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 1-10 of 10 results.

A055813 Primes arising in A064205.

Original entry on oeis.org

2, 5, 19, 263, 373, 1033, 65551, 123833, 283553, 292993, 2197633, 5778673, 12795193, 10500613, 19980283, 32213033, 36978853, 72494813, 140724953, 164119063, 183105509, 193710277, 320210941, 450065801, 268354393, 496922891, 506906473, 1333406021
Offset: 1

Views

Author

Labos Elemer, Nov 22 2001

Keywords

Examples

			a(3) = 19 since, for A064205(3) = 8, sigma(8)+tau(8) = 15+4 = 19, a prime number. - _Michael De Vlieger_, Dec 17 2016

Links

Crossrefs

Cf. A007503, A064205.

Programs

  • Mathematica
    Select[Table[Total@ Map[DivisorSigma[#, n] &, {0, 1}], {n, 10^7}], PrimeQ] (* Michael De Vlieger, Dec 17 2016 *)
  • PARI
    lista(nn) = for (n=1, nn, if (isprime(p=sigma(n) + numdiv(n)), print1(p, ", "))); \\ Michel Marcus, Dec 18 2016

Formula

a(n) = A007503(A064205(n)). - Amiram Eldar, Dec 06 2022

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

A229264 Primes in A065387 in the order of their appearance.

Original entry on oeis.org

2, 19, 19, 79, 103, 113, 257, 523, 509, 1151, 1279, 1193, 1579, 2273, 3061, 2389, 2693, 2843, 5003, 4831, 5119, 7411, 5693, 5623, 8623, 6323, 10139, 8933, 18401, 14957, 20411, 20479, 21191, 20123, 29683, 28211, 36833, 55021, 57203, 68743, 48761, 66533, 62423
Offset: 1

Views

Author

Paolo P. Lava, Sep 18 2013

Keywords

Examples

			Third term of A038344 is 9 and sigma(9) + phi(9) = 13 + 6 = 19 is prime.

Links

Crossrefs

Cf. A000005, A000010, A000203, A009087, A023194, A038344, A055813, A062700, A064205, A065387, A115919, A141242, A229265, A229266, A229267, A229268.

Programs

  • Maple
    with(numtheory); P:=proc(q) local a, n; for n from 1 to q do a:=sigma(n)+phi(n);
if isprime(a) then print(a); fi; od; end: P(10^6);
  • Mathematica
    Select[Table[DivisorSigma[1,n]+EulerPhi[n],{n,30000}],PrimeQ] (* Harvey P. Dale, Apr 30 2018 *)
  • PARI
    lista(kmax) = {my(f, s); for(k = 1, kmax, f = factor(k); s= sigma(f) + eulerphi(f); if(isprime(s), print1(s, ", ")));} \\ Amiram Eldar, Nov 19 2024

Extensions

Name corrected by Amiram Eldar, Nov 19 2024

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);

A272060 Numbers k such that sigma((k-1)/2) + tau((k-1)/2) is prime.

Original entry on oeis.org

3, 5, 17, 257, 325, 1025, 65537, 82945, 202501, 250001, 2829125, 7496645, 10240001, 13675205, 16000001, 27060805, 48469445, 71402501, 133448705, 150062501, 156250001, 172186885, 182250001, 343064485, 354117125, 453519617, 467943425, 1235663105
Offset: 1

Views

Author

Jaroslav Krizek, Apr 19 2016

Keywords

Comments

Numbers k such that A000203((k-1)/2) + A000005((k-1)/2) is a prime q.
Corresponding values of primes q are in A055813.
Prime terms are in A272061.
The first 5 known Fermat primes from A019434 are in this sequence.

Examples

			sigma((17-1)/2) + tau((17-1)/2) = sigma(8) + tau(8) = 15 + 4 = 19; 19 is prime, so 17 is in the sequence.

Links

Crossrefs

Cf. A000005, A000203, A007503, A019434, A055813, A064205, A272061.

Programs

  • Magma
    [n: n in [3..1000000] | IsPrime(NumberOfDivisors((n-1) div 2) + SumOfDivisors((n-1) div 2)) and (n-1) mod 2 eq 0];
  • Mathematica
    Select[Range[3, 10^7, 2], PrimeQ[DivisorSigma[1, #] + DivisorSigma[0, #]] &[(# - 1)/2] &] (* Michael De Vlieger, Apr 20 2016 *)
  • PARI
    isok(n) = isprime(sigma((n-1)/2) + numdiv((n-1)/2));
lista(nn) = forstep (n=3, nn, 2, if (isok(n), print1(n, ", "))); \\ Michel Marcus, Apr 19 2016
  • PARI
    is(n)=my(f=factor(n\2)); n>2 && isprime(sigma(f)+numdiv(f)) && isprime(n) \\ Charles R Greathouse IV, Apr 29 2016

Formula

a(n) = 2*A064205(n) + 1.

A272061 Primes p such that sigma((p-1)/2) + tau((p-1)/2) is prime.

Original entry on oeis.org

3, 5, 17, 257, 65537, 453519617, 1372257937, 1927561217, 21320672257, 76001667857, 138388464037, 1216026685697, 2085136000001, 8503056000001, 30118144000001, 35427446793217, 37015056000001, 83037656250001, 87329473560577, 97850397828097, 222330465562501, 233952748524197
Offset: 1

Views

Author

Jaroslav Krizek, Apr 19 2016

Keywords

Comments

Primes p such that A007503((p-1)/2) is a prime q.
Corresponding values of primes q: 2, 5, 19, 263, 65551, 496922891, ...
Prime terms from A272060.
The first 5 known Fermat primes from A019434 are in this sequence.
Primes of the form 2*m+1 with m a term of A064205. - Michel Marcus, Apr 25 2016

Examples

			sigma((17-1)/2) + tau((17-1)/2) = sigma(8) + tau(8) = 15 + 4 = 19; 19 is prime, so 17 is in the sequence.

