A277127 -id:A277127 - cp's OEIS frontend

A278021 Numbers n such that n - lambda(n) is prime, where lambda = A002322.

4, 9, 15, 25, 33, 35, 49, 65, 69, 77, 87, 91, 95, 115, 119, 121, 123, 143, 159, 169, 185, 187, 215, 221, 247, 249, 255, 259, 267, 287, 289, 295, 299, 319, 323, 329, 339, 341, 361, 365, 377, 393, 395, 407, 413, 415, 427, 437, 455, 473, 485, 511, 515, 519

Offset: 1

Robert Israel, Nov 08 2016

nonn

All terms are composite.
4 is the only even term.
For odd primes p, 3*p is a term iff p is in A005384.

			25-lambda(25) = 25-20 = 5 is prime so 25 is in the sequence.

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

Cf. A002322, A005384, A010051, A277127.

Contains A001248. Contained in union of A001248 and A024556.

select(t -> isprime(t - numtheory:-lambda(t)), [$1..10000]);

A010051(n - A002322(n)) = 1.

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

A277312 Smallest k such that k - lambda(k) = prime(n), where lambda(k) = A002322(k).

Original entry on oeis.org

4, 9, 25, 49, 15, 169, 289, 361, 33, 841, 961, 1369, 1681, 1849, 69, 65, 87, 3721, 4489, 115, 5329, 91, 123, 7921, 9409, 10201, 10609, 159, 11881, 12769, 16129, 215, 18769, 19321, 185, 22801, 24649, 26569, 249, 221, 267, 32761, 329, 37249, 38809, 39601, 247, 259, 339, 52441

Offset: 1

Thomas Ordowski, Oct 09 2016

nonn

a(n) is the smallest k such that A277127(k) = A000040(n).
a(n) <= prime(n)^2, because p^2 - lambda(p^2) = p prime.
Conjecture: a(n) = prime(n)^2 for infinitely many n.
For n > 1, a(n) is an odd composite. - Robert Israel, Oct 14 2016

Robert Israel, Table of n, a(n) for n = 1..446

Cf. A000040, A002322, A053194, A277127, A278021.

N:= 100: # to get a(1)..a(N)
A:= Vector(N):
A[1]:= 4:
count:= 1:
for k from 9 by 2 while count < N do
  r:= k - numtheory:-lambda(k);
  if isprime(r) then
    n:= numtheory:-pi(r);
    if n <= N and A[n] = 0 then
      count:= count+1;
      A[n]:= k;
    fi
   fi
od:
convert(A,list); # Robert Israel, Oct 14 2016

Table[k = 1; While[k - CarmichaelLambda@ k != Prime@ n, k++]; k, {n, 50}] (* Michael De Vlieger, Oct 14 2016 *)

a(n) = {my(k = 1); while (k - lcm(znstar(k)[2]) != prime(n), k++); k;} \\ Michel Marcus, Oct 09 2016

More terms from Altug Alkan, Oct 09 2016

A290281 Numbers k such that (k-1) mod phi(k) = lambda(k), where phi = A000010 and lambda = A002322.

Original entry on oeis.org

6601, 11972017, 34657141, 67902031, 139952671, 258634741, 2000436751, 8801128801, 9116583841, 9462932431, 38069223721, 326170416001, 359316634951, 1860929324101, 2022188518351, 2283475947391, 2648686458601, 2697891108151, 4513362899761, 5020030521001, 5472940991761, 6163867710001, 7507903975951, 19288340548471

Offset: 1

Robert Israel and Thomas Ordowski, Jul 25 2017

nonn

Numbers k such that A215486(k) = A002322(k).
Subsequence of the Carmichael numbers (A002997).
Composite numbers k such that (k-1) == lambda(k) (mod phi(k)).
Composite numbers k such that A277127(k) == 1 (mod A000010(k)).
Problem: are there infinitely many such numbers?
Conjecture: these are numbers k such that phi(k) + lambda(k) = k - 1. Checked up to 2^64. - Amiram Eldar and Thomas Ordowski, Dec 06 2019

Amiram Eldar, Table of n, a(n) for n = 1..239 (terms below 10^22 calculated using data from Claude Goutier; terms 1..79 from Robert Israel)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Index entries for sequences related to Carmichael numbers.

