A024556 -id:A024556 - cp's OEIS frontend

A231372 Squarefree composite numbers k such that 10 is a primitive root for all prime factors of k.

119, 133, 161, 203, 323, 329, 391, 413, 427, 437, 493, 551, 667, 679, 763, 791, 799, 893, 917, 1003, 1037, 1043, 1081, 1121, 1159, 1169, 1253, 1267, 1351, 1357, 1363, 1403, 1561, 1603, 1631, 1649, 1711, 1769, 1799, 1841, 1843, 1853, 1883, 1921, 2071, 2147, 2191

Offset: 1

Arkadiusz Wesolowski, Nov 08 2013

nonn

If k is the smallest integer satisfying 10^k == 1 (mod p), we say that 10 has order k (mod p). If n is the product of distinct primes p_i, the period of 1/n in base b is the least common multiple of the orders of b (mod p_i), provided b and n are relatively prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Primitive Root.
Wikipedia, Decimal.

Subsequence of A024556.

Cf. A001913, A231370, A231371.

q[n_] := CompositeQ[n] && SquareFreeQ[n] && AllTrue[FactorInteger[n][[;;,1]],  MultiplicativeOrder[10, #] == # - 1 &]; Select[Range[2000], q] (* Amiram Eldar, Oct 03 2021 *)

A124569 Numbers k such that k, k+2, k+4 and k+6 are squarefree.

Original entry on oeis.org

1, 11, 13, 15, 17, 29, 31, 33, 35, 37, 51, 53, 55, 65, 67, 83, 85, 87, 89, 91, 101, 103, 105, 107, 109, 127, 137, 139, 155, 157, 159, 161, 177, 179, 181, 191, 193, 195, 197, 199, 209, 211, 213, 215, 217, 227, 229, 231, 233, 235, 247, 249, 251, 253, 263, 265, 267

Offset: 1

Zak Seidov, Dec 27 2006

nonn

Numbers k, k+2, k+4 and k+6 are four successive odd squarefree numbers.

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

Cf. A007675, A024556.

filter:= proc(n) andmap(numtheory:-issqrfree,[n,n+2,n+4,n+6]) end proc:
select(filter, [seq(i,i=1..1000,2)]); # Robert Israel, Jun 08 2021

okQ[n_] := AllTrue[n + {0, 2, 4, 6}, SquareFreeQ];
Select[Range[1, 1000, 2], okQ] (* Jean-François Alcover, May 18 2023 *)

isok(n) = issquarefree(n) && issquarefree(n+2) && issquarefree(n+4) && issquarefree(n+6); \\ Michel Marcus, Oct 11 2013

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

Original entry on oeis.org

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.

A055456 a(n) is the smallest number which is not the sum of exactly 1 or of n earlier terms.

Original entry on oeis.org

1, 3, 2, 13, 4, 5, 6, 7, 8, 9, 10, 11, 12, 183, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Henry Bottomley, May 19 2000

			a(3)=2 because 1 is already in the sequence, 2 has not yet appeared (i.e., is not the sum of 1 earlier term), and the sum of 3 earlier terms is 3, 5 or 7.
a(4)=13 because 1, 2, and 3 have already appeared and the sum of 4 earlier terms could be any integer from 4 through 12.

Cf. A035334. a(n) is not n-1 iff n-1 is in A024556 or equivalently in A002065.

If n-1 > 0 has not already appeared in sequence then a(n) = n-1, otherwise a(n) = n^2 - n + 1.

A068992 Composite numbers k such that Sum_{d|k} sigma(d)/tau(d) is an integer.

Original entry on oeis.org

10, 15, 21, 26, 30, 33, 34, 35, 39, 49, 51, 55, 57, 58, 60, 65, 69, 70, 74, 75, 77, 78, 82, 85, 87, 91, 93, 95, 98, 102, 105, 106, 110, 111, 115, 119, 120, 122, 123, 129, 130, 133, 141, 143, 145, 146, 147, 155, 156, 159, 161, 165, 169, 170, 174, 177, 178, 182, 183

Offset: 1

Benoit Cloitre, Apr 06 2002

For p prime, contains p^2 if and only if p == 1 (mod 6). - Robert Israel, May 14 2019

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

Contains A024556.

Cf. A000005, A000203.

N:= 1000:
V:= Vector(N):
for d from 1 to N do
  r:= numtheory:-sigma(d)/numtheory:-tau(d);
  C:= [seq(i,i=d..N,d)];
  V[C]:= V[C] +~ r
od:
select(t -> not(isprime(t)) and V[t]::integer, [$2..N]); # Robert Israel, May 14 2019

q[n_] := CompositeQ[n] && IntegerQ @ DivisorSum[n, Divide @@ DivisorSigma[{1, 0}, #] &]; Select[Range[200], q] (* Amiram Eldar, Jun 08 2022 *)

A215739 Even and odd composite squarefree numbers, interlaced.

Original entry on oeis.org

6, 15, 10, 21, 14, 33, 22, 35, 26, 39, 30, 51, 34, 55, 38, 57, 42, 65, 46, 69, 58, 77, 62, 85, 66, 87, 70, 91, 74, 93, 78, 95, 82, 105, 86, 111, 94, 115, 102, 119, 106, 123, 110, 129, 114, 133, 118, 141, 122, 143, 130, 145, 134, 155, 138, 159, 142, 161, 146, 165

Offset: 1

Zak Seidov, Aug 22 2012

nonn

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

Cf. A120944.

