A239875 Friendly club of (first Wieferich prime)-1 and (second Wieferich prime)-1: numbers n such that sigma(n)/n = 112/39.

1092, 3510, 19344, 316992, 11504844, 1308463104, 335004893184, 5360108961792

Offset: 1

Author

Jeppe Stig Nielsen, Mar 28 2014

Keywords

• nonn
• more

Comments

Dobson (see Link section) noted that A001220(1)-1 and A001220(2)-1 are friendly numbers, i.e., they have the same abundancy index (namely, 112/39), hence belong to the same "club" of friendly numbers.

Numbers n such that sigma(n)/n = 112/39 where sigma is A000203.

Clearly, all terms are multiples of 39. Note that 39 itself satisfies sigma(39)/39 = 56/39, half the desired value. Then if s is a perfect number (s is in A000396) and if s is coprime to 39, we have that 39*s belongs to the sequence, since sigma(39*s)/(39*s) = (sigma(39)/39)*(sigma(s)/s) = (56/39)*2 = 112/39. Michel Marcus made me aware of this fact. All even perfect numbers s other than s=6 satisfy the requirement, so this gives a lot of new terms. Other terms in this sequence (of the form 39 times perfect numbers) are thus: 335004893184, 5360108961792, 89927877317458132992, 103679783671223438041533012022199844864 etc.

The only known terms that do not come from 39 times a perfect number, are a(2)=3510 and a(5)=11504844. Their cofactors with 39 are not coprime to 39. Are there more numbers a(k) such that gcd(a(k)/39, 39) > 1?

How often is a(k)+1 a prime number? For k=1 and k=2 we get Wieferich primes. The number 39*A000396(8) belongs to the sequence, and 39*A000396(8)+1 is a prime (though not a Wieferich prime).

a(9) > 10^13. - Giovanni Resta, Jul 13 2015

Examples

			19344 is included because sigma(19344)/19344 = 55552/19344 = 112/39.
		

Links

• John Blythe Dobson, A note on the two known Wieferich Primes

Programs

PARI

forstep(n=39,10^10,39,if(sigma(n)*39==n*112,print1(n, ", ")))

Extensions

a(7)-a(8) from Giovanni Resta, Jul 13 2015