# [seqfan] A subset of primitive friendly pairs

michel.marcus at free.fr michel.marcus at free.fr
Fri Jun 29 09:40:29 CEST 2012

```Hello Seqfans

I would like to tell you about a subset of sequence A096366 that I have been studying.

The first pair of this subset is defined as the first primitive friendly pair 6 28.
Both members are divisible by 2.
2*3 2^2*7

The next pair is the smallest primitive friendly pair of numbers
that are divisible by 3 and coprime to 2.
3^3*5 3^2*7*13

All pairs are the smallest primitive friendly pair of numbers
that are divisible by a prime p, and coprime to the primes below p.

Here is my current list from p=2 to 31.
2*3 2^2*7
3^3*5 3^2*7*13
5*7*13*31 5^2*13^2*19*31^3*37*61
7*19*23*37*137 7^2*19^2*37^3*73*127*137^2
11^2*13*17*19*31*83*331 11^6*13^2*17^2*31^2*43*61*103*307*331^2*2617*5233*45319
13^2*31*61^2*97 13*31^2*61*83*331
17^3*19*29*43*127 17^4*19^2*37*43^3*127^2*271*5419*11093*44371*88741
19^2*127 19^4*151*911
23^2*37*79*137 23*37^3*73*137^2
29*31*37^2*41*43^2*67^3*79*151*433*449*461^3*547*631^2*106261 29^3*37^3*41^3*43^3*53*67*79^2*151^2*163*211*421*461^4*547^2*613*631*1093*23971
31^2*37^2*47*53*61*67^3*73*83^2*97*137*331*367^2*433*449*3463 31*37^3*47^2*53^3*61^2*67^2*73^2*83*97^2*137^2*281*317*367*1801*3169

In decimal
6 28
135 819
14105 5397553488925
15506071 155910789068784883
432712085377 468952332085139186546370744026318507437
1890948943 20936431529
14783271043 91765283361830966873857001143707378257
45847 17927087081
211838579 1596235637603
147560225903398137982300169126840969637180767467 6783307574272986865966984972702428059049774647525155647068831944517
12060713581457342807125295910808091355523729 125711636487189214276491626589291318493362872103247

The pair for p=2 are the first two perfect numbers as explained in A096366.
Pairs for p=13, 19, 23 can also be found at http://wwwhomes.uni-bielefeld.de/achim/mpn.html

These pairs are quite probably the smallest for p=2, 3, 19.
It can happen that a candidate pair is superseded by a pair where the small number is smaller, with the large number much larger.
For instance, the pair 5^3*11*13 5^4*11^2*19*71 17875 102018125
has been replaced by the 14105 5397553488925.
Is it possible to find pairs for p > 31?
Is it possible to find smaller pairs?

Is this interesting for OEIS?
If yes, would it make sense to enter them with 2 entries like A050972 and A050973?

Best regards.
Michel
```