A096366 -id:A096366 - cp's OEIS frontend

A233039 Larger member of primitive friendly pairs ordered by smallest maximal element.

28, 200, 224, 234, 270, 496, 496, 819, 936, 1488, 1638, 3724, 6200, 6200, 6860, 6975, 8128, 8128, 8128, 10976, 13104, 18600, 21600, 24384, 24384, 24800, 27000, 27000, 29792, 40131, 40640, 43008, 50274, 54000, 54400, 58032, 87750, 93100, 154791, 160524

Offset: 1

Michel Marcus, Dec 03 2013

nonn

Subsequence of A050973.
Friends m and n are primitive friendly if and only if they have no common prime factor of the same multiplicity (see A096366).
Perfect numbers greater than 6 (A000396) belong to this sequence as they form primitive friendly pairs (PFPs) with smaller perfect, so that the n-th perfect number will appear n-1 times in the sequence.
PFPs are quite useful to derive new greater amicable pairs from existing ones (see A230148).

			28 forms a friendly pair with the lesser integer 6, and this pair cannot be derived from a smaller pair, so it is primitive and 28 belongs to the sequence.
140 forms also a pair with 30, hence 140 belongs to A050973. But the pair (30, 140) can be derived from (6, 28) by multiplying both members by 5, so it is not primitive; hence 140 does not belong to the sequence.

Donovan Johnson, Table of n, a(n) for n = 1..300

Cf. A050973, A096366, A214131, A214133.

vp(f) = {maxp = f[#f~, 1]; v = vector(primepi(maxp)); for (j=1, #f~, v[primepi(f[j, 1])] = f[j, 2];);v;}
ispfp(vpn, vpi) = {for (k=1, min(#vpn, #vpi), if (vpi[k] && (vpn[k] == vpi[k]), return (0));); return (1);}
lista(nn) = {for (n=2, nn, ab = sigma(n)/n; vpn = vp(factor(n)); for (i=2, n-1, if (sigma(i)/i == ab, if (ispfp(vpn, vp(factor(i))), print1(n, ", ")););););} \\ Michel Marcus, Dec 03 2013

A214132 Lesser term of smallest primitive friendly pair such that both terms are divisible by the n-th prime p and coprime to the primes below p.

Original entry on oeis.org

6, 135, 14105, 15506071, 432712085377, 1890948943, 14783271043, 45847, 211838579, 147560225903398137982300169126840969637180767467, 12060713581457342807125295910808091355523729

Offset: 1

Michel Marcus, Jul 05 2012

For n>2, instances are known up to 11 (p=31).

By definition, gcd(a(n), A002110(n-1)) = 1. - Michel Marcus, Jan 19 2014

			For n=1, the perfect numbers 6=2*3 and 28=2^2*7 (A214133) are primitive friendly.
For n=2, 135=3^3*5 and 819=3^2*7*13 (A214133) are primitive friendly.

Achim Flammenkamp, The Multiply Perfect Numbers Page (for p = 13, 17, 19)
Michel Marcus, Table A214132 and A214133 for n = 1..11

Cf. A214133. Subset of sequence A096366.

A214133 Greater term of smallest primitive friendly pair such that both terms are divisible by the n-th prime p and coprime to the primes below p.

Original entry on oeis.org

28, 819, 5397553488925, 155910789068784883, 468952332085139186546370744026318507437, 20936431529, 91765283361830966873857001143707378257, 17927087081, 1596235637603

Offset: 1

Michel Marcus, Jul 05 2012

nonn

By definition, GCD(a(n), A002110(n-1)) = 1. - Michel Marcus, Jan 19 2014

			For n=1, the perfect numbers 6=2*3 (A214132) and 28=2^2*7 are primitive friendly.
For n=2, 135=3^3*5 (A214132) and 819=3^2*7*13 are primitive friendly.

Cf. A214132. Subset of sequence A096366.

a(7) corrected and a(8)-a(9) added by Michel Marcus, Jun 18 2016

A095751 Conjectured list of integers known to be friendly but not known to be primitive friendly.

Original entry on oeis.org

66, 78, 102, 114, 120, 132, 138, 150, 174, 186, 204, 222, 228, 246, 252, 258, 276, 282, 294, 300, 308, 312, 318, 330, 348, 354, 364, 366, 372

Offset: 1

Walter Nissen, Jul 09 2004

There may be other integers in the sequence within the range of those given, but they have yet to be calculated and moreover, some of these given may prove to be primitive friendly.
Abundancy is defined as the ratio of the multiplicative sum-of-divisors function to the integer itself: abund(n) = sigma(n)/n. E.g., abund(10) = sigma(10) / 10 = (1+2+5+10)/10 = 1.8 = 9/5.
Integers m and n are friendly iff they have the same abundancy. E.g., abund(12) = abund(234) = 7/3 ===> 12 and 234 are friends.
Friends m and n are primitive friendly iff they have no common prime factor of the same multiplicity.

