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 1-6 of 6 results.

A275701 Numbers n whose abundance is 26: sigma(n) - 2n = 26.

Original entry on oeis.org

80, 1184, 6464, 29312, 78975, 510464, 557192, 137431875584, 549741658112, 8796036399104, 35184258842624, 2251798907715584
Offset: 1

Views

Author

Timothy L. Tiffin, Aug 05 2016

Keywords

Comments

Any term x = a(m) can be combined with any term y = A275702(n) to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2. Although this property is a necessary condition for two numbers to be amicable, it is not a sufficient one. So far, these two sequences have produced only one amicable pair: (x,y) = (1184,1210) = (a(2),A275702(5)) = (A063990(3),A063990(4)). If more are ever found, then they will also exhibit y-x = 26.
Notice that:
a(1) = 80 = 5* 16 = (2*4^2-27)*(4^2)
a(2) = 1184 = 37* 32 = (4^3-27)*(4^3)/2
a(3) = 6464 = 101* 64 = (2*4^3-27)*(4^3)
a(4) = 29312 = 229*128 = (4^4-27)*(4^4)/2
a(6) = 510464 = 997*512 = (4^5-27)*(4^5)/2.
If p = 2*4^k-27 is prime and n = p*(p+27)/2, then it is not hard to show that sigma(n) - 2*n = 26. The values of k in A275767 will guarantee that p is prime (A275749). Similarly, if q = 4^k-27 is prime and n = q*(q+27)/2, then sigma(n) - 2*n = 26. The values of k in A274519 will guarantee that q is prime (A275750). So, the following values will be in this sequence and provide upper bounds for the next eight terms:
(2*4^9-27)*(4^9) = 137431875584 >= a(8)
(4^10-27)*(4^10)/2 = 549741658112 >= a(9)
(4^11-27)*(4^11)/2 = 8796036399104 >= a(10)
(2*4^11-27)*(4^11) = 35184258842624 >= a(11)
(4^13-27)*(4^13)/2 = 2251798907715584 >= a(12)
(4^25-27)*(4^25)/2 = 633825300114099501099609227264 >= a(13)
(4^28-27)*(4^28)/2 = 2596148429267412841487728652582912 >= a(14)
(4^29-27)*(4^29)/2 = 41538374868278617137133892585652224 >= a(15).
a(8) > 10^9. - Michel Marcus, Sep 15 2016
a(8) > 2*10^9. - Michel Marcus, Dec 31 2016
a(13) > 10^18. - Hiroaki Yamanouchi, Aug 23 2018

Examples

			a(1) = 80, since sigma(80)-2*80 = 186-160 = 26.
a(2) = 1184, since sigma(1184)-2*1184 = 2394-2368 = 26.
a(3) = 6464, since sigma(6464)-2*6464 = 12954-12928 = 26.

Links

Crossrefs

Cf. A033880, A063990, A274519, A275702 (deficiency 26), A275749, A275750, A275767.
Cf. A223609 (abundance 10), ..., A223613 (abundance 24).

Programs

  • Magma
    [n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq 26]; // Vincenzo Librandi, Sep 16 2016
  • Mathematica
    Select[Range[10^7], DivisorSigma[1, #] - 2 # == 26 &] (* Vincenzo Librandi, Sep 16 2016 *)
  • PARI
    isok(n) = sigma(n) - 2*n == 26; \\ Michel Marcus, Sep 15 2016

Extensions

a(8)-a(12) from Hiroaki Yamanouchi, Aug 23 2018

A104072 Primes of the form 2^n + 5^2.

Original entry on oeis.org

29, 41, 89, 281, 1049, 1048601, 4194329, 17179869209, 1180591620717411303449, 4951760157141521099596496921, 5192296858534827628530496329220121, 332306998946228968225951765070086169
Offset: 1

Views

Author

Roger L. Bagula, Mar 02 2005

Keywords

Comments

Primes of the form 4^n + 4! + 1. - Vincenzo Librandi, Nov 13 2010
Indeed, calculating mod 3 we have 2^n + 5^2 = (-1)^n + 1 = 0 if n is odd, so n must be even to yield a prime. - M. F. Hasler, Nov 13 2010
Those even values of n are given in A157006. Since n = 2k, these prime numbers also have the form 4^k + 25, where k is given in A204388. - Timothy L. Tiffin, Aug 06 2016
These primes a(m) can be used to generate numbers having deficiency 26. The formula a(m)*(a(m)-25)/2 produces those terms in A275702 having rightmost digit 8. - Timothy L. Tiffin, Aug 09 2016

