A233929 -id:A233929 - cp's OEIS frontend

A097015 Smallest k such that sigma(k) + 1 is divisible by primorial(n).

1, 1, 2401, 2614689, 36324729, 36324729, 2411675443849, 2411675443849, 12361036649679601

Offset: 0

Labos Elemer, Aug 19 2004

nonn

10^19 < a(9) <= 725298909352131113041. Terms a(3) through a(8) all have the prime signature p^4*q^2*r^2. Any x such that sigma(x) = -1 (mod 30) must have at least eight prime factors. However, for all n, there are solutions with fewer than three distinct prime factors. More generally, for any k > 1, let p be a prime of the form mk+1; then sigma(p^(k-2)) = -1 (mod k). For a(9), 725298909352131113041 is the least solution with eight prime factors. I have not been able to rule out a smaller solution with more prime factors. - David Wasserman, Dec 14 2007

Cf. A002110 (primorial), A084301, A084303, A000203, A097011, A097012, A097014, A233929.

a(n) = A233929(A002110(n)). - Andrew Howroyd, Dec 12 2024

More terms from David Wasserman, Dec 14 2007

a(0)=1 prepended by Andrew Howroyd, Dec 12 2024

A234257 Integers n such that the smallest x with sigma(x) == -1 mod n is n - 2.

Original entry on oeis.org

4, 5, 7, 9, 15, 20, 21, 31, 39, 45, 63, 69, 75, 103, 111, 139, 151, 159, 165, 175, 195, 199, 201, 213, 231, 243, 259, 279, 283, 285, 315, 319, 333, 339, 349, 351, 355, 369, 375, 381, 399, 403, 411, 423, 459, 463, 465, 489, 501, 511, 525, 543, 549, 565, 579

Offset: 1

Michel Marcus, suggested by Benoit Cloitre, Dec 22 2013

nonn

See examples section for an analysis of first few terms, using A233929(n), the smallest x satisfying sigma(x) == n - 1 modulo n.
Thus many terms will be a prime+2: 4, 5, 7, 9, 15, 21, 31, 39, 45, ... So far, 20 is the only term that is not a prime +2.
But not all primes are represented; the first instance is 13, a prime+2, that is not in the sequence. This is because, for n=13, A233929(13) would have been 11 if one did not have 6 before with sigma(6) = sigma(11) = 12 so also congruent to 13-1.

			Take n=4, A233929(4) is 2, since sigma(2)=3 == 3 modulo 4, and we have 4 - 2 = 2.
Take n=5, A233929(5) is 3, since sigma(3)=4 == 4 modulo 5, and we have 5 - 3 = 2.
The same happens for n=7, 9, and 15, A233929(n) being the primes equal to n-2: 5, 7 and 13.
For n=20, A233929(20) is 18, since sigma(18) = 39 == 19 modulo 20, but 20 is not a prime+2.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A233929.

for(n=3, 579, for(x=1, n-2, if(sigma(x)%n==n-1, if(x==n-2, print1(n ", ")); next(2)))) \\ Donovan Johnson, Jan 06 2014

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A097015 Smallest k such that sigma(k) + 1 is divisible by primorial(n).

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A234257 Integers n such that the smallest x with sigma(x) == -1 mod n is n - 2.

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