A050973 -id:A050973 - cp's OEIS frontend

A236355 Terms of A050973 that give maximum record values for A050973(k)/A050972(k).

28, 224, 234, 496, 6200, 8128, 174592, 544635, 2886100, 33550336

Offset: 1

Michel Marcus, Jan 23 2014

The corresponding terms in A050972 are: 6, 40, 12, 6, 30, 6, 40, 6, ...

Note the subsequence 28, 496, 8128, 33550336: friends of 6 (cf A000396).

			28/6 < 224/40 < 234/12 < 496/6 < 6200/30 < 8128/6 < 174592/40 < ....

Eric Weisstein's World of Mathematics, Friendly Pair.

Cf. A050972, A050973, A236372.

isrmax(i, rmax) = {si = sigma(i); if (gcd(si, i) == 1, return (0)); s = si/i; for (j=1, i\rmax, if ((sigma(j)/j) == s, if ((newr = i/j) > rmax, return (newr)););); return (0);}
lista(nn) = {rmax = 1; for (i = 1, nn, if ((newr = isrmax(i, rmax)), rmax = newr; print1(i, ", ");););}

A236372 Terms of A050973 that give minimum record values for A050973(k)/A050972(k).

Original entry on oeis.org

28, 200, 936, 7856640, 12103000, 8004519424

Offset: 1

Michel Marcus, Jan 24 2014

The corresponding terms in A050972 are: 6, 80, 864, 7344000, 11804800, 7908221230.

Note that from n=1 to 4 sigma(a(n))/a(n) is increasing, but decreasing for n=5 and 6. Is this going on afterwards? - Michel Marcus, Feb 08 2014

			28/6 > 200/80 > 936/864 > 7856640/7344000 > 12103000/11804800 > ...

Eric Weisstein's World of Mathematics, Friendly Pair.

Cf. A050972, A050973, A236355.

isrmin(i, rmin) = {si = sigma(i); if (gcd(si, i) == 1, return (0)); s = si/i; forstep (j=i-1, 1+i\rmin, -1, if ((sigma(j)/j) == s, if ((newr = i/j) < rmin, return (newr)););); return (0);}
lista(nn) = {rmin = 1000; for (i=1, nn, if ((newr = isrmin(i, rmin)), rmin = newr; print1(i, ", ");););}

a(6) from Michel Marcus, Feb 08 2014

A263118 Indices of the primitive friendly pairs in the sequence of friendly pairs (A050973, A050972) ordered by smallest maximal element.

Original entry on oeis.org

1, 3, 4, 5, 6, 10, 11, 18, 20, 29, 33, 70, 115, 116, 133, 136, 155, 156, 157, 212, 255, 360, 414, 468, 470, 477, 518, 519, 578, 771, 787, 830, 971, 1039, 1046, 1121, 1687, 1793, 2983, 3092, 3359, 3360, 3570, 4084, 4190, 4255, 5281, 7032, 7141, 7167, 8248, 8385, 8386, 8630, 8890

Offset: 1

Michel Marcus, Oct 10 2015

nonn

Friends x and y are primitive friendly if and only if they have no common prime factor with the same multiplicity, that is, if A165430(x, y) = 1.

			The first pair (6, 28) is primitive since 6=2*3 and 28=2^2*7; their only common prime factor, 2, appears with different exponents, so 1 is a term.
The second pair (30, 140) is not primitive since 30=5*6 and 140=5*28; the prime factor 5 appears in each with the same exponent, so 2 is not a term.

Walter Nissen, Primitive Friendly Integers and Exclusive Multiples, Up for the Count!
Eric Weisstein's World of Mathematics, Friendly Pair.

Cf. A050972, A050973, A165430, A233039.

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d);}
ugcd(x,y) = vecmax(setintersect(udivs(x), udivs(y)));
lista(vp, vg) = {for (n=1, #vp, if (ugcd(vp[n], vg[n])==1, print1(n, ", ")););} \\ where vp and vg are A050972 and A050973

A233039(n) = A050973(a(n)).

A263152 a(n) is the greatest common unitary divisor of the friendly pairs, A050972(n) and A050973(n).

