Karsten Meyer - cp's OEIS frontend

A200525 Zeisel numbers with p(0)=4.

385, 2585, 7315, 8911, 27001, 39905, 48391, 87283, 192211, 196285, 319705, 410089, 425585, 441091, 624605, 679855, 1310185, 1899163, 2460439, 2586971, 2777041, 6654005, 7042411, 7501261, 8291459, 9516637, 10484585, 11141671, 12527281, 13015891, 13788319

Offset: 1

Karsten Meyer, Nov 18 2011

nonn

Pick any integers A and B and consider the linear recurrence relation given by p(0) = 4, p(i + 1) = A*p(i) + B. If for some n > 2, p(1), p(2), ..., p(n) are distinct primes, then the product of these primes is called a Zeisel number.

			a=2, b=-3 => p(1) = (4*2)+(-3) = 5; p(2) = (5*2)+(-3) = (7); p(3) = (7*2)+(-3) = 11 => 5*7*11 = 385.
a=2, b=5 => p(1) = (4*2)+5 = 13; p(2) = (13*2)+5 = 31; p(3) = (31*2)+5 = 67 => 13*31*67 = 27001.

Eric Weisstein's World of Mathematics, Zeisel Number.
Wikipedia, Zeisel number

Cf. A051015.

n0=4
do m=1 to 53
  a=2*m
  do b=(1-(4*a)) to 999
    n1=(n0*a)+b
    n2=(n1*a)+b
    n3=(n2*a)+b
    z=n1*n2*n3
    say n0 a b
    lineout("zeisel_4.txt",z||" = "||n1||"*"||n2||"*"||n3||"      "||a||" "||b||" n0="||n0)
    end
  end

A191864 a(n) = (a(n-1) + a(n-4)) * (a(n-2) - a(n-3)) with a(1)=1, a(2)=2, a(3)=3 and a(4)=4.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 10, 28, 99, 1908, 136178, 246396654, 33083692025310, 8147205746460109635768, 269537638338486762080764802762484576, 2195978587041305889551144566841383797948181151148527903340

Offset: 1

Karsten Meyer, Jun 18 2011

nonn

			a(5) = (4+1)*(3-2) = 5 ; a(6) = (5+2)*(4-3) = 7

RecurrenceTable[{a[1]==1,a[2]==2,a[3]==3,a[4]==4,a[n]==(a[n-1]+ a[n-4])(a[n-2]- a[n-3])},a,{n,20}] (* Harvey P. Dale, Mar 08 2020 *)

a=vector(20,i,i);for(n=6,#a,a[n]=(a[n-1]+a[n-4])*(a[n-2]-a[n-3]));a \\ Charles R Greathouse IV, Jun 21 2011

a(n) = (a(n-1) + a(n-4)) * (a(n-2) - a(n-3)) with a(1)=1, a(2)=2, a(3)=3 and a(4)=4

a(n) = k^(phi^n + o(1)) with k = 1.06164666362... and phi = (1+sqrt(5))/2. [Charles R Greathouse IV, Jun 21 2011]

A178958 Numbers n from A181780 that are not in A181781.

Original entry on oeis.org

15, 28, 35, 39, 51, 52, 55, 63, 66, 70, 75, 76, 87, 95, 99, 111, 112, 115, 119, 123, 124, 130, 135, 143, 147, 148, 154, 155, 159, 171, 172, 176, 183, 186, 187, 190, 195, 196, 203, 207, 208, 215, 219, 232, 235, 238, 244, 246, 255, 267, 268, 275, 276, 279, 280, 286, 287, 291, 292, 295, 299

Offset: 1

Karsten Meyer, Dec 31 2010

nonn

Numbers that are Fermat pseudoprimes to some base a (2<=a<=n-2) not Euler pseudoprimes to any base a (2<=a<=n-2).

