in A326057.
Disjoint union of A372566 and A379479.
Subsequences: A000396. See also A378980.

PARI
```
is_A379477 = A379476;
```

{k such that gcd(A252748(k), A286385(k)) > 1}.

A379217 Quotient (A003961(k)-sigma(k)) / (2*k-A003961(k)) computed for those k for which this quotient is an integer, where A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

Original entry on oeis.org

0, 0, 1, -2, -1, 1, -3, 18, 9, -1, 17, 3, 1, -35, -7, -15, 57, -1, 339, -381, 3, -7, -969, -1213, -1, 3, 3, -979, 419, 29, -42735, 21, 731232, 3, 1445, 2809731, -4566981, 557, -19691, -1, 5, 544371, 5, -475, -1784691, 9051, 176870849, 808683, 280791301, 1803, -891775, -3679, -3733533, -444406677, 731480523, 275091

Offset: 1

Antti Karttunen, Dec 20 2024

sign

Terms in A378980 that correspond here with -1's are perfect numbers (A000396).

Antti Karttunen, Table of n, a(n) for n = 1..69 (based on b-file of A378980 computed by _Amiram Eldar_)
Index entries for sequences related to prime indices in the factorization of n.
Index entries for sequences related to sigma(n).

Cf. A000396, A003961, A252748, A378980, A378981, A379216.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A378981(n) = ((A003961(n)-sigma(n))%((2*n)-A003961(n)));
is_A378980(n) = !A378981(n);
for(n=1,2^25,if(is_A378980(n),print1(((A003961(n)-sigma(n))/((2*n)-A003961(n))),", ")));

a(n) = A286385(A378980(n)) / A379216(n) = A286385(A378980(n)) / -A252748(A378980(n)).

A387416 Numbers k for which A003961(k) - sigma(k) is a multiple of k, where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 2, 18, 3528

Offset: 1

Antti Karttunen, Sep 03 2025

If it exists, a(5) > 3288334336.

			1 is included as A003961(1) - A000203(1) = 0 is a multiple of 1.
2 is included as A003961(2) - A000203(2) = 0 is a multiple of 2.
18 is included as A003961(18) - A000203(18) = 75 - 39 = 36 = 2 * 18.
3528 is included as A003961(3528) - A000203(3528) = 81675 - 11115 = 70560 = 20 * 3528.

Cf. A000203, A003961, A286385.

Cf. also A378980.

{k | A286385(k) == 0 (mod k)}.

cp's OEIS Frontend

A379477 Numbers k such that A003961(k)-2k and A003961(k)-sigma(k) have a common divisor > 1, where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

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A379217 Quotient (A003961(k)-sigma(k)) / (2*k-A003961(k)) computed for those k for which this quotient is an integer, where A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

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PARI

Formula

A387416 Numbers k for which A003961(k) - sigma(k) is a multiple of k, where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

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