Examples

			From _Timothy L. Tiffin_, Aug 07 2016: (Start)
a(1) = 2^2  + 5^2 =       4 + 25 =      29.
a(2) = 2^4  + 5^2 =      16 + 25 =      41.
a(3) = 2^6  + 5^2 =      64 + 25 =      89.
a(4) = 2^8  + 5^2 =     256 + 25 =     281.
a(5) = 2^10 + 5^2 =    1024 + 25 =    1049.
a(6) = 2^20 + 5^2 = 1048576 + 25 = 1048601. (End)

Crossrefs

Cf. A157006, A204388, A275702.

Programs

  • Mathematica
    a = Delete[Union[Flatten[Table[If [PrimeQ[2^n + 25] == True, 2^n + 25, 0], {n, 1, 400}]]], 1]
Select[2^Range[0,120]+25,PrimeQ] (* Harvey P. Dale, Jun 20 2017 *)

Formula

a(m) = 2^(A157006(m)) + 5^2 = 4^(A204388(m)) + 25. - Timothy L. Tiffin, Aug 07 2016
If n == 0 mod 4, then a(m) == 1 mod 10. If n == 2 mod 4, then a(m) == 9 mod 10. - Timothy L. Tiffin, Aug 09 2016

A275997 Numbers k whose deficiency is 64: 2k - sigma(k) = 64.

Original entry on oeis.org

134, 284, 410, 632, 1292, 1628, 4064, 9752, 12224, 22712, 66992, 72944, 403988, 556544, 2161664, 2330528, 8517632, 13228352, 14563832, 15422912, 20732792, 89472632, 134733824, 150511232, 283551872, 537903104, 731670272, 915473696, 1846850576, 2149548032, 2159587616
Offset: 1

Views

Author

Timothy L. Tiffin, Aug 16 2016

Keywords

Comments

Any term x = a(m) in this sequence can be used with any term y in A275996 to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2, which is a necessary (but not sufficient) condition for two numbers to be amicable.
The smallest amicable pair is (220, 284) = (A275996(2), a(2)) = (A063990(1), A063990(2)), where 284 - 220 = 64 is the abundance of 220 and the deficiency of 284.
The amicable pair (66928, 66992) = (A275996(7), a(11)) = (A063990(18), A063990(19)), where 66992 - 66928 = 64 is the deficiency of 66992 and the abundance of 66928.
Contains numbers 2^(k-1)*(2^k + 63) whenever 2^k + 63 is prime. - Max Alekseyev, Aug 27 2025

Examples

			a(1) = 134, since 2*134 - sigma(134) = 268 - 204 = 64.

Links

Crossrefs

Deficiency k: A191363 (k=2), A125246 (k=4), A141548 (k=6), A125247 (k=8), A101223 (k=10), A141549 (k=12), A141550 (k=14), A125248 (k=16), A223608 (k=18), A223607 (k=20), A223606 (k=22), A385255(k=24), A275702 (k=26), A387352 (k=32).
Abundance k: A088831 (k=2), A088832 (k=4), A087167 (k=6), A088833 (k=8), A223609 (k=10), A141545 (k=12), A141546 (k=14), A141547 (k=16), A223610 (k=18), A223611 (k=20), A223612 (k=22), A223613 (k=24), A275701 (k=26), A175989 (k=32), A275996 (k=64), A292626 (k=128).
Cf. A002025, A063990.

Programs

  • Mathematica
    Select[Range[10^7], 2 # - DivisorSigma[1, #] == 64 &] (* Michael De Vlieger, Jan 10 2017 *)
  • PARI
    isok(n) = 2*n - sigma(n) == 64; \\ Michel Marcus, Dec 30 2016

Extensions

a(23)-a(31) from Jinyuan Wang, Mar 02 2020

A292626 Numbers k whose abundance is 128: sigma(k) - 2*k = 128.

