cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A248083 Subsequence of lesser of 2 terms of A095301 that are 2 apart.

Original entry on oeis.org

5850, 8370, 12400, 12402, 13804, 15930, 21328, 25506, 26908, 29428, 32058, 36988, 41886, 45162, 46564, 52108, 53730, 59668, 61290, 62944, 64818, 66220, 71008, 71370, 72772, 74788, 76410, 82600, 83970, 85876, 91026, 92428, 97468, 99090, 103662, 104130, 105028
Offset: 1

Views

Author

Michel Marcus, Oct 01 2014

Keywords

Comments

A095301 lists integers that have a smaller friend, that is, there exists a smaller integer with same abundancy, sigma(x)/x. In A095301 there are terms with a difference of 2, motivating this sequence.

Examples

			5850 is in the sequence because 5850 and 5852, that differ by 2, are both in A095301. That is, for both X=5850 and Y=5852, there exists 2 smaller z (x and y) such that sigma(z)/z = sigma(Z)/Z: 300 and 1254.

Crossrefs

Cf. A095301, A248084 (similar, with 3 consecutive terms).

A248084 Subsequence of lesser of 3 terms of A095301 that are 2 apart.

Original entry on oeis.org

12400, 151396, 503800, 831400, 1050028, 1134196, 1461796, 1486600, 1637296, 1762288, 1814200, 1987468, 2424940, 2444596, 2449744, 3237388, 3427396, 3452200, 3779800, 4082596, 4574608, 4799788, 4862284, 4887088, 5065396, 5674732, 5745400, 6048196, 6375796
Offset: 1

Views

Author

Michel Marcus, Oct 01 2014

Keywords

Comments

A095301 lists integers that have a smaller friend, that is, there exists a smaller integer with same abundancy, sigma(x)/x. In A095301 there are some groups of 3 terms with a common difference of 2, motivating this sequence.
A search, up to 10^9, for similar groups of 4 terms provided no such instance.
84346341351244 and 2652126430251504 belong to the group stated above. - Hiroaki Yamanouchi, Oct 16 2014
This sequence lists the terms A248083(n) such that A248083(n+1)=A248083(n)+2. - M. F. Hasler, Oct 03 2014

Examples

			12400 is in the sequence because 12400, 12402 and 12404, with a difference of 2 between each, are in A095301, with smaller friends: 150, 636 and 2658. As a result of this, both 12400 and 12402 can be found in A248083.

Crossrefs

Cf. A095301, A248083 (similar, with only 2 consecutive terms).

A050973 Larger member of friendly pairs ordered by smallest maximal element.

Original entry on oeis.org

28, 140, 200, 224, 234, 270, 308, 364, 476, 496, 496, 532, 600, 644, 672, 700, 812, 819, 868, 936, 1036, 1148, 1170, 1204, 1316, 1400, 1484, 1488, 1488, 1540, 1638, 1638, 1638, 1652, 1708, 1800, 1820, 1876, 1988, 2016, 2044, 2200, 2212, 2324
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

Perfect numbers greater than 6 (A000396) belong to this sequence as they form friendly pairs with smaller perfect, so that the n-th perfect number will appear n-1 times in the sequence. - Michel Marcus, Dec 03 2013
If we remove duplicates from the sequence we get A095301. - Jeppe Stig Nielsen, Jul 08 2015
It is possible to derive a friendly pair from 2 existing pairs (a_n,b_n) and (a_k,b_k); if (a_n,b_k) and (a_k,b_n) (resp. (a_k,b_k) and (a_n,b_n)) are coprime, then (a_n*b_k,a_k*b_n) (resp. (a_k*b_k,a_n*b_n)) is a friendly pair. For instance one can derive (32760,30240) from (819,135) and (224,40). Moreover, since 32760/35 and 30240/35 are both coprime to 35, one can also derive the primitive friendly pair (936,864). - Michel Marcus, Oct 09 2015

Links

Crossrefs

Cf. A050972, A074902, A095301.

