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 1-3 of 3 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

A275750 Prime numbers of the form 4^k - 27.

Original entry on oeis.org

37, 229, 997, 1048549, 4194277, 67108837, 1125899906842597, 72057594037927909, 288230376151711717, 1361129467683753853853498429727072845797, 1393796574908163946345982392040522594123749, 1725436586697640946858688965569256363112777243042596638790631055949797
Offset: 1

Views

Author

Timothy L. Tiffin, Aug 07 2016

Keywords

Comments

Values of the exponent k are given in A274519. If the exponent is odd, then the rightmost digit of a(n) will be 7. If the exponent is even, then the rightmost digit of a(n) will be 9.
As a result of the recent extensions to A274519 by Vincenzo Librandi,
a(13) = 4^305 - 27 > 4.2491 * 10^183
a(14) = 4^515 - 27 > 1.1505 * 10^310
a(15) = 4^2029 - 27 > 3.7994 * 10^1221
a(16) = 4^2393 - 27 > 5.3648 * 10^1440
a(17) = 4^2605 - 27 > 2.3242 * 10^1568
a(18) = 4^3530 - 27 > 1.8696 * 10^2125
a(19) = 4^4036 - 27 > 8.2058 * 10^2429
a(20) = 4^4750 - 27 > 6.0947 * 10^2859
a(21) > 4^5000 - 27 > 1.9950 * 10^3010.
These primes a(m) can be used to generate numbers having abundance 26. The formula a(m)*(a(m)+27)/2 produces some of the terms in A275701.

Examples

			a(1) = 4^A274519(1) - 27 = 4^3  - 27 =       64 - 27 =       37.
a(2) = 4^A274519(2) - 27 = 4^4  - 27 =      256 - 27 =      229.
a(3) = 4^A274519(3) - 27 = 4^5  - 27 =     1024 - 27 =      997.
a(4) = 4^A274519(4) - 27 = 4^10 - 27 =  1048576 - 27 =  1048549.
a(5) = 4^A274519(5) - 27 = 4^11 - 27 =  4194304 - 27 =  4194277.
a(6) = 4^A274519(6) - 27 = 4^13 - 27 = 67108864 - 27 = 67108837.

Links

Crossrefs

Cf. A274519, A275701, A275749.

Programs

Formula

a(n) = 4^A274519(n) - 27.

A275767 Numbers k for which 2*4^k - 27 is prime.

Original entry on oeis.org

2, 3, 9, 11, 291, 1263, 2661, 3165, 8973, 8999, 27479, 42689
Offset: 1

Views

Author

Timothy L. Tiffin, Aug 07 2016

Keywords

Comments

The prime numbers that these exponents generate are given in A275749.
Since 2*4^(2k) - 27 = 2*16^k - 27 == (2*1^k - 27) mod 5 = -25 mod 5 == 0 mod 5, no even number greater than 2 will be in this sequence.
a(8) > 5000. - Vincenzo Librandi, Aug 08 2016

Examples

			a(1) = 2, since 2*4^2 - 27 = 32 - 27 = 5, which is prime.
a(2) = 3, since 2*4^3 - 27 = 128 - 27 = 101, which is prime.
a(3) = 9, since 2*4^9 - 27 = 524288 - 27 = 524261, which is prime.
a(4) = 11, since 2*4^11 - 27 = 8388608 - 27 = 8388581, which is prime.

Links

Crossrefs

Cf. A274519, A275749.

Programs

  • Magma
    [n: n in [2..1000] |IsPrime(2*4^n-27)]; // Vincenzo Librandi, Aug 08 2016
  • Mathematica
    Select[Range[2, 1000], PrimeQ[2 4^# - 27] &] (* Vincenzo Librandi, Aug 08 2016 *)
  • Python
    from sympy import isprime
def afind(limit, startk=2):
    alst, pow4 = [], 4**startk
    for k in range(startk, limit+1):
        if isprime(2*pow4 - 27): print(k, end=", ")
        pow4 *= 4
afind(1300) # Michael S. Branicky, Sep 22 2021

Extensions

a(6)-a(8) from Vincenzo Librandi, Aug 08 2016
a(9)-a(10) from Michael S. Branicky, Sep 22 2021
a(11) from Michael S. Branicky, Apr 05 2023
a(12) from Michael S. Branicky, Oct 25 2024
Showing 1-3 of 3 results.

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