Cf. A000010, A002322, A002997, A215486.

Subsequence of A264012.

# Using data files for A002997
count:= 0:
for cfile in ["carmichael-16","carmichael17","carmichael18"] do
do
    S:= readline(cfile);
    if S = 0 then break fi;
    L:= map(parse, StringTools:-Split(S));
    n:= L[1]; pm:= map(`-`,L[2..-1],1);
    phin:= convert(pm,`*`);
    lambdan:= ilcm(op(pm));
    if n-1 - lambdan mod phin = 0 then
      count:= count+1; A[count]:= n;
    fi
od:
   fclose(cfile);
od:
seq(A[i],i=1..count); # Robert Israel, Jul 26 2017

Select[Range[10^8], Divisible[# - 1, (lam = CarmichaelLambda[#])] && Mod[# - 1, EulerPhi[#]] == lam &] (* Amiram Eldar, Dec 06 2019 *)

A276674 Numbers n such that x - lambda(x) = n has no solution, where lambda(x) = A002322(x).

Original entry on oeis.org

21, 28, 45, 46, 51, 64, 65, 77, 82, 85, 91, 106, 111, 126, 129, 133, 136, 148, 155, 166, 172, 175, 185, 189, 205, 208, 209, 217, 221, 225, 231, 232, 235, 237, 244, 247, 267, 273, 274, 276, 286, 291, 298, 305, 316, 319, 326, 333, 339, 341, 358, 362, 364, 365, 371

Offset: 1

Thomas Ordowski, Oct 03 2016

nonn

Problem: are there infinitely many such numbers?
Note that all these numbers are composite, because p - lambda(p) = 1 and p^2 - lambda(p^2) = p prime.
If x - lambda(x) = n > 1, then x <= n^2.
Conjecture: if x - lambda(x) = 2*m > 0, then x <= 4*m.
Noncototients among these numbers are 172, 232, 244, 274, 298, 326, 362, ...

Cf. A002322, A005278 (see links). Complement of A277127.

lista(nn) = {v = vecsort(vector(nn^2, n, n - lcm(znstar(n)[2])), ,8); for (n=1, nn, if (! vecsearch(v, n), print1(n, ", ")););} \\ Michel Marcus, Oct 03 2016

use ntheory ":all"; sub A { my $l=shift; my %C; undef $C{$-carmichael_lambda($)} for 1..$l*$l; my @R = grep { !exists $C{$} } 1..$l; @R; } say for A(500); # _Dana Jacobsen, Apr 27 2017

More terms from Michel Marcus, Oct 03 2016

A330446 Composite numbers k such that 2^(k-1) == - lambda(k) (mod k), where lambda is the Carmichael lambda function (A002322).

Original entry on oeis.org

140, 1054, 1068, 4844, 11209, 19856, 24949, 28390, 78184, 423796, 769516, 4283544, 5935168, 13116053, 122189752, 441252296, 528500308, 636697392, 669629030, 669778082, 1228748591

Offset: 1

Amiram Eldar and Thomas Ordowski, Dec 15 2019

Composite numbers k such that A062173(k) = A277127(k).
The odd terms are 11209, 24949, 13116053, ...
Note that if p is an odd prime, then 2^(p-1) == - lambda(p) (mod p), because lambda(p) = p-1.

			140 is a term since it is composite and 2^(140-1) == 140 - lambda(140) == 128 (mod 140).

Cf. A002322, A062173, A277127.

Select[Range[10^6], CompositeQ[#] && PowerMod[2, # - 1, #] == # - CarmichaelLambda[#] &]

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A278021 Numbers n such that n - lambda(n) is prime, where lambda = A002322.

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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).

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A277312 Smallest k such that k - lambda(k) = prime(n), where lambda(k) = A002322(k).

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A290281 Numbers k such that (k-1) mod phi(k) = lambda(k), where phi = A000010 and lambda = A002322.

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A276674 Numbers n such that x - lambda(x) = n has no solution, where lambda(x) = A002322(x).

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A330446 Composite numbers k such that 2^(k-1) == - lambda(k) (mod k), where lambda is the Carmichael lambda function (A002322).

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