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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A328370 Quasi-amicable pairs.

Original entry on oeis.org

48, 75, 140, 195, 1050, 1925, 1575, 1648, 2024, 2295, 5775, 6128, 8892, 16587, 9504, 20735, 62744, 75495, 186615, 206504, 196664, 219975, 199760, 309135, 266000, 507759, 312620, 549219, 526575, 544784, 573560, 817479, 587460, 1057595, 1000824, 1902215, 1081184, 1331967, 1139144, 1159095, 1140020, 1763019
Offset: 1

Views

Author

Omar E. Pol, Oct 14 2019

Keywords

Comments

Also called betrothed pairs, or quasiamicable pairs, or reduced amicable pairs.
A pair of numbers x and y is called quasi-amicable if sigma(x) = sigma(y) = x + y + 1, where sigma(n) is the sum of the divisors of n.
All known quasi-amicable pairs have opposite parity.
First differs from A005276 at a(6).
According to Hisanori Mishima (see link) there are 404 quasi-amicable pairs where the smaller part is less than 10^10. See A126160 for more values. - Peter Luschny, Nov 18 2019

Examples

			Initial quasi-amicable pairs:
    48,   75;
   140,  195;
  1050, 1925;
  1575, 1648;
  2024, 2295;
...
The sum of the divisors of 48 is 1 + 2 + 3 + 4 + 6 + 8 + 12 + 16 + 24 + 48 = 124. On the other hand the sum of the divisors of 75 is 1 + 3 + 5 + 15 + 25 + 75 = 124. Note that 48 + 75 + 1 = sigma(48) = sigma(75) = 124. The smallest quasi-amicable pair is (48, 75), so a(1) = 48 and a(2) = 75.

Links

Crossrefs

A003502 and A003503 interleaved.
Cf. A000203, A005276, A126160, A179612, A259180.

Programs

  • Maple
    with(numtheory): aList := proc(searchbound)
local r, n, m, L: L := []:
for m from 1 to searchbound do
   n := sigma(m) - m - 1:
   if n <= m then next fi;
   r := sigma(n) - n - 1:
   if r = m then L := [op(L), m, n] fi;
od; L end:
aList(10000); # Peter Luschny, Nov 18 2019

Formula

a(2*n-1) = A003502(n); a(2*n) = A003503(n).

A166385 Primes between the lower and upper member of the n-th pair of betrothed numbers.

Original entry on oeis.org

53, 59, 61, 67, 71, 73, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Oct 13 2009

Keywords

Comments

Primes in one of the intervals [A003502(k),A003503(k)], k>=1.

Examples

			The group of primes from 53 to 73 is in the sequence, because they are between 48 and 75.
The group of primes from 149 to 193 is in the sequence, because they are between 140 and 195.

Crossrefs

Cf. A000040, A005276.

Extensions

1063 inserted by R. J. Mathar, Oct 21 2009

A166386 The count of primes between the (2*n-1)-th betrothed number and the (2n)-th betrothed number.

Original entry on oeis.org

6, 10, 72, 34, 35, 41, 70, 415, 1138, 824, 548, 3693, 270
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Oct 13 2009

Keywords

Crossrefs

Cf. A000040, A005276.

Extensions

a(4) corrected by R. J. Mathar, Oct 16 2009

A281265 Variation on betrothed numbers.

Original entry on oeis.org

6160, 11697, 12220, 16005, 23500, 28917, 68908, 76245, 249424, 339825, 425500, 434784, 570405, 649990, 660825, 678376, 697851, 871585, 1017856, 1077336, 1238380, 1252216, 1340865, 1483785, 1568260, 1754536, 1823925, 1899261, 2067625, 2166136, 2362360, 2479065
Offset: 1

Views

Author

Paolo P. Lava, Apr 13 2017

Keywords

Comments

Members of a pair (x,y) such that sigma(x) = sigma(y) = x + y - 1, where sigma = A000203.
The first time a pair ordered by its first element is not adjacent is x = 425500, y = 570405 which correspond to a(11) and a(13), respectively.

Examples

			sigma(6160) = sigma(11697) = 6160 + 11697 - 1 = 17856.

Crossrefs

Cf. A000203, A005276, A007992, A015630.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,b,n; for n from 1 to q do
a:=sigma(n)-n+1; b:=sigma(a)-a+1; if b=n and a<>b then print(n);
fi; od; end: P(10^9);

Formula

A007992 U A015630.
Previous Showing 11-14 of 14 results.

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