			4^(15-1) == 1 (mod 15), but 4^((15-1)/2) == 4 (mod 15)

A181780

fsp(n)=
{ /* whether n is Fermat pseudoprime to any base a where 2<=a<=n-2 */
    for (a=2,n-2,
        if ( gcd(a,n)!=1, next() );
        if ( (Mod(a,n))^(n-1)==+1, return(1) )
    );
    return(0);
}
esp(n)=
{ /* whether n is Euler pseudoprime to any base a where 2<=a<=n-2 */
    local(w);
    if ( n%2==0, return(0) );
    for (a=2,n-2,
        if ( gcd(a,n)!=1, next() );
        w = abs(component((Mod(a,n))^((n-1)/2),2));
        if ( (w==1) || (w==n-1), return(1) )
    );
    return(0);
}
for(n=3,300, if(isprime(n),next()); if( fsp(n) && (!esp(n)) , print1(n,", ") ); );

A178705 Odd composite numbers q such that there exists a, 2<=a<=q-2, such that a^d == 1 mod q where d = A000265(q-1). Thus q is a strong pseudoprime in base a.

Original entry on oeis.org

49, 91, 121, 133, 169, 175, 217, 231, 247, 259, 301, 325, 341, 343, 361, 385, 403, 427, 435, 451, 469, 475, 481, 511, 529, 553, 559, 561, 589, 595, 637, 645, 651, 671, 679, 703, 715, 721, 763, 775, 781, 793, 805, 817, 841, 847, 861, 871, 889, 891, 925, 931, 949, 961, 973, 1001, 1015, 1027, 1035, 1045

Offset: 1

Karsten Meyer, Dec 26 2010

nonn

Odd composite numbers q such that gcd(A000010(q), A000265(q-1)) > 1. - Robert Israel, Dec 20 2017

			18^3 == 1 mod 49

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

Cf. A000010, A000265.

filter:= proc(n)
  if isprime(n) then return false fi;
  igcd((n-1)/2^padic:-ordp(n-1,2), numtheory:-phi(n)) > 1
end proc:
select(filter, [seq(i,i=9..2000,2)]); # Robert Israel, Dec 20 2017

filterQ[n_] := If[PrimeQ[n], False, GCD[(n-1)/2^IntegerExponent[n-1, 2], EulerPhi[n]] > 1];
Select[Range[9, 2000, 2], filterQ] (* Jean-François Alcover, Sep 25 2020, after Robert Israel *)

a^d == 1 mod q

Corrected by Robert Israel, Dec 20 2017

A178530 Numbers k with the property that there exist nonnegative integers a and b such that k = concat(a,b) = a^2+b^2.

Original entry on oeis.org

0, 1, 100, 101, 1233, 8833, 10100, 990100, 5882353, 94122353, 1765038125, 2584043776, 7416043776, 8235038125, 116788321168, 123288328768, 876712328768, 883212321168, 7681802663025, 8896802846976, 13793103448276, 15348303604525, 84651703604525, 86206903448276, 91103202846976, 92318202663025, 106058810243728

Offset: 1

Karsten Meyer, Dec 23 2010

The sum of two numbers a1 and a2 that share a common b has the form of 10^j. Example: 12 + 88 = 100
The ordered pair of the final digit of a and b is always one of (0,0), (0,1), (0,5), (0,6), (2,3), (8,3), (2,8), or (8,8).
If b has k decimal digits, then (2a - 10^k)^2 + (2b - 1)^2 = 10^(2k) + 1 giving a way for efficient computation of many terms. - Max Alekseyev, Aug 17 2013

			0 = 0^2+0^2 [this seems a bit far-fetched. - _N. J. A. Sloane_, Dec 23 2010]
1=0^2+1^2 [ditto]
100=10^2+0^2.
101=10^2+1^2.
1233=12^2+33^2.

Max Alekseyev, Table of n, a(n) for n = 1..200

See A055616, A064942, A101311 for closely related sequences.

Sort[Reap[Do[n=a^2+b^2; If[n==FromDigits[Join[IntegerDigits[a], IntegerDigits[b]]], Sow[n]], {a,0, 1000}, {b, 0, 1000}]][[2, 1]]]

Edited by N. J. A. Sloane, Dec 23 2010
a(11)-a(14) from Nathaniel Johnston, Jan 03 2011
Terms a(15) onward from Max Alekseyev, Aug 17 2013

A181782 Odd composite numbers n that are strong pseudoprimes to some base a, 2 <= a <= n-2.