Original entry on oeis.org

1, 5, 1, 1, 1, 1, 11, 13, 17, 1, 1, 19, 3, 23, 3, 25, 29, 1, 31, 1, 37, 41, 5, 43, 47, 7, 53, 3, 1, 55, 7, 2, 1, 59, 61, 9, 65, 67, 71, 9, 73, 11, 79, 83, 85, 11, 5, 5, 89, 11, 13, 95, 97, 101, 103, 13, 11, 107, 109, 113, 115, 4, 121, 17, 7, 125, 13, 127, 131

Offset: 1

Michel Marcus, Oct 11 2015

nonn

Dividing both A050972(n) and A050973(n) by a "greater than 1" divisor of a(n), if any, will give a smaller friendly pair.

If a(n) is greater than 1, dividing both A050972(n) and A050973(n) will give a primitive friendly pair.

			The greatest common unitary divisor of the first friendly pair (6, 28) is 1, hence a(1) = 1.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A050972, A050973, A263118.

Cf. A165430 (greatest common unitary divisor).

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d);}
ugcd(x,y) = vecmax(setintersect(udivs(x), udivs(y)));
lista(vp, vg) = {for (n=1, #vp, print1(ugcd(vp[n], vg[n])", ")); } \\ where vp and vg are A050972 and A050973

a(n) = A165430(A050972(n), A050973(n)).

a(A263118(n)) = 1, the primitive friendly pairs.

A007770 Happy numbers: numbers whose trajectory under iteration of sum of squares of digits map (see A003132) includes 1.

Original entry on oeis.org

1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, 103, 109, 129, 130, 133, 139, 167, 176, 188, 190, 192, 193, 203, 208, 219, 226, 230, 236, 239, 262, 263, 280, 291, 293, 301, 302, 310, 313, 319, 320, 326, 329, 331, 338

Offset: 1

N. J. A. Sloane, A.R.McKenzie(AT)bnr.co.uk

Sometimes called friendly numbers, but this usage is deprecated.
Gilmer shows that the lower density of this sequence is < 0.1138 and the upper density is > 0.18577. - Charles R Greathouse IV, Dec 21 2011
Corrected the upper and lower density inequalities in the comment above. - Nathan Fox, Mar 14 2013
Grundman defines the heights of the happy numbers by the number of iterations needed to reach the 1: 0, 5, 1, 2, 4, 3, 3, 2, 3, 4, 4, 2, 5, 3, 3, 2, 4, 4, 3, 1, ... (A090425(n) - 1). E.g., for n=2 the height of 7 is 5 because it needs 5 iterations: 7 -> 49 -> 97 -> 130 -> 10 -> 1. - R. J. Mathar, Jul 09 2017
El-Sedy & Siksek prove that this sequence contains arbitrarily long subsequences of consecutive terms; that is, the upper uniform density of this sequence is 1. - Charles R Greathouse IV, Sep 12 2022

			1 is OK. 2 --> 4 --> 16 --> 37 --> ... --> 4, which repeats with period 8, so never reaches 1, so 2 (and 4) are unhappy.
A correspondent suggested that 98 is happy, but it is not. It enters a cycle 98 -> 145 -> 42 -> 20 -> 4 -> 16 ->37 ->58 -> 89 -> 145 ...

L. E. Dickson, History of the Theory of Numbers, Vol, I: Divisibility and Primality, AMS Chelsea Publ., 1999.
R. K. Guy, Unsolved Problems Number Theory, Sect. E34.
J. N. Kapur, Reflections of a Mathematician, Chap. 34 pp. 319-324, Arya Book Depot New Delhi 1996.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 25-26.