Links

Crossrefs

Cf. A000005, A000203, A007503, A055813, A064205, A272060.

Programs

  • Magma
    [n: n in [3..1000000] | IsPrime(n) and IsPrime(NumberOfDivisors((n-1) div 2) + SumOfDivisors((n-1) div 2)) and (n-1) mod 2 eq 0];
  • Maple
    with(numtheory): A272061:=n->`if`(isprime(n) and isprime(sigma((n-1)/2)+tau((n-1)/2)), n, NULL): seq(A272061(n), n=3..10^5); # Wesley Ivan Hurt, Apr 20 2016
  • Mathematica
    Select[Prime[Range[10000]],PrimeQ[DivisorSigma[1,(#-1)/2] + DivisorSigma[0,(#-1)/2]] & ] (* Robert Price, Apr 21 2016 *)
  • PARI
    isok(n) = isprime(sigma((n-1)/2) + numdiv((n-1)/2));
lista(nn) = forprime (p=3, nn, if (isok(p), print1(p, ", "))); \\ Michel Marcus, Apr 19 2016
  • PARI
    is(n)=my(f=factor(n\2)); isprime(sigma(f)+numdiv(f)) && isprime(n) \\ Charles R Greathouse IV, Apr 28 2016

Extensions

a(7)-a(8) from Michel Marcus, Apr 24 2016
a(9) from Charles R Greathouse IV, Apr 29 2016
a(10) from Charles R Greathouse IV, Apr 29 2016
a(11)-a(20), using A064205 bfile, added by Michel Marcus, Nov 23 2022
a(21)-a(22) from Amiram Eldar, Dec 06 2022

A065116 Numbers k such that sigma(k) + tau(k) and sigma(k) - tau(k) are primes.

Original entry on oeis.org

8, 162, 512, 32768, 41472, 3748322, 5120000, 6837602, 8000000, 35701250, 75031250, 78125000, 91125000, 907039232, 10660336128, 11911961250, 21234895362, 41265473762, 55965865922, 209642370242, 835707290112, 1148179179938, 1821331173888, 2097152000000
Offset: 1

Views

Author

Robert G. Wilson v, Nov 12 2001

Keywords

Comments

Intersection of A064205 and A065061.
From Kevin P. Thompson, Jun 20 2022: (Start)
Terms 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) = 363 and tau(162) = 10 are numbers whose sum (373) and difference (353) are both primes.

Links

Crossrefs

Cf. A064205, A065061.

Programs

  • Mathematica
    Do[ds1 = DivisorSigma[1, n]; ds0 = DivisorSigma[0, n]; If[ PrimeQ[ds1 + ds0] && PrimeQ[ds1 - ds0], Print[n]], {n, 1, 10^7} ]

Extensions

a(10)-a(19) from Donovan Johnson, Jul 09 2010
a(20)-a(24) from Donovan Johnson, Aug 23 2013

A358342 Lesser of twin primes p such that sigma((p-1)/2) + tau((p-1)/2) is a prime.

Original entry on oeis.org

3, 5, 17, 65537, 1927561217, 6015902625062501, 12370388895062501, 835920078368222501, 6448645485213008897, 50973659693056000001, 54332889713542767617, 64304984013657011717, 112112769248058062501, 147337258721536000001
Offset: 1

Views

Author

Jaroslav Krizek, Nov 10 2022

Keywords

Comments

Lesser of twin primes p such that A000203((p-1)/2) + A000005((p-1)/2) is a prime q.
The first 4 terms are Fermat primes from A019434.
Corresponding values of primes q: 2, 5, 19, 65551, 2248681529, ...
Subsequence of A272060 and A272061.
Lesser of twin primes of the form 2*m+1 with m a term of A064205.
There are no other terms <= 10^14.
All the terms above 3 are in A145824. - Amiram Eldar, Jan 05 2023

Examples

			17 and 19 are twin primes; sigma((17-1)/2) + tau((17-1)/2) = sigma(8) + tau(8) = 15 + 4 = 19; 19 is prime, so 17 is in the sequence.

Crossrefs

Intersection of A001359 and A272061.
Cf. A000005 (tau), A000203 (sigma), A019434, A064205, A145824, A272060.

Programs

  • Magma
    [n: n in [3..10^7] | IsPrime(n) and IsPrime(n+2) and IsPrime(&+Divisors((n-1) div 2) + #Divisors((n-1) div 2))]
  • Mathematica
    Join[{3}, Select[4*Range[25000]^2 + 1, PrimeQ[#] && PrimeQ[# + 2] && PrimeQ[DivisorSigma[1, (# - 1)/2] + DivisorSigma[0, (# - 1)/2]] &]]
(* or *)
A272061 = Cases[Import["https://oeis.org/A272061/b272061.txt", "Table"], {, }][[;; , 2]]; Select[A272061, PrimeQ[# + 2] &] (* Amiram Eldar, Jan 05 2023 *)
  • PARI
    isok(p) = if (isprime(p) && isprime(p+2), my(f=factor((p-1)/2)); isprime(sigma(f)+numdiv(f))); \\ Michel Marcus, Nov 23 2022
Showing 1-10 of 10 results.

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