Original entry on oeis.org

25, 49, 65, 85, 91, 121, 125, 133, 145, 169, 175, 185, 205, 217, 221, 231, 247, 259, 265, 289, 301, 305, 325, 341, 343, 361, 365, 377, 385, 403, 425, 427, 435, 445, 451, 469, 475, 481, 485, 493, 505, 511, 529, 533, 545, 553, 559, 561, 565, 589, 595, 625, 629, 637, 645, 651, 671, 679, 685, 689, 697

Offset: 1

Karsten Meyer, Nov 10 2010

nonn

			49 is a strong pseudoprime to the bases 18, 19, 30 and 31, so 49 is in the sequence.

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

Cf. A141768.

/* function sppq() from http://www.jjj.de/pari/rabinmiller.gpi */
sppq(n,a)=
{ /* Return whether n is a strong pseudoprime to base a (Rabin Miller) */
    local(q, t, b, e);
    q = n-1;  t = 0;  while ( 0==bitand(q,1), q\=2; t+=1 );
    /* here  n==2^t*q+1 */
    b = Mod(a, n)^q;
    if ( 1==b, return(1) );
    e = 1;
    while ( eJoerg Arndt, Dec 27 2010 */

select( is_A181782(n)={bittest(n,0) && !isprime(n) && for(a=2,n-2, my(t=valuation(n-1,2), b=Mod(a,n)^(n>>t)); b==1&&return(1); while(t-->0 && b!=-1 && b!=1, b=b^2); b==-1&&return(1))}, [1..700]) \\ Defines is_A181782(): select(...) gives a check and illustration for free. Inside the for loop is the exact equivalent of the sppq() function above. - M. F. Hasler, Nov 26 2018

Definition corrected by Max Alekseyev, Nov 12 2010

Terms corrected by Joerg Arndt, Dec 27 2010

A181781 Numbers n that are Euler pseudoprimes to some base b, 2 <= b <= n-2.

Original entry on oeis.org

21, 25, 33, 45, 49, 57, 65, 69, 77, 85, 91, 93, 105, 117, 121, 125, 129, 133, 141, 145, 153, 161, 165, 169, 175, 177, 185, 189, 201, 205, 209, 213, 217, 221, 225, 231, 237, 245, 247, 249, 253, 259, 261, 265, 273, 285, 289, 297, 301, 305, 309, 321, 325, 329, 333, 341, 343, 345

Offset: 1

Karsten Meyer, Nov 12 2010

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Karsten Meyer, Tabelle Pseudoprimzahlen (15-4999)
Karsten Meyer, Rexx program
Eric W. Weisstein, MathWorld: Euler Pseudoprime

Cf. A006970, A181780.

isEulPSP := proc(n,b) if isprime(n) then false; else m := modp(b &^ ((n-1)/2),n) ; if m= 1 or m = n-1 then true; else false; end if; end if;end proc:
isA181781 := proc(n) for b from 2 to n-2 do if isEulPSP(n,b) then return true; end if; end do: return false;end proc:
for n from 3 to 800 do if isA181781(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, May 30 2011

fQ[n_?PrimeQ, b_] = False; fQ[n_, b_] := Block[{p = PowerMod[b, (n - 1)/2, n]}, p == Mod[1, n] || p == Mod[-1, n]]; gQ[n_] := AnyTrue[Range[2, n - 2], fQ[n, #] &]; Select[2 Range[172] + 1, gQ] (* Michael De Vlieger, Sep 09 2015, after Jean-François Alcover at A006970, Version 10 *)

Definition corrected by Max Alekseyev, Nov 12 2010

Edited definition to be consistent with OEIS style. - N. J. A. Sloane, Nov 13 2010

A181780 Numbers n which are Fermat pseudoprimes to some base b, 2 <= b <= n-2.

Original entry on oeis.org

15, 21, 25, 28, 33, 35, 39, 45, 49, 51, 52, 55, 57, 63, 65, 66, 69, 70, 75, 76, 77, 85, 87, 91, 93, 95, 99, 105, 111, 112, 115, 117, 119, 121, 123, 124, 125, 129, 130, 133, 135, 141, 143, 145, 147, 148, 153, 154, 155, 159, 161, 165, 169, 171, 172, 175, 176

Offset: 1

Karsten Meyer, Nov 12 2010

nonn

A nonprime number n is a Fermat pseudoprime to base b if b^(n-1) = 1 (mod n).