Jud McCranie, Table of n, a(n) for n = 1..143071
Breeanne Baker Swart, Susan Crook, Helen G. Grundman, Laura Hall-Seelig, May Mei, and Laurie Zack, Fixed Points of Augmented Generalized Happy Functions II: Oases and Mirages, arXiv:1908.02194 [math.NT], 2019.
T. Cai and X. Zhou, On the height of happy numbers, Rocky Mt. J. Math. 38 (6) (2008) 1921.
E. El-Sedy and S. Siksek, On happy numbers, Rocky Mountain J. Math. 30 (2000), 565-570.
Justin Gilmer, On the density of happy numbers, arXiv:1110.3836 [math.NT], 2011-2015.
H. G. Grundmann, Semihappy Numbers, J. Int. Seq. 13 (2010), 10.4.8.
B. L. Mayer and L. H. A. Monteiro, On the divisors of natural and happy numbers: a study based on entropy and graphs, AIMS Mathematics (2023) Vol. 8, Issue 6, 13411-13424.
Luca Onnis, On a variant of the happy numbers and their generalizations, arXiv:2203.03381 [math.GM], 2022.
Hao Pan, Consecutive happy numbers, arXiv:math/0607213 [math.NT], 2006.
W. Schneider, Happy Numbers (Includes list of terms below 10000)
R. Styer, Smallest Examples of Strings of Consecutive Happy Numbers, J. Int. Seq. 13 (2010), 10.6.3.
Eric Weisstein's World of Mathematics, Happy Number
Eric Weisstein's World of Mathematics, Digitaddition
Wikipedia, Happy number

Cf. A003132 (the underlying map), A001273, A035497 (happy primes), A046519, A031177, A002025, A050972, A050973, A074902, A103369, A035502, A068571, A072494, A124095, A219667, A239320 (base 3), A240849 (base 5).

Cf. A090425 (required iterations including start and end).

a007770 n = a007770_list !! (n-1)
a007770_list = filter ((== 1) . a103369) [1..]
-- Reinhard Zumkeller, Aug 24 2011

f[n_] := Total[IntegerDigits[n]^2]; Select[Range[400], NestWhile[f, #, UnsameQ, All] == 1 &] (* T. D. Noe, Aug 22 2011 *)
Select[Range[1000],FixedPoint[Total[IntegerDigits[#]^2]&,#,10]==1&] (* Harvey P. Dale, Oct 09 2011 *)
(* A example with recurrence formula to test if a number is happy *)
a[1]=7;
a[n_]:=Sum[(Floor[a[n-1]/10^k]-10*Floor[a[n-1]/10^(k+1)]) ^ (2) ,{k, 0,
      Floor[Log[10,a[n-1]]] }]
Table[a[n],{n,1,10}] (* José de Jesús Camacho Medina, Mar 29 2014 *)

ssd(n)=n=digits(n);sum(i=1,#n,n[i]^2)
is(n)=while(n>6,n=ssd(n));n==1 \\ Charles R Greathouse IV, Nov 20 2012

select( {is_A007770(n)=while(6M. F. Hasler, Dec 20 2024

def ssd(n): return sum(int(d)**2 for d in str(n))
def ok(n):
  while n not in [1, 4]: n = ssd(n) # iterate until fixed point or in cycle
  return n==1
def aupto(n): return [k for k in range(1, n+1) if ok(k)]
print(aupto(338)) # Michael S. Branicky, Jan 07 2021

From Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 23 2009: (Start)
1) Every power 10^k is a member of the sequence.
2) If n is member the numbers obtained by placing zeros anywhere in n are members.
3) If n is member each permutation of digits of n gives another member.
4) If the repeated process of summing squared digits give a number which is already a member of sequence the starting number belongs to the sequence.
5) If n is a member the repunit consisting of n 1's is a member.
6) If n is a member delete any digit d, new number consisting of remaining digits of n and d^2 1's placed everywhere to n is a member.
7) It is conjectured that the sequence includes an infinite number of primes (see A035497).
8) For any starting number the repeated process of summing squared digits ends with 1 or gives an "8-loop" which ends with (37,58,89,145,42,20,4,16,37) (End)

A074902 Known friendly numbers.