Programs

  • PARI
    lista(nn) = {for (n=1, nn, ab = sigma(n)/n; for (i=2, n-1, if (sigma(i)/i == ab, print1(n, ", "));););} \\ Michel Marcus, Dec 03 2013

A094759 Least k <= n such that n*sigma(k) = k*sigma(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

Views

Author

Amarnath Murthy, May 30 2004

Keywords

Comments

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

Links

Crossrefs

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).

Programs

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

Extensions

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

A212609 The larger companion to the deficient numbers in A212608.

Original entry on oeis.org

819, 6975, 9009, 13923, 15561, 18837, 23751, 25389, 30303, 33579, 35217, 38493, 40131, 43407, 48321, 49959, 54873, 58149, 59787, 64701, 67977, 72891, 79443, 82719, 84357, 87633, 89271, 90675, 92547, 99099, 104013, 107289, 112203, 113841, 118575, 122031
Offset: 1

Views

Author

T. D. Noe, May 23 2012

Keywords

Comments

These numbers are in order; their companions are not.
This is a subsequence of A095301. - Michel Marcus, Sep 28 2014

Crossrefs

Cf. A095301, A212608.

A259917 All friendly numbers, with smallest member of each club listed just before the second-smallest one.

Original entry on oeis.org

6, 28, 30, 140, 80, 200, 40, 224, 12, 234, 84, 270, 66, 308, 78, 364, 102, 476, 496, 114, 532, 240, 600, 138, 644, 120, 672, 150, 700, 174, 812, 135, 819, 186, 868, 864, 936, 222, 1036, 246, 1148, 60, 1170, 258, 1204, 282, 1316, 560, 1400, 318, 1484, 1488, 330
Offset: 1

Views

Author

Jeppe Stig Nielsen, Jul 08 2015

Keywords

Comments

Run through all natural numbers i = 1, 2, 3, ... in order, and record for each the abundancy index sigma(i)/i. When we reach an abundancy that has been seen before, output first the "old" number which had that abundancy (unless that number has already been output earlier), and output secondly the current i.
By construction, no number can occur more than once in the sequence.
Friendly numbers that are not smallest in their club, appear in increasing order. Friendly numbers that are smallest in their club, appear just before the second-smallest member.
If we were to "forget" to output the smallest member in each club, we would get instead A095301.
Oppositely, if we output the smallest members only, we get instead A259918.
It is not known whether the number 10 belongs to this sequence.

Links

Crossrefs

Cf. A050973, A259918, A095301, A014567, A074902.
Terms form a subset of A069059.

Programs

  • PARI
    known=List(); for(i=1,10^5,a=sigma(i)/i; match=0; for(j=1,#known,if(known[j][1]==a,match=j;break())); if(match,old=known[match][2]; if(old,print1(old,", "); known[match]=[a,0]); print(i,","),listput(known,[a,i])))

A259918 Friendly numbers that are smallest in their clubs, ordered by the second-smallest member of the clubs.

Original entry on oeis.org

6, 30, 80, 40, 12, 84, 66, 78, 102, 114, 240, 138, 120, 150, 174, 135, 186, 864, 222, 246, 60, 258, 282, 560, 318, 330, 354, 366, 720, 390, 402, 426, 360, 438, 880, 474, 498, 510, 440, 534, 132, 1040, 570, 582, 606, 618, 520, 924, 642, 654, 678, 690, 540, 726
Offset: 1

Views

Author

Jeppe Stig Nielsen, Jul 08 2015

Keywords

Comments

Subsequence of A259917 (see that entry).

Links

Crossrefs

Cf. A259917, A050973, A095301, A014567.

Programs

  • PARI
    known=List();for(i=1,10^5,a=sigma(i)/i;match=0;for(j=1,#known,if(known[j][1]==a,match=j;break()));if(match,old=known[match][2];if(old,print1(old,", ");known[match]=[a,0]),listput(known,[a,i])))
Showing 1-7 of 7 results.

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