It appears that these n are pseudoprimes for an even number of bases. When n is the product of two distinct primes, it appears that there are exactly two such bases x and y with x + y = n. See A211455, A211456, and A211457. - T. D. Noe, Apr 12 2012

			15 is Fermat pseudoprime to base 4 and 11, so it is a Fermat pseudoprime.

Karsten Meyer and T. D. Noe, Table of n, a(n) for n = 1..10000 (first 5978 terms from Karsten Meyer)
Karsten Meyer, Tabelle Pseudoprimzahlen (15-4999)
Karsten Meyer, Rexx program for this sequence
Eric Weisstein's World of Mathematics, Fermat Pseudoprime
Index entries for sequences related to pseudoprimes

Cf. A039769, A181781, A211455, A211456, A211457, A211458, A227180, A280199.

Even terms give A039772. - Thomas Ordowski, Dec 28 2016

t = {}; Do[s = Select[Range[2, n-2], PowerMod[#, n-1, n] == 1 &]; If[s != {}, AppendTo[t, n]], {n, Select[Range[213], ! PrimeQ[#] &]}]; t (* T. D. Noe, Nov 07 2011 *)
(* The following program is much faster than the one above. See A227180 for indications of a proof of this assertion. *) Select[Range[213], ! IntegerQ[Log[3, #]] && ! PrimeQ[#] && GCD[# - 1, EulerPhi[#]] > 1 &] (* Emmanuel Vantieghem, Jul 06 2013 *)

fsp(n)=
{ /* whether n is Fermat pseudoprime to any base a where 2<=a<=n-2 */
    for (a=2,n-2,
        if ( gcd(a,n)!=1, next() );
        if ( (Mod(a,n))^(n-1)==+1, return(1) )
    );
    return(0);
}
for(n=3,300, if(isprime(n),next());  if ( fsp(n) , print1(n,", ") ); );
\\ Joerg Arndt, Jan 08 2011

is(n)=if(isprime(n), return(0)); my(f=factor(n)[,1]); prod(i=1, #f, gcd(f[i]-1, n-1)) > 2 \\ Charles R Greathouse IV, Dec 28 2016

Rexx
```
See Meyer link.
```

For any odd a(m), a(m) = A211456(m) + A211457(m). - Thomas Ordowski, Dec 09 2013

Used a comment line to give a more explicit definition. - N. J. A. Sloane, Nov 12 2010

Definition corrected by Max Alekseyev, Nov 12 2010

A111305 Composite numbers k such that a^(k-1) == 1 (mod k) only when a == 1 (mod k).

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 54, 56, 58, 60, 62, 64, 68, 72, 74, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 114, 116, 118, 120, 122, 126, 128, 132, 134, 136, 138, 140, 142, 144

Offset: 1

Karsten Meyer, Nov 02 2005

nonn

These unCarmichael numbers fail the Fermat primality test as often as possible.
All such numbers are even: for odd n, (-1)^(n-1) = 1.
The even numbers not in this sequence are 2 and A039772.
If c is a Carmichael number, then 2c is in the sequence. Also, the sequence is A209211 without the first two terms. - Emmanuel Vantieghem, Jul 03 2013

			10 is a term because 3^9 == 3 (mod 10), 7^9 == 7 (mod 10), 9^9 == 9 (mod 10).

Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867v1 [math.NT], May 04 2013.

Cf. A002997, A039772, A209211, A227180.

Select[Range[4, 144],Count[Table[PowerMod[b, # - 1, #], {b, 1, # - 1}], 1] == 1 &] (* Geoffrey Critzer, Apr 11 2015 *)

is(n)=for(a=2,n-1, if(Mod(a,n)^(n-1)==1, return(0))); !isprime(n) \\ Charles R Greathouse IV, Dec 22 2016

Edited by Don Reble, May 16 2006

A112540 Numbers m such that (15m-4, 15m-2, 15m+2, 15m+4) is a prime quadruple.