Module[{nn=200,csfn,od,ev,len},csfn=Select[Range[nn],CompositeQ[ #] && SquareFreeQ[ #]&];od=Select[csfn,OddQ];ev=Select[csfn,EvenQ];len=Min[ Length[ od],Length[ev]];Riffle[Take[ev,len],Take[od,len]]] (* Harvey P. Dale, Nov 20 2022 *)

For k=1,2,.., a(2k-1)=A039956(k+1), a(2k)=A024556(k); A039956 Even squarefree numbers, A024556 Odd squarefree composite numbers.

A276980 Odd squarefree numbers n > 1 such that lambda(n)^2 = phi(n), where lambda is the Carmichael lambda function and phi is Euler's totient function.

Original entry on oeis.org

273, 1729, 2109, 2255, 4433, 4641, 4697, 5673, 6643, 6935, 7667, 8103, 8729, 10235, 11543, 14497, 16385, 16523, 17507, 18915, 20033, 22649, 23579, 26691, 29309, 29393, 34799, 35853, 35929, 37209, 37829, 39277, 42653, 45551, 55699, 56163, 68735, 68901, 69167, 69977, 70993, 73505, 75361, 76373

Offset: 1

Thomas Ordowski and Altug Alkan, Apr 11 2017

nonn

Such a number n must have at least three prime factors.
Are there infinitely many such numbers?
Among them are some Carmichael numbers: 1729, 75361, ... (A002997).

			273 = 3 * 7 * 13, so phi(273) = 2 * 6 * 12 = 144 = 12^2 and lambda(273) = lcm(2, 6, 12) = 12, hence 273 is in the sequence.
Notice that phi(315) = 144 and lambda(315) = 12 also. However, mu(315) = 0 since 315 = 3^2 * 5 * 7, so for that reason 315 is not in the sequence.

Robert G. Wilson v, Table of n, a(n) for n = 1..1024

Subsequence of A024556.

Cf. A000010, A002322, A002997.

samePsiSqPhiQ[n_] := SquareFreeQ[n] && CarmichaelLambda[n]^2 == EulerPhi[n]; Select[1 + 2 Range@50000, samePsiSqPhiQ] (* Robert G. Wilson v, Apr 14 2017 *)

is(n) = n>1 && n%2!=0 && issquarefree(n) && lcm(znstar(n)[2])^2==eulerphi(n) \\ Felix Fröhlich, Apr 22 2017

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

A309005 Odd squarefree composite numbers m such that m+2 is prime.

Original entry on oeis.org

15, 21, 35, 39, 51, 57, 65, 69, 77, 87, 95, 105, 111, 129, 155, 161, 165, 177, 195, 209, 221, 231, 237, 249, 255, 267, 291, 305, 309, 329, 335, 345, 357, 365, 371, 377, 381, 395, 399, 407, 417, 429, 437, 447, 455, 465, 485, 489, 497, 501, 519, 545, 555, 561, 591, 597, 611

Offset: 1

David James Sycamore, Jul 05 2019

nonn

The squarefree terms of A241809 and A136354 are in this sequence.

			15 = 3*5 is the smallest squarefree composite number m such that m+2 is prime; 15+2=17.

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

Cf. A024556, A241809, A136354.

[n: n in [2..611] | IsPrime(n+2) and  not IsPrime(n) and IsSquarefree(n)]; // Vincenzo Librandi, Jul 07 2019

with(NumberTheory):
N := 500;
for n from 2 to N do
if IsSquareFree(n) and not mod(n, 2) = 0 and not isprime(n) and isprime(n+2) then print(n);
end if:
  end do:

Select[Range[15, 611, 2], And[CompositeQ@ #, SquareFreeQ@ #, PrimeQ[# + 2]] &] (* Michael De Vlieger, Jul 08 2019 *)
Select[Prime[Range[2,150]]-2,SquareFreeQ[#]&&CompositeQ[#]&] (* Harvey P. Dale, Dec 03 2022 *)

isok(n) = isprime(n+2) && (n%2) && (n>1) && !isprime(n) && issquarefree(n); \\ Michel Marcus, Jul 05 2019

A321617 Last term of the first occurrence of n consecutive odd squarefree composite numbers.

Original entry on oeis.org

15, 35, 95, 219, 221, 903, 905, 1357

Offset: 1

Hugo Pfoertner, Dec 19 2018

See A322493.

Cf. A024556, A322493.

Join[{15},With[{osc=Select[Range[1,1401,2],CompositeQ[#]&&SquareFreeQ[ #]&]}, Flatten[Table[Select[Partition[osc,n,1],Union[Differences[#]] == {2}&,1],{n,8}],1]][[All,-1]]] (* Harvey P. Dale, Feb 10 2019 *)

cp's OEIS Frontend

A231372 Squarefree composite numbers k such that 10 is a primitive root for all prime factors of k.

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A124569 Numbers k such that k, k+2, k+4 and k+6 are squarefree.

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

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A055456 a(n) is the smallest number which is not the sum of exactly 1 or of n earlier terms.

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A068992 Composite numbers k such that Sum_{d|k} sigma(d)/tau(d) is an integer.

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A215739 Even and odd composite squarefree numbers, interlaced.

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A276980 Odd squarefree numbers n > 1 such that lambda(n)^2 = phi(n), where lambda is the Carmichael lambda function and phi is Euler's totient function.

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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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