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.

A003502 The smaller of a betrothed pair.

Original entry on oeis.org

48, 140, 1050, 1575, 2024, 5775, 8892, 9504, 62744, 186615, 196664, 199760, 266000, 312620, 526575, 573560, 587460, 1000824, 1081184, 1139144, 1140020, 1173704, 1208504, 1233056, 1236536, 1279950, 1921185, 2036420, 2102750, 2140215, 2171240, 2198504, 2312024
Offset: 1

Views

Author

Robert G. Wilson v

Keywords

Examples

			48 is a term because sigma(48) - 48 - 1 = 124 - 48 - 1 = 75 and 48 < 75 and sigma(75) - 75 - 1 = 124 - 75 - 1 = 48. - _David A. Corneth_, Jan 24 2019

References

  • R. K. Guy, Unsolved Problems in Number Theory, B5.

Links

Crossrefs

Cf. A000203, A003503, A005276.

Programs

  • Mathematica
    aapQ[n_] := Module[{c=DivisorSigma[1, n]-1-n}, c!=n&&DivisorSigma[ 1, c]-1-c == n]; Transpose[Union[Sort[{#, DivisorSigma[1, #]-1-#}]&/@Select[Range[2, 10000], aapQ]]] [[1]] (* Amiram Eldar, Jan 24 2019 after Harvey P. Dale at A007992 *)
  • PARI
    is(n) = m = sigma(n) - n - 1; if(m == 0 || n >= m, return(0)); n == sigma(m) - m - 1 \\ David A. Corneth, Jan 24 2019

Extensions

Computed by Fred W. Helenius (fredh(AT)ix.netcom.com)
Extended by T. D. Noe, Dec 29 2011

A015630 Augmented amicable pairs (larger member of each pair).

Original entry on oeis.org

11697, 16005, 28917, 76245, 339825, 570405, 871585, 697851, 678376, 1340865, 2067625, 1823925, 1483785, 1899261, 2479065, 2580105, 4895241, 4740505, 5736445, 3171556, 4791916, 6516237, 4416976, 7524525, 9868075, 7589745
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Let f(n) = 1 + sum of aliquot divisors of n; these are pairs (n,m) with f(n)=m, f(m)=n.
The terms of the sequence are sorted in the order of the smaller (omitted) member of each pair. [Harvey P. Dale, Feb 29 2012]

Links

Crossrefs

Cf. A007992.

Programs

  • Mathematica
    aap[n_]:=Module[{p=Total[Most[Divisors[n]]]+1},If[p!=n&&n==Total[Most[ Divisors[p]]]+1,{p,n},0]]; Transpose[Union[Sort/@DeleteCases[aap/@ Range[10000000],0]]][[2]] (* Harvey P. Dale, Feb 29 2012 *)

A095702 Smallest "n-augmented" amicable number: the smallest positive integer k such that m = sigma(k) - k + n > k and k = sigma(m) - m + n, where sigma(k) is the sum of the divisors of k.

Original entry on oeis.org

220, 6160, 24, 180, 20, 6, 224, 2632, 40, 10, 16, 28, 340, 14, 15, 42, 66, 3696, 208, 20, 21, 54, 264, 24, 68, 26, 88, 120, 102, 30, 4030, 56, 33, 34, 35, 60, 110, 38, 280, 40, 354, 66, 476, 44, 130, 46, 408, 92, 1276, 96, 51, 52, 354, 78, 55, 120, 57, 58, 852, 60, 170
Offset: 0

Views

Author

Jack Brennen, Jul 06 2004

Keywords

Examples

			a(1)= 6160 because sigma(6160)-6160+1 == 11697, sigma(11697)-11697+1 == 6160 and 6160 is the smallest integer for which this holds.

Links

Crossrefs

Cf. A000203, A002025, A002046, A007992, A015630.

