A050530 -id:A050530 - cp's OEIS frontend

A053339 Squarefree terms of A050530 with 3 prime divisors.

255, 435, 455, 561, 595, 665, 705, 795, 805, 885, 957, 1001, 1105, 1295, 1309, 1335, 1463, 1495, 1551, 1605, 1615, 1645, 1729, 1749, 1855, 1885, 1947, 1955, 2001, 2055, 2065, 2091, 2093, 2185, 2235, 2345, 2387, 2405, 2465, 2555, 2703, 2717, 2755, 2821

Offset: 1

Labos Elemer, Jan 05 2000

nonn

			435 = 3*5*29 and 435 - Phi(435) = 3*5 + 3*29 + 5*29 - 3 - 5 - 29 + 1 = 211, the 47th prime. [corrected by _Jon E. Schoenfield_, May 30 2018]

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A051953, A050530.

Select[Select[Range[3000],PrimeQ[#-EulerPhi[#]]&],SquareFreeQ[3] && PrimeOmega[#]==3&] (* Harvey P. Dale, Jun 23 2013 *)

isok(n) = isprime(n-eulerphi(n)) && issquarefree(n) && (omega(n)==3); \\ Michel Marcus, May 31 2018

Numbers k = pqr such that A051953(k) = k - EulerPhi(k) is a prime of polynomial form pq + pr + qr - p - q - r + 1.

A053340 Terms of A050530 with four prime divisors.

Original entry on oeis.org

5865, 8645, 10005, 10465, 13685, 15045, 15295, 16269, 18285, 20445, 22015, 24871, 26845, 27965, 28405, 28815, 29733, 30705, 31031, 31255, 33215, 35245, 36105, 37037, 37145, 37365, 37765, 37995, 38985, 39831, 40579, 41041, 43435, 44135

Offset: 1

Labos Elemer, Jan 05 2000

nonn

			The 3rd entry is 10005=3*5*23*29, also in A050530 and A051953(10005)=5077 is a prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A051953, A050530.

n=pqrs, having 4 prime divisors, squarefree and n-phi(n) is a prime

A068080 Integers n such that n + phi(n) is a prime.

Original entry on oeis.org

1, 2, 3, 7, 15, 19, 31, 33, 35, 37, 51, 65, 69, 77, 79, 85, 91, 95, 97, 133, 139, 141, 143, 145, 157, 159, 161, 177, 187, 199, 209, 211, 213, 215, 217, 229, 235, 247, 255, 267, 271, 299, 303, 307, 319, 331, 335, 337, 339, 341, 345, 365, 367, 371, 379, 391, 393

Offset: 1

Amarnath Murthy, Feb 17 2002

The subsequence of prime terms is given by A005382. - Michel Marcus, Aug 22 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A050530.

[n: n in [1..400] |IsPrime(n+EulerPhi(n))]; // Vincenzo Librandi, Dec 19 2015

Select[Range[400], PrimeQ[# + EulerPhi[#]] &] (* Carl Najafi, Aug 22 2011 *)

isok(n) = isprime(n+eulerphi(n)); \\ Michel Marcus, Aug 22 2015

Edited and extended by Robert G. Wilson v, Feb 18 2002

A121048 a(n) = n + phi(n), where phi is the Euler totient function.

Original entry on oeis.org

2, 3, 5, 6, 9, 8, 13, 12, 15, 14, 21, 16, 25, 20, 23, 24, 33, 24, 37, 28, 33, 32, 45, 32, 45, 38, 45, 40, 57, 38, 61, 48, 53, 50, 59, 48, 73, 56, 63, 56, 81, 54, 85, 64, 69, 68, 93, 64, 91, 70, 83, 76, 105, 72, 95, 80, 93, 86, 117, 76, 121, 92, 99, 96, 113, 86, 133, 100, 113

Offset: 1

Jonathan Vos Post, Aug 08 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000010, A000040, A050530, A051953, A059956, A068080.

[n + EulerPhi(n): n in [1..70]]; // Vincenzo Librandi, Jul 13 2012

Mathematica
```
Table[n+EulerPhi[n],{n,1,70}]
```

a(n) = n+eulerphi(n); \\ Michel Marcus, Sep 19 2022

a(n) = n + phi(n) = n + A000010(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 + 1/zeta(2) = 1.607927... . - Amiram Eldar, Dec 16 2023

A290089 Filter-sequence for the prime signature of cototient: a(1) = 0; for n > 1, a(n) = A101296(A051953(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 5, 1, 5, 2, 5, 1, 6, 1, 6, 3, 6, 1, 7, 2, 4, 3, 7, 1, 4, 1, 7, 2, 6, 2, 8, 1, 6, 4, 8, 1, 9, 1, 8, 4, 8, 1, 10, 2, 9, 2, 6, 1, 11, 4, 10, 4, 9, 1, 6, 1, 10, 5, 10, 2, 4, 1, 11, 3, 4, 1, 12, 1, 4, 4, 8, 2, 8, 1, 12, 5, 9, 1, 13, 4, 6, 2, 12, 1, 9, 2, 12, 4, 12, 2, 14, 1, 8, 4, 13, 1, 9, 1, 8, 4

Offset: 1

Antti Karttunen, Aug 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A051953, A101296, A277906, A290087.