Original entry on oeis.org

860, 5336, 6536, 9656, 16256, 55796, 70864, 98048, 361556, 776096, 2227616, 4145216, 4498136, 4632896, 8124416, 13086016, 34869056, 38546576, 150094976, 172960856, 196066256, 962085536, 1080008576, 1733780336, 1844788112, 2143256576, 2531343872, 2986104064, 9677743616, 11276687456, 17104503968, 20680182272, 21568135616
Offset: 1

Views

Author

Fabian Schneider, Sep 20 2017

Keywords

Links

Crossrefs

Subsequence of A259174.
Deficiency k: A191363 (k=2), A125246 (k=4), A141548 (k=6), A125247 (k=8), A101223 (k=10), A141549 (k=12), A141550 (k=14), A125248 (k=16), A223608 (k=18), A223607 (k=20), A223606 (k=22), A385255(k=24), A275702 (k=26), A387352 (k=32), A275997 (k=64).
Abundance k: A088831 (k=2), A088832 (k=4), A087167 (k=6), A088833 (k=8), A223609 (k=10), A141545 (k=12), A141546 (k=14), A141547 (k=16), A223610 (k=18), A223611 (k=20), A223612 (k=22), A223613 (k=24), A275701 (k=26), A175989 (k=32), A275996 (k=64).

Programs

  • Mathematica
    fQ[n_] := DivisorSigma[1, n] == 2 n + 128; Select[ Range@ 10^8, fQ] (* Robert G. Wilson v, Nov 19 2017 *)
  • PARI
    isok(n) = sigma(n) - 2*n == 128; \\ Michel Marcus, Sep 20 2017

Extensions

a(9)-a(18) from Michel Marcus, Sep 20 2017
a(19)-a(24), a(26), a(29)-a(30), a(33) from Robert G. Wilson v, Nov 20 2017
Missing terms a(25), a(27)-a(28), a(31)-a(32) inserted and terms a(34) onward added by Max Alekseyev, Aug 30 2025

A385255 Numbers m whose deficiency is 24: sigma(m) - 2*m = -24.

Original entry on oeis.org

124, 9664, 151115727458150838697984
Offset: 1

Views

Author

Max Alekseyev, Jul 29 2025

Keywords

Comments

Contains numbers 2^(k-1)*(2^k + 23) for k in A057203. First three terms have this form.

Crossrefs

Deficiency k: A191363 (k=2), A125246 (k=4), A141548 (k=6), A125247 (k=8), A101223 (k=10), A141549 (k=12), A141550 (k=14), A125248 (k=16), A223608 (k=18), A223607 (k=20), A223606 (k=22), A275702 (k=26).
Abundance k: A088831 (k=2), A088832 (k=4), A087167 (k=6), A088833 (k=8), A223609 (k=10), A141545 (k=12), A141546 (k=14), A141547 (k=16), A223610 (k=18), A223611 (k=20), A223612 (k=22), A223613 (k=24), A275701 (k=26).
Cf. A057203.

A387352 Numbers m with deficiency 32: sigma(m) - 2*m = -32.

Original entry on oeis.org

250, 376, 1276, 12616, 20536, 396916, 801376, 1297312, 8452096, 33721216, 40575616, 59376256, 89397016, 99523456, 101556016, 150441856, 173706136, 269096704, 283417216, 500101936, 1082640256, 1846506832, 15531546112, 34675557856, 136310177392, 136783784608
Offset: 1

Views

Author

Max Alekseyev, Aug 27 2025

Keywords

Comments

Contains numbers 2^(k-1)*(2^k + 31) for k in A247952.

Links

Crossrefs

Deficiency k: A191363 (k=2), A125246 (k=4), A141548 (k=6), A125247 (k=8), A101223 (k=10), A141549 (k=12), A141550 (k=14), A125248 (k=16), A223608 (k=18), A223607 (k=20), A223606 (k=22), A385255(k=24), A275702 (k=26), A275997 (k=64).
Abundance k: A088831 (k=2), A088832 (k=4), A087167 (k=6), A088833 (k=8), A223609 (k=10), A141545 (k=12), A141546 (k=14), A141547 (k=16), A223610 (k=18), A223611 (k=20), A223612 (k=22), A223613 (k=24), A275701 (k=26), A175989 (k=32), A275996 (k=64), A292626 (k=128).
Cf. A247952.
Showing 1-6 of 6 results.

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