Programs

  • Mathematica
    a[n_] := Module[{k = n + 1, s}, While[( s = DivisorSigma[1, k] - k + n) <= k || DivisorSigma[1, s] - s + n != k, k++]; k]; Array[a, 61, 0] (* Amiram Eldar, Dec 24 2020 *)
  • PARI
    for(g=0,60,x=g+1;while(1,a=sigma(x)-x+g;if((a-x)*a,if(sigma(a)-a+g==x,print1(x,",");break));x+=1))

A306867 Lesser of augmented bi-unitary amicable pair.

Original entry on oeis.org

434784, 1100176, 1252216, 1754536, 2166136, 2362360, 3064096, 6224890, 7626136, 7851256, 7950096, 9026235, 9581320, 12494856, 13324311, 14192080, 15218560, 15243424, 15422536, 19028296, 19466560, 19555360, 29180466, 36716680, 37542190, 40682824, 44044000, 44588896
Offset: 1

Views

Author

Amiram Eldar, Mar 14 2019

Keywords

Comments

A pair m < n is an augmented bi-unitary amicable pair if bsigma(m) = bsigma(n) = m + n - 1, where bsigma(n) is the sum of bi-unitary divisors of n (A188999).
The larger members are in A306868.

Examples

			434784 is in the sequence since it is the lesser of the amicable pair (434784, 871585): bsigma(434784) = bsigma(871585) = 1306368 = 434784 + 871585 - 1.

Links

Crossrefs

Cf. A007992, A126174, A188999, A292980, A306868.

Programs

  • Mathematica
    fun[p_, e_]:=If[OddQ[e], (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); f[n_] := bsigma[n] - n + 1; s={}; Do[m = f[n]; If[m > n && f[m] == n, AppendTo[s, n]], {n, 1, 10^7}]; s

A306872 Lesser of augmented unitary amicable pair.

Original entry on oeis.org

6224890, 37542190, 56523810, 101304490, 135657795, 233990890, 5304907426, 8473747030, 8483430670, 9220653310, 11033448910, 12139959910, 13108452735, 13849730895, 16697472870, 19644687195, 20321234206, 23076788295, 40575765615, 55636542346, 89094853155, 101786530846
Offset: 1

Views

Author

Amiram Eldar, Mar 14 2019

Keywords

Comments

A pair m < n is an augmented unitary amicable pair if usigma(m) = usigma(n) = m + n - 1, where usigma(n) is the sum of unitary divisors of n (A034460).
The larger members are in A306873.

Examples

			6224890 is in the sequence since it is the lesser of the amicable pair (6224890, 7336455): usigma(6224890) = usigma(7336455) = 13561344 = 6224890 + 7336455 - 1.

Links

Crossrefs

Cf. A002952, A007992, A034460, A126174, A306873.

Programs

  • Mathematica
    us[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n;  s={}; Do[m = us[n] + 1; If[m > n && us[m] == n - 1, AppendTo[s, n]], {n, 1, 10^9}]; s

A126161 Number of augmented amicable pairs (m,n) with m

Original entry on oeis.org

0, 0, 0, 1, 4, 9, 36, 84, 188, 420, 930, 1931
Offset: 1

Views

Author

Ant King, Dec 20 2006

Keywords

Examples

			a(6)=9 because there are 9 augmented amicable pairs with m<=10^6

Links

Crossrefs

Cf. A007992, A015630.

Programs

  • Mathematica
    s[n_]:=DivisorSigma[1,n]-n; AugmentedAmicableNumberQ[n_]:=If[s[s[n]+1]==n-1 && !DivisorSigma[1,n]==2n-1,True,False]; AugmentedAmicablePairList[ k_]:=(bnlist=Select[Range[k], AugmentedAmicableNumberQ[ # ]&]; newprlist= Table[Sort[{bnlist[[n]],s[bnlist[[n]]]+1}],{n,1,Length[bnlist]}]; augamprlist=Union[newprlist,newprlist]); data=AugmentedAmicablePairList[10^7]; Table[Length[Select[data,First[ # ]<10^k &]],{k,1,7}]

Formula

An augmented amicable pair (m,n) is a pair of integers m, n with m

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.
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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