A344700 -id:A344700 - cp's OEIS frontend

A344752 Quotient A306927(k) / A344705(k) computed for such k >= 1 that the quotient is a nonnegative natural number. (The k are given by A344700.)

0, 1, 2, 1, 3, 1, 12, 21, 2, 3, 36, 4, 4, 61, 2, 3, 5, 243, 36, 2, 1, 70, 345, 90, 5, 5, 38, 11, 3, 131, 25, 87, 172, 2, 9, 5, 228, 5, 20, 43, 18, 5, 10, 304, 3, 1035, 31, 1301, 7, 172, 8, 554, 60, 15, 295, 59, 14, 150, 110, 7, 2439, 258, 371, 5, 549, 8, 13, 15, 63, 1134, 24, 900, 23, 50, 7, 4, 27, 1292, 254, 6681, 5, 18

Offset: 1

Antti Karttunen, May 28 2021

nonn

Cf. A000203, A001615, A306927, A344700, A344705.

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
isA344700_and_give_quotient(n) = { my(t=A001615(n), s=sigma(n), u = (n+t)-s); if(((u>0)&&(0==((t-n)%u))), ((t-n)/u), 0); };
for(n=1,2^17,x=isA344700_and_give_quotient(n); if(x>0||(1==n), print1(x,", ")));

A344755 Numbers k such that A344753(k) is a multiple of A048250(k), and k is a multiple of A344753(k)/A048250(k).

Original entry on oeis.org

6, 28, 150, 496, 528, 1980, 4560, 8128, 8736, 11400, 19872, 20664, 75840, 82080, 253080, 254880, 741744, 1627290, 5130300, 5607360, 7529760, 19645440, 20718720, 33550336, 35092512, 45643392, 45995040, 56424960, 86944320, 169910136, 174013920, 180442080, 196378992, 242040960, 304577280, 314511360, 326611440, 451344960

Offset: 1

Antti Karttunen, May 29 2021

nonn

Numbers k for which A344753(k)/A048250(k) is a divisor of k.

Perfect numbers (A000396, including also any hypothetical odd terms) are included as only on them A001615 coincides with A344753, and because A001615(n) = A003557(n)*A048250(n), with A003557(n) being a divisor of n.

Cf. A000396 (subsequence), A001065, A001615, A003557, A306927, A048250, A344753.
Subsequence of A344754.
Cf. also A344700.

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A344753(n) = sumdiv(n,d,(dA344755(n) = { my(t=A344753(n),u=A048250(n)); ((0==(t%u))&&(0==(n%(t/u)))); };

A344705 a(n) = n + A001615(n) - sigma(n), where A001615 is the Dedekind psi-function, and sigma(n) gives the sum of divisors of n; difference between psi and the sum of proper divisors.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 5, 8, 10, 11, 8, 13, 14, 15, 9, 17, 15, 19, 14, 21, 22, 23, 12, 24, 26, 23, 20, 29, 30, 31, 17, 33, 34, 35, 17, 37, 38, 39, 22, 41, 42, 43, 32, 39, 46, 47, 20, 48, 47, 51, 38, 53, 42, 55, 32, 57, 58, 59, 36, 61, 62, 55, 33, 65, 66, 67, 50, 69, 70, 71, 21, 73, 74, 71, 56, 77, 78, 79, 38, 68, 82

Offset: 1

Antti Karttunen, May 28 2021

First negative term occurs as a(1440) = -18.

Antti Karttunen, Table of n, a(n) for n = 1..16560
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000203, A001065, A001615, A033879, A244963, A291209 (positions of zeros), A344700, A344704, A344753.

Cf. A013661, A082020.

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

A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d));
A344705(n) = ((n + A001615(n)) - sigma(n));

a(n) = A001615(n) - A001065(n) = n - A244963(n) = n + A001615(n) - sigma(n).
a(n) = A033879(n) + A306927(n).
a(n) = n + A344753(n) - 2*A001065(n).
Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n*log(n)), where c = 15/Pi^2 + 1 - Pi^2/6 = 0.874883... . - Amiram Eldar, Dec 08 2023

A344754 Numbers k such that A344753(k) is a multiple of A048250(k).

Original entry on oeis.org

1, 6, 24, 28, 54, 96, 112, 150, 153, 216, 294, 384, 448, 486, 496, 528, 672, 726, 864, 1014, 1080, 1377, 1500, 1536, 1734, 1792, 1944, 1980, 1984, 2112, 2166, 2250, 2376, 2688, 3174, 3456, 3672, 3750, 4320, 4374, 4560, 4753, 5046, 5292, 5766, 6000, 6048, 6144, 6720, 7168, 7776, 7936, 8128, 8214, 8448, 8700, 8736, 9024

Offset: 1

Antti Karttunen, May 29 2021

nonn

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

Cf. A001065, A001615, A306927, A048250, A344753.
Subsequences: A000396, A344755.
Cf. also A344700.

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A344753(n) = sumdiv(n,d,(dA344754(n) = (0==(A344753(n)%A048250(n)));

A344704 a(n) = gcd(A001615(n)-n, sigma(n)-(A001615(n)+n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 12, 1, 2, 1, 20, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 6, 1, 4, 3, 2, 1, 4, 1, 1, 3, 2, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 11, 1, 1, 6, 1, 10, 3, 2, 1, 3, 1, 2, 1, 4, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 3, 4, 1, 18, 7, 4, 1, 2, 5, 12, 1, 5, 3, 1, 1, 6, 1, 2, 3

Offset: 1

Antti Karttunen, May 28 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A001615, A244963, A306927, A344700, A344705.

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A344704(n) = gcd(A001615(n)-n, sigma(n)-(A001615(n)+n));

a(n) = gcd(A306927(n), n-A244963(n)) = gcd(A001615(n)-n, sigma(n)-(A001615(n)+n)).

cp's OEIS Frontend

A344752 Quotient A306927(k) / A344705(k) computed for such k >= 1 that the quotient is a nonnegative natural number. (The k are given by A344700.)

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A344755 Numbers k such that A344753(k) is a multiple of A048250(k), and k is a multiple of A344753(k)/A048250(k).

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A344705 a(n) = n + A001615(n) - sigma(n), where A001615 is the Dedekind psi-function, and sigma(n) gives the sum of divisors of n; difference between psi and the sum of proper divisors.

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A344754 Numbers k such that A344753(k) is a multiple of A048250(k).

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A344704 a(n) = gcd(A001615(n)-n, sigma(n)-(A001615(n)+n)).

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