Cf. A000040 (the positions of 1's), A050530 (the positions of 2's).

a(1) = 0; for n > 1, a(n) = A101296(A051953(n)).

A051961 Smallest number w such that A051953(w) = w - phi(w) is the n-th prime.

Original entry on oeis.org

4, 9, 25, 15, 35, 33, 65, 51, 95, 161, 87, 217, 185, 123, 215, 329, 371, 177, 427, 335, 213, 511, 395, 581, 1501, 485, 303, 515, 321, 545, 255, 635, 917, 411, 1529, 447, 1057, 1099, 455, 1169, 1211, 537, 1991, 573, 965, 591, 435, 2743, 1115, 681, 665

Offset: 1

Labos Elemer, Jan 05 2000

nonn

			The 31st term is 255 since 255 - phi(255) = 127, the 31st prime, and no number less than 255 has this property.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A050530, A051953.

With[{c=Table[n-EulerPhi[n],{n,4000}]},Table[Position[c,p,1,1],{p,Prime[ Range[ 60]]}]]//Flatten (* Harvey P. Dale, Sep 14 2020 *)

a(n) = {my(k = 1); while(k - eulerphi(k) != prime(n), k++); k;} \\ Michel Marcus, Feb 02 2015

A050530(a(n)) = prime(n) and a(n) is the least number with this property.

a(n) = A063507(A000040(n)). - Michel Marcus, Feb 02 2015

A053343 Semiprimes of the form pq where p < q and p + q - 1 is prime.

Original entry on oeis.org

15, 33, 35, 51, 65, 77, 87, 91, 95, 119, 123, 143, 161, 177, 185, 209, 213, 215, 217, 221, 247, 259, 287, 303, 321, 329, 335, 341, 371, 377, 395, 403, 407, 411, 427, 437, 447, 469, 473, 485, 511, 515, 527, 533, 537, 545, 551, 573, 581, 591, 611, 629, 635

Offset: 1

Labos Elemer, Jan 05 2000

Squarefree terms of A050530 with 2 prime divisors.

All terms are odd. - Muniru A Asiru, Aug 29 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Hacène Belbachir, Oussama Igueroufa, Combinatorial interpretation of bisnomial coefficients and Generalized Catalan numbers, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 47-54.

Subsequence of A291318.

Cf. A050530, A290434, A290435.

A053343:=List(Filtered(Filtered(List(Filtered(List([1..10^5],Factors),i->Length(i)=2),Set),j->Length(j)=2),i->IsPrime(Sum(i)-1)),Product); # Muniru A Asiru, Aug 29 2017

With[{nn=70}, Take[Times@@@Select[Subsets[Prime[Range[nn]], {2}], PrimeQ[Total[#] - 1] &]//Union, nn]] (* Vincenzo Librandi, Aug 23 2017 *)

list(lim)=my(v=List()); forprime(p=5,lim\3, forprime(q=3,min(lim\p,p-2), if(isprime(p+q-1), listput(v,p*q)))); Set(v) \\ Charles R Greathouse IV, Aug 23 2017

n=pq such that n-phi(n) = pq-(p-1)(q-1) = p+q-1 is prime.

New name from Vincenzo Librandi Aug 23 2017

A068081 Numbers n such that n + phi(n) and n - phi(n) are prime.

Original entry on oeis.org

15, 33, 35, 51, 65, 77, 91, 95, 143, 161, 177, 209, 213, 215, 217, 247, 255, 303, 335, 341, 371, 411, 427, 435, 447, 455, 533, 545, 561, 573, 591, 611, 665, 707, 713, 717, 779, 803, 871, 917, 933, 965, 1001, 1041, 1067, 1105, 1115, 1133, 1157, 1159, 1211

Offset: 1

Amarnath Murthy, Feb 17 2002

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A050530, A068080.

