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 1-4 of 4 results.

A065061 Numbers k such that sigma(k) - tau(k) is a prime.

Original entry on oeis.org

3, 8, 162, 512, 1250, 8192, 31250, 32768, 41472, 663552, 2531250, 3748322, 5120000, 6837602, 7558272, 8000000, 15780962, 33554432, 35701250, 42762752, 45334242, 68024448, 75031250, 78125000, 91125000, 137149922, 243101250, 512000000, 907039232, 959570432
Offset: 1

Views

Author

Jason Earls, Nov 06 2001

Keywords

Comments

From Kevin P. Thompson, Jun 20 2022: (Start)
Terms greater than 3 must be twice a square (see A064205).
No terms are congruent to 4 or 6 (mod 10) (see A064205).
(End)

Examples

			162 is a term since sigma(162) - tau(162) = 363 - 10 = 353, which is prime.

Links

Crossrefs

I.e., A065608(n) is prime. Cf. A064205.
Cf. A000005, A000203, A023194, A038344, A055813, A062700, A115919, A141242, A229264, A229265, A229266, A229268.

Programs

  • Mathematica
    Do[ If[ PrimeQ[ DivisorSigma[1, n] - DivisorSigma[0, n]], Print[n]], {n, 1, 10^7}]
  • PARI
    { n=0; for (m=1, 10^9, if (isprime(sigma(m) - numdiv(m)), write("b065061.txt", n++, " ", m); if (n==100, return)) ) } \\ Harry J. Smith, Oct 05 2009
  • Python
    from itertools import count, islice
from sympy import isprime, divisor_sigma as s, divisor_count as t
def agen(): # generator of terms
    yield 3
    yield from (k for k in (2*i*i for i in count(1)) if isprime(s(k)-t(k)))
print(list(islice(agen(), 30))) # Michael S. Branicky, Jun 20 2022

Extensions

a(17)-a(28) from Harry J. Smith, Oct 05 2009
a(29)-a(30) from Kevin P. Thompson, Jun 20 2022

A229268 Primes of the form sigma(k) - tau(k), where sigma(k) = A000203(k) and tau(k) = A000005(k).

Original entry on oeis.org

2, 11, 353, 1013, 2333, 16369, 58579, 65519, 123733, 1982273, 7089683, 5778653, 12795053, 10500593, 22586027, 19980143, 24126653, 67108837, 72494713, 90781993, 106199593, 203275951, 164118923, 183105421, 320210549, 259997173, 794091653, 1279963973
Offset: 1

Views

Author

Paolo P. Lava, Sep 18 2013

Keywords

Examples

			Second term of A065061 is 8 and sigma(8) - tau(8) = 15 - 4 = 11 is prime.

Links

Crossrefs

Cf. A000005, A000010, A000203, A009087, A023194, A038344, A055813, A062700, A064205, A065608, A141242, A229264, A229266

Programs

  • Maple
    with(numtheory); P:=proc(q) local a,n; a:= sigma(n)-tau(n); for n from 1 to q do
if isprime(a) then print(a); fi; od; end: P(10^6);
  • Mathematica
    Join[{2}, Select[(DivisorSigma[1, #] - DivisorSigma[0, #]) & /@ (2*Range[20000]^2), PrimeQ]] (* Amiram Eldar, Dec 06 2022 *)

Formula

a(n) = A000203(A065061(n)) - A000005(A065061(n)). - Michel Marcus, Sep 21 2013
a(n) = A065608(A065061(n)). - Amiram Eldar, Dec 06 2022

Extensions

More terms from Michel Marcus, Sep 21 2013

A229266 Primes of the form sigma(k) + tau(k) + phi(k), where sigma(k) = A000203(k), tau(k) = A000005(k) and phi(k) = A000010(k).

Original entry on oeis.org

3, 23, 557, 1289, 2447, 3779, 9209, 10331, 11351, 18367, 14051, 34351, 42953, 67883, 95717, 96587, 134807, 164249, 193057, 310553, 253159, 321397, 383723, 548213, 657311, 499151, 630023, 516251, 732181, 713927, 927013, 932431, 784627, 906473, 855331, 1121987
Offset: 1

Views

Author

Paolo P. Lava, Sep 18 2013

Keywords

Examples

			The third term of A229265 is 200 and sigma(200) +  tau(200) + phi(200) = 465 + 12 + 80 = 557 is prime.

Crossrefs

Cf. A000005, A000010, A000203, A009087, A023194, A038344, A055813, A062700, A064205, A115919, A141242, A229264, A229265, A065061, A229268.

Programs

  • Maple
    with(numtheory); P:=proc(q) local a, n; for n from 1 to q do a:=sigma(n)+tau(n)+phi(n);
if isprime(a) then print(a); fi; od; end: P(10^6);
  • Mathematica
    Select[Table[DivisorSigma[0,n]+DivisorSigma[1,n]+EulerPhi[n],{n,10^6}],PrimeQ] (* Harvey P. Dale, Oct 03 2023 *)

A229265 Numbers k such that sigma(k) + tau(k) + phi(k) is a prime, where sigma(k) = A000203(k), tau(k) = A000005(k) and phi(k) = A000010(k).

Original entry on oeis.org

1, 8, 200, 512, 968, 1458, 3200, 4232, 5618, 5832, 6962, 10368, 16928, 26912, 36992, 40328, 53792, 61952, 84050, 101250, 110450, 140450, 147968, 220448, 247808, 249218, 253472, 257762, 279752, 282752, 320000, 336200, 344450, 359552, 361250, 445568, 472392, 512072
Offset: 1

Views

Author

Paolo P. Lava, Sep 18 2013

Keywords

Examples

			sigma(200) = 465, tau(200) = 12, phi(200) = 80 and 465 + 12 + 80 = 557 is prime.

Crossrefs

Cf. A000005, A000010, A000203, A009087, A023194, A038344, A055813, A062700, A064205, A115919, A141242, A229264, A229266-A229268.

Programs

  • Maple
    with(numtheory); P:=proc(q) local n; for n from 1 to q do
if isprime(sigma(n)+tau(n)+phi(n)) then print(n); fi; od; end: P(10^6);
Showing 1-4 of 4 results.

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