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.

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A211677 First number k whose value of sigma(k)/k appears n times.

Original entry on oeis.org

1, 28, 496, 1638, 24384, 2886100, 13035330, 29410290, 4426793280
Offset: 1

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Author

T. D. Noe, May 09 2012

Keywords

Comments

The values of sigma(k)/k are 1, 2, 2, 8/3, 8/3, 96/35, 32/9, 32/9, 32/7. Note that these values are nondecreasing. Is that always the case? In the table below, all numbers in the same row are friendly to each other.
a(10) <= 27477725184. a(11) <= 88071903612. a(12) <= A027687(12). - Donovan Johnson, Aug 06 2012
For n>1, these are the smallest numbers to appear consecutively (n-1) times in A050973. - Michel Marcus, Jan 28 2014

Examples

			These are the values of k such that sigma(k)/k appears n times:
n   k values
1:  1
2:  6, 28
3:  6, 28, 496
4:  84, 270, 1488, 1638
5:  84, 270, 1488, 1638, 24384
6:  210, 17360, 43400, 284480, 2229500, 2886100
7:  3780, 66960, 167400, 406224, 1097280, 6656832, 13035330
8:  3780, 66960, 167400, 406224, 1097280, 6656832, 13035330, 29410290
9:  164989440, 270138960, 318729600, 326781000, 481572000, 623397600, 675347400, 995248800, 4426793280 - _Donovan Johnson_, Aug 06 2012
These numbers appear in A211679.

Links

Crossrefs

Cf. A000203 (sigma), A050973, A211679.

Extensions

a(7)-a(8) from Donovan Johnson, May 10 2012
a(9) from Michel Marcus and Donovan Johnson, Aug 06 2012

A214701 Number of different values taken by sigma(k)/k for k from 1 up to n.

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, 27, 28, 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
Offset: 1

Views

Author

Michel Marcus, Jul 26 2012

Keywords

Comments

The first 26 values of sigma(n)/n are all different. a(28)= 27 since (6, 28) being the smallest friendly pair, 2=abundancy(28) already appeared before. When n belongs to A050973, then a(n) = a(n-1).

Examples

			a(2)= 2 because sigma(n)/n takes 2 different values up to 2 : 1 and 3/2.

Links

Programs

  • Mathematica
    s = {}; Table[s = Union[s, {DivisorSigma[1, n]/n}]; Length[s], {n, 100}] (* T. D. Noe, Jul 26 2012 *)

Formula

a(n)/n tends to 0.98208.. approximately when n increases. - Michel Marcus, Jul 07 2015

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])))

A339090 GCD of friendly pairs.

Original entry on oeis.org

2, 10, 40, 8, 6, 6, 22, 26, 34, 2, 4, 38, 120, 46, 24, 50, 58, 9, 62, 72, 74, 82, 30, 86, 94, 280, 106, 12, 6, 110, 42, 18, 6, 118, 122, 360, 130, 134, 142, 72, 146, 440, 158, 166, 170, 88, 10, 20, 178, 66, 520, 190, 194, 202, 206, 104, 66, 214, 218, 226, 230
Offset: 1

Views

Author

Ruediger Jehn, Nov 23 2020

Keywords

Comments

The first 10000 friendly pairs (copied from the b-files of A050972 and A050973) all have either 2 or 3 (or both) as a common divisor and hence their GCD is divisible by 2 or 3. However this does not hold for all friendly pairs. GCD(3472, 544635) = 7. These two numbers belong to the club where sigma(x)/x = 16/7 (also 42 belongs to this club).

Examples

			The first friendly pair is (6, 28) and therefore a(1) = gcd(6, 28) = 2.

Crossrefs

Cf. A050972, A050973.

Formula

a(n) = gcd(A050972(n), A050973(n)).
Previous Showing 11-15 of 15 results.

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