Original entry on oeis.org

6, 12, 24, 28, 30, 40, 42, 56, 60, 66, 78, 80, 84, 96, 102, 108, 114, 120, 132, 135, 138, 140, 150, 168, 174, 186, 200, 204, 210, 222, 224, 228, 234, 240, 246, 252, 258, 264, 270, 273, 276, 280, 282, 294, 300, 308, 312, 318, 330, 348, 354, 360, 364, 366, 372

Offset: 1

N. J. A. Sloane, Sep 15 2002

nonn

The sequence is not known to be complete up to 372, since there are many small numbers, including 10, 14, 15 and 20, which have not been proved to be solitary. If any other numbers up to 372 are friendly, the smallest corresponding values of m are > 10^30.
A positive integer n is 'friendly' if abundancy(n) = abundancy(m) for some positive integer m not equal to n, where abundancy(n) = sigma(n)/n (cf. A000203); otherwise n is 'solitary'. (The name "friendly" is also sometimes mistakenly used with other meanings; cf. A063990 and A007770.)
All perfect numbers are friendly numbers, but they are only friendly with each other (a perfect number being defined as having abundancy index of 2.) - Daniel Forgues, Jun 23 2009
Triangle A211679 has rows that list the first numbers that have n-1 smaller friends. Sequence A211677 lists just the last number in each row. - T. D. Noe, May 10 2012

			24 is in the sequence since abundancy(24) = abundancy(91963648) = 5/2.

Claude W. Anderson and Dean Hickerson, Problem 6020: Friendly Integers, Amer. Math. Monthly 84 (1977) pp. 65-66.
Sagar Mandal, Prime Divisors of 10's Friends: A Generalization of Prior Bounds, arXiv:2412.02701 [math.GM], 2024. See p. 10.
Sagar Mandal, Exploring the relationships between the divisors of friends of 10, Bull. Calcutta Math. Soc. (2025) Vol. 48, No. 1-3, 21-32.
Sagar Mandal and Sourav Mandal, Upper bounds for the prime divisors of friends of 10-II, arXiv:2412.02701 [math.GM], 2024. See p. 8.
Eric Weisstein's World of Mathematics, Friendly Pair
Eric Weisstein's World of Mathematics, Friendly Number

Union of A050972 and A050973. Cf. A014567.

Edited by Dean Hickerson, Sep 19 2002

A050972 Smaller member of friendly pairs ordered by smallest maximal element.

Original entry on oeis.org

6, 30, 80, 40, 12, 84, 66, 78, 102, 6, 28, 114, 240, 138, 120, 150, 174, 135, 186, 864, 222, 246, 60, 258, 282, 560, 318, 84, 270, 330, 84, 270, 1488, 354, 366, 720, 390, 402, 426, 360, 438, 880, 474, 498, 510, 440, 30, 140, 534, 132, 1040, 570, 582, 606

Offset: 1

Eric W. Weisstein

nonn

Conjecture: If a(n) is in A005153, then A050973(n) is in A005153. It is natural to ask if there exists a generalization of the indicator function for A005153, call it m(n), such that m(n) = 1 for n in A005153, 0 < m(n) < 1 otherwise, and m(a(n)) <= m(A050973(n)) for all n. See also A017666. - Jaycob Coleman, Sep 27 2014

Donovan Johnson, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Friendly Pair.

Cf. A050973 (larger member of pair), A074902 (friendly numbers).

A095301 Numbers n such that there is some k < n with nsigma(k) = ksigma(n).

Original entry on oeis.org

28, 140, 200, 224, 234, 270, 308, 364, 476, 496, 532, 600, 644, 672, 700, 812, 819, 868, 936, 1036, 1148, 1170, 1204, 1316, 1400, 1484, 1488, 1540, 1638, 1652, 1708, 1800, 1820, 1876, 1988, 2016, 2044, 2200, 2212, 2324, 2380, 2464, 2480, 2492, 2574, 2600

Offset: 1

Klaus Brockhaus, Jun 01 2004

nonn

Original name: Numbers n such that A094759(n) < n.
Agrees with A050973 without duplicates.
Also numbers n such that the value sigma(n)/n has already been reached before n. If n belongs to the sequence then A214701(n) = A214701(n-1). - Michel Marcus, Aug 19 2012

			A094759(28) = 6 < 28, hence 28 is in the sequence.

B. Spearman and K. S. Williams, Handbook of Estimates in the Theory of Numbers, Carleton Math. Lecture Note Series No. 14, 1975; see p. 3.2, Eq. (3.9).

Michel Marcus and Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 180 terms from Michel Marcus)
P. Erdős, Remarks on number theory II: Some problems on the sigma function, Acta Arith., 5 (1959), 171-177.

