A325812 -id:A325812 - cp's OEIS frontend

A325813 a(n) = gcd(A034448(n)-n, n-A048146(n)), where A034448 and A048146 are respectively the sum of unitary and non-unitary divisors of n.

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, 12, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 6, 1, 4, 3, 2, 1, 4, 1, 7, 3, 6, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 3, 1, 2, 1, 12, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 3, 4, 1, 18, 7, 4, 1, 2, 5, 12, 1, 1, 21, 1, 1, 6, 1, 2, 3

Offset: 1

Antti Karttunen, May 23 2019

nonn

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

Cf. A034460, A323166, A325812, A325814.

Cf. also A325385.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
A325813(n) = gcd(A034460(n), A325814(n));

a(n) = gcd(A034460(n), A325814(n)).

A325814 a(n) = n - A048146(n), where A048146 is the sum of non-unitary divisors of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 6, 10, 11, 4, 13, 14, 15, 2, 17, 9, 19, 8, 21, 22, 23, 0, 20, 26, 15, 12, 29, 30, 31, 2, 33, 34, 35, -5, 37, 38, 39, 4, 41, 42, 43, 20, 27, 46, 47, -8, 42, 35, 51, 24, 53, 18, 55, 8, 57, 58, 59, 12, 61, 62, 39, 2, 65, 66, 67, 32, 69, 70, 71, -33, 73, 74, 55, 36, 77, 78, 79, -4, 42, 82, 83, 20

Offset: 1

Antti Karttunen, May 23 2019

sign

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

Cf. A000203, A033879, A034448, A034460, A048146, A064591 (positions of zeros), A228058, A325812, A325813, A325824.
Cf. also A325314.
Cf. A002117, A013661.

Table[n - DivisorSum[n, # &, ! CoprimeQ[#, n/#] &], {n, 84}] (* Michael De Vlieger, May 27 2019 *)

A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));

a(n) = n - A048146(n).
a(n) = A033879(n) + A034460(n).
a(A228058(n)) = A325824(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1/2 - zeta(2) * (1 - 1/zeta(3)) / 2 = 0.3617493553... . - Amiram Eldar, Feb 22 2024

A325822 Terms k of A228058 for which A325814(k) is a multiple of A034460(k).

Original entry on oeis.org

477, 3725, 29161, 107797, 166753, 205409, 500837, 535277, 780625, 1610389, 5649841, 6968125, 10292809, 10633429, 24231241, 32771201, 38322857, 40028661, 104861501, 170384117, 183593125, 277405641, 326081953, 488265625, 491716541, 704531953, 797338489, 836737393, 2053219321, 2359421369, 3012238153

Offset: 1

Antti Karttunen, May 23 2019

nonn

Such terms A228058(n) that A325823(n) is a divisor of A325824(n).
If any odd perfect number exists, then it must occur in this sequence.
This is not a subsequence of A325376: 107797 is the first term that does not occur there.

Index entries for sequences where any odd perfect numbers must occur

Cf. A034460, A228058, A325376, A325812, A325814, A325823, A325824.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
for(n=1,oo, if(isA228058(n) && !(A325814(n)%A034460(n)), print1(n, ", ")));

cp's OEIS Frontend

A325813 a(n) = gcd(A034448(n)-n, n-A048146(n)), where A034448 and A048146 are respectively the sum of unitary and non-unitary divisors of n.

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A325814 a(n) = n - A048146(n), where A048146 is the sum of non-unitary divisors of n.

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A325822 Terms k of A228058 for which A325814(k) is a multiple of A034460(k).

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