			66 is a friend of 308, 5456, 89408 and 369053696, but all of these are divisible by 11 and not 121, while 66 is not known to be primitive friendly.
280 is not a term because although 280 = 2^3*5*7 and 1553357978368 = 2^8*7^2*19^2*37*73*127 have the same abundancy they have no common prime factors of the same multiplicity and so are primitive friendly. It should be noted that 18620 = 2^2*5*7^2*19 also has the same abundancy. - _Suyash Pandit_, Sep 24 2023

Hickerson, Dean; "Re: Friendly number", post to sci.math newsgroup, 2000, available through groups.google.com.

Claude W. Anderson and Dean Hickerson, Problem 6020: Friendly Integers, Amer. Math. Monthly 84 (1977) pp. 65-66.
Walter Nissen, Primitive Friendly Integers and Exclusive Multiples, 2004 post to NMBRTHRY mailing list

Cf. A014567, A074902, A095738, A095739, A096366.

Terms 280 and 360 removed by Suyash Pandit, Sep 24 2023

Added "Conjectured" to definition following comments from the Editors. - N. J. A. Sloane, Oct 09 2023

A214220 Smallest squarefree deficient number with n prime factors that is part of primitive friendly pair.

Original entry on oeis.org

273, 1995, 356235, 6768465, 215561445, 8381978605, 889577580507, 9176928067615, 977229739621135, 159778696591499755, 12961232730855705065, 5133226984135500020155, 527303555325724107882055, 36714792413117114527127897, 96369422480705367222515908615

Offset: 3

Michel Marcus, Jul 07 2012

nonn

			For n=3, 273 (3*7*13) and 2876211 (3^2*13^2*31*61) have no common prime factor with same exponent.

Michel Marcus, Table of n, a(n) for n = 3..22
Michel Marcus, Prime factors of the friendly pairs for n=3..22

Cf. A096366.

A175907 Known friendly squarefree numbers.

Original entry on oeis.org

6, 30, 42, 66, 78, 102, 114, 138, 174, 186, 210, 222, 246, 258, 273, 282, 318, 330, 354, 366, 390, 402, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 642, 654, 678, 690, 714, 762, 786, 798, 806, 822, 834, 858, 870, 894, 906, 930, 942, 966, 978, 1002, 1038

Offset: 1

Thomas Kellar, Oct 14 2010, Oct 15 2010

nonn

From Walter Nissen, May 28 2011: (Start)
As with most aspects of friendly and solitary numbers, this sequence is not known to be complete. A friend could possibly be found for 10, for example; same doubtful status as an odd perfect number.
Note that not all friendly numbers will be found among the primitive friendly numbers listed in link "Primitive Friendly Pairs", and this would be true even if those were not limited to small examples.
Other terms are 1330, 1995, and 49166.
(End)

			6, being 2 * 3, is squarefree. Having abundancy = 2, 6 is friendly with all the other perfect numbers. Ergo, it is in the sequence. ( 1 ), 2, 3, and 5, being prime powers, are solitary. 4 is a square. Ergo, a(1) is 6.

Oystein Ore, Number Theory and Its History, McGraw-Hill, 1948, reprinted 1988, section 5-3, pp. 96-100.

Claude W. Anderson and Dean Hickerson, Problem 6020: Friendly Integers, Amer. Math. Monthly 84 (1977) pp. 65-66.
Richard Laatsch, Measuring the abundancy of integers, Mathematics Magazine 59 (2) (1986) 84-92.
Walter Nissen, Primitive Friendly Pairs
Walter Nissen, Abundancy: some resources
Eric Weisstein's World of Mathematics, Solitary Number

Cf. A005117, A014567, A074902 (known friendly numbers), A095751, A096366, A140688.

{ for (j=1,2000, if (issquarefree(j), t=sigma(j)/j; for (i=1,1000000, p=sigma(i)/i; if(p == t && j != i, print(j," ",i); ); ); ); ); quit; } \\ provides useful suggestions, but not definitive, Walter Nissen, May 28 2011

Added 273 as it is friendly with 2876211; 273 is a counterexample to the conjecture that 6 divides a(n). - Walter Nissen, May 28 2011

Added 806 as it is friendly with 2449562488893. - Suyash Pandit, Jan 24 2024

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A233039 Larger member of primitive friendly pairs ordered by smallest maximal element.

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A214132 Lesser term of smallest primitive friendly pair such that both terms are divisible by the n-th prime p and coprime to the primes below p.

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A214133 Greater term of smallest primitive friendly pair such that both terms are divisible by the n-th prime p and coprime to the primes below p.

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A095751 Conjectured list of integers known to be friendly but not known to be primitive friendly.

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A214220 Smallest squarefree deficient number with n prime factors that is part of primitive friendly pair.

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A175907 Known friendly squarefree numbers.

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