Cf. A000203, A094759, A050973.

for(n=1,2600,s=sigma(n);k=1;while(n*sigma(k)!=k*s,k++);if(k

allab = []; nb = 0; for (i=1, n, ab = sigma(i)/i; already = 0; if (length(allab) > 0, for (j=1, length(allab), if (ab == allab[j], already = 1; break););); if (already == 1,  nb++; print1(i, ","), allab = concat(allab, ab););)
/* Michel Marcus, Aug 19 2012 */

New name from Charles R Greathouse IV, Jul 21 2015

A094759 Least k <= n such that nsigma(k) = ksigma(n), where sigma(n) is the sum of divisors of n (A000203).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 6, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Amarnath Murthy, May 30 2004

nonn

Conjecture: There are infinitely many terms such that a(n)A050973 has those n, A050972 has the a(n).
See A095301 for a version of A050973 that do not duplicate every n that has several smaller k of the same abundancy. - Jeppe Stig Nielsen, Jul 09 2015
That conjecture is an easy fact: Since, e.g., (6,28) is a friendly pair, then so is (6k,28k) for any multiplier k with gcd(42,k)=1. So any n=28k, where gcd(42,k)=1, satisfies a(n)A095301 does not have asymptotic density zero. - Jeppe Stig Nielsen, Jul 09 2015
This sequence is related to Theorem 1 on p. 173 of the Erdős link in the following way. For a given x, let us consider the set of integers such that a(n) <= x, which is equivalent to removing duplicates from the current sequence. This set would begin with: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, ... So this set has the same number of elements as the number of distinct terms numbers of the form sigma(n)/n with 1 <= n <=x. Then by Erdős, it is c1*x + o(x), with 6/Pi^2 < c1 < 1. With x = 10^7, we find c1 ~= 0.98208... - Michel Marcus, Jul 21 2015
a(n) is the least k which has the same abundancy index as n, that is, minimal k for which sigma(k)/k = sigma(n)/n. - Antti Karttunen, Jul 10 2019

Robert Israel, Table of n, a(n) for n = 1..10000
P. Erdős, Remarks on number theory II: Some problems on the sigma function, Acta Arith., 5 (1959), 171-177.

Cf. A000203, A050972, A050973, A094757, A094758, A326200.
Cf. A095301 for n such that a(n) < n.
Cf. A000396 (positions of 6's), A005820 (positions of 120's).

N:= 100: # to get a(1) to a(N)
for n from 1 to N do
   v:= numtheory:-sigma(n)/n;
   if not assigned(R[v]) then R[v]:= n fi;
   A[n]:= R[v];
od:
seq(A[n],n=1..N); # Robert Israel, Jul 21 2015

Table[Module[{k=1,sn=DivisorSigma[1,n]},While[n DivisorSigma[1,k]!=k*sn,k++];k],{n,80}] (* Harvey P. Dale, Aug 03 2025 *)

for(n=1,74,s=sigma(n);k=1;while(n*sigma(k)!=k*s,k++);print1(k,","));

Edited and extended by Don Reble and Klaus Brockhaus, May 31 2004

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

Original entry on oeis.org

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

cp's OEIS Frontend

A236355 Terms of A050973 that give maximum record values for A050973(k)/A050972(k).

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A236372 Terms of A050973 that give minimum record values for A050973(k)/A050972(k).

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A263118 Indices of the primitive friendly pairs in the sequence of friendly pairs (A050973, A050972) ordered by smallest maximal element.

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A263152 a(n) is the greatest common unitary divisor of the friendly pairs, A050972(n) and A050973(n).

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A007770 Happy numbers: numbers whose trajectory under iteration of sum of squares of digits map (see A003132) includes 1.

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A074902 Known friendly numbers.

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A050972 Smaller member of friendly pairs ordered by smallest maximal element.

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A095301 Numbers n such that there is some k < n with n*sigma(k) = k*sigma(n).

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A095301 Numbers n such that there is some k < n with nsigma(k) = ksigma(n).

A094759 Least k <= n such that nsigma(k) = ksigma(n), where sigma(n) is the sum of divisors of n (A000203).