A068081:=[];; for n in [1,3..10^4+1] do if IsPrime(n+Phi(n)) and IsPrime(n-Phi(n)) then Add(A068081,n); fi; od; A068081;  # Muniru A Asiru, Aug 31 2017

with(numtheory): for n from 1 by 2 to 10^4 do if [isprime(n+phi(n)),
isprime(n-phi(n))]=[true,true] then print(n); fi; od; # Muniru A Asiru, Aug 31 2017

Select[ Range[1500], PrimeQ[ # + EulerPhi[ # ]] && PrimeQ[ # - EulerPhi[ # ]] & ]
epQ[n_]:=Module[{ep=EulerPhi[n]},AllTrue[n+{ep,-ep},PrimeQ]]; Select[ Range[ 1500],epQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 22 2016 *)

is(n)=my(t=eulerphi(n)); isprime(n-t) && isprime(n+t) \\ Charles R Greathouse IV, Jan 25 2017

Edited and extended by Robert G. Wilson v, Feb 18 2002

A277254 Numbers k such that p = k - phi(k) < q = k - lambda(k), and p and q are both primes, where phi(k) = A000010(k) and lambda(k) = A002322(k).

Original entry on oeis.org

15, 33, 35, 65, 77, 87, 91, 95, 119, 123, 143, 185, 215, 221, 247, 255, 259, 287, 329, 341, 377, 395, 407, 427, 437, 455, 473, 485, 511, 515, 537, 573, 595, 635, 705, 713, 717, 721, 749, 767, 779, 793, 795, 803, 805, 815, 817, 843, 869, 871, 885, 899, 923, 965, 1001

Offset: 1

Thomas Ordowski, Oct 07 2016

nonn

Numbers k such that p = A051953(k) < q = A277127(k), and p and q are both primes.
If k is such number, then b^p == b^q (mod k) for every integer b.
Problem: are there infinitely many such numbers?
Suppose p^2 divides k. Then p divides k - phi(k), and so the only way k - phi(k) can be prime is if k = p^2. But then k - phi(k) = k - A002322(k). Hence all terms in this sequence are squarefree. - Charles R Greathouse IV, Oct 08 2016
All terms are odd composites. - Robert Israel, Oct 09 2016
It seems that gpf(k) < p = k - phi(k). - Thomas Ordowski, Oct 09 2016

			For n=15, A051953(15) = 7, A277127(15) = 11, 7 < 11 and both are primes, thus 15 is included in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Subsequence of A033949 and of A024556.

Cf. A000010, A002322, A050530, A051953, A277127.

filter:= proc(n) uses numtheory;
  local p,q;
  p:= n-phi(n);
  q:= n-lambda(n);
  pRobert Israel, Oct 09 2016

Select[Range[10^3], And[#1 < #2, Times @@ Boole@ PrimeQ@ {#1, #2} == 1] & @@ {# - EulerPhi@ #, # - CarmichaelLambda@ #} &] (* Michael De Vlieger, Oct 08 2016 *)

is(n)=my(f=factor(n),p=n-eulerphi(f),q=n-lcm(znstar(f)[2])); p < q && isprime(p) && isprime(q) \\ Charles R Greathouse IV, Oct 08 2016

More terms from Altug Alkan, Oct 07 2016

A051999 Minimal value w such that A051953(w) = w - phi(w) is prime and w has n prime divisors.

Original entry on oeis.org

4, 15, 255, 5865, 170085, 5437705, 226473065, 10380578845, 494390700895, 43592479037107

Offset: 1

Labos Elemer, Jan 05 2000

a(11) <= 2995513256722805. - Donovan Johnson, Feb 06 2010

			a(1)=2^2, a(2)=3*5, a(3)=3*5*17, a(4)=3*5*17*23, a(5)=3*5*17*23*29, a(6)=5437705=5*7*13*17*19*37 with 1,2,3,4,5,6 prime divisors, respectively. The generated primes of form w - phi(w) are as follows: 2, 7, 127, 3049, 91237, 2452721.

Cf. A050530, A051953.

a(7)-a(10) from Donovan Johnson, Feb 06 2010

cp's OEIS Frontend

A053339 Squarefree terms of A050530 with 3 prime divisors.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A053340 Terms of A050530 with four prime divisors.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A068080 Integers n such that n + phi(n) is a prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A121048 a(n) = n + phi(n), where phi is the Euler totient function.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A290089 Filter-sequence for the prime signature of cototient: a(1) = 0; for n > 1, a(n) = A101296(A051953(n)).

Views

Author

Keywords

Links

Crossrefs

Formula

A051961 Smallest number w such that A051953(w) = w - phi(w) is the n-th prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A053343 Semiprimes of the form pq where p < q and p + q - 1 is prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Mathematica

PARI

Formula

Extensions

A068081 Numbers n such that n + phi(n) and n - phi(n) are prime.

Views

Author

Keywords

Links

Crossrefs