Original entry on oeis.org

1, 7, 13, 55, 99, 125, 139, 217, 231, 377, 629, 867, 1043, 1049, 1071, 1203, 1261, 1295, 1401, 1485, 1687, 2115, 2323, 2919, 3423, 3689, 4199, 4481, 4633, 4815, 5151, 5313, 5403, 5515, 5921, 6523, 6609, 6741, 7323, 7769, 7953, 8147, 9031, 9611, 10485, 11047

Offset: 1

Karsten Meyer, Dec 16 2005

Also (4p + 16)/60 such that (p, p+2, p+6 and p+8) is a prime quadruple for p >= 11. - Michel Lagneau, Jul 02 2012
The density of these four-prime groups is approximately equal to (log x)^-3.45 (but not (log x)^-4). - Xueshi Gao, Jun 01 2014
All of the terms of this sequence are either 1, 7 or 13 modulo 14. - Rodolfo Ruiz-Huidobro, Dec 27 2019
From Eric Snyder, Jun 23 2021: (Start)
Building on the comment of R. Ruiz-Huidobro above, all terms of the sequence are congruent to one of {-1, 0 ,1} (mod 7). The appearance of mod 14 stems from the fact that all entries in this list must be odd. Equivalently, a(n) cannot be +- 2 or +- 3 (mod 7). This can be generalized for all larger primes:
All primes p >= 7 can be expressed as 15k +- q in a least absolute residue system, with q in {2, 4} if k is odd, and q in {1,7} if k is even.
For all primes 15k +- q >= 7, four residues +-r, +-s (mod p) exist such that, if for any p >= 7, [(m == +- r (mod p) or m == +- s (mod p)), and (m != k)], then m is not in this sequence. For the different values of p = 15k +- q, the values of +-r and +-s are as follows:
    For p = 15k +- 1 (k even), r = +- 2k, s = +- 4k
    For p = 15k +- 2 (k odd), r = +- k, s = +- 2k
    For p = 15k + 4 (k odd), r = +- k, s = +- (7k + 2)
    For p = 15k - 4 (k odd), r = +- k, s = +- (7k - 2)
    For p = 15k + 7 (k even), r = +- (4k + 2), s = +- (8k + 4)
    For p = 15k - 7 (k even), r = +- (4k - 2), s = +- (8k - 4)
These can be used to create an Eratosthenes-like sieve for the prime decades. (End)

			m = 7 yields the quadruple (15*7 - 4 = 101, 15*7 - 2 = 103, 15*7 + 2 = 107, 15*7 + 4 = 109), so 7 is a term of the sequence.

Dana Jacobsen, Table of n, a(n) for n = 1..10000
Eric Snyder, An Eratosthenean Sieve for the Prime Decades

Cf. A007530, A014561.

[n: n in [0..2*10^4] | IsPrime(15*n-4) and IsPrime(15*n-2) and IsPrime(15*n+2) and IsPrime(15*n+4)]; // Vincenzo Librandi, Dec 28 2015

A112540:=n->`if`(isprime(15*n-4) and isprime(15*n-2) and isprime(15*n+2) and isprime(15*n+4),n,NULL); seq(A112540(n), n=1..20000); # Wesley Ivan Hurt, Jul 26 2014

Select[Range[6610], PrimeQ[15# - 4] && PrimeQ[15# - 2] && PrimeQ[15# + 2] && PrimeQ[15# + 4]&] (* T. D. Noe, Nov 16 2006 *)

for(n=1, 1e4, if(isprime(15*n-4) && isprime(15*n-2) && isprime(15*n+2) && isprime(15*n+4), print1(n, ", "))) \\ Felix Fröhlich, Jul 26 2014

use ntheory ":all"; say for map { (4*$+16)/60 } sieve_prime_cluster(11,15*10000, 2,6,8); # _Dana Jacobsen, Dec 15 2015

from sympy import isprime
def ok(m): return all(isprime(15*m+k) for k in [-4, -2, 2, 4])
print(list(filter(ok, range(11111)))) # Michael S. Branicky, Jun 24 2021

Corrected by T. D. Noe, Nov 16 2006

cp's OEIS Frontend

User: Karsten Meyer

A200525 Zeisel numbers with p(0)=4.

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Rexx

A191864 a(n) = (a(n-1) + a(n-4)) * (a(n-2) - a(n-3)) with a(1)=1, a(2)=2, a(3)=3 and a(4)=4.

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A178958 Numbers n from A181780 that are not in A181781.

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A178705 Odd composite numbers q such that there exists a, 2<=a<=q-2, such that a^d == 1 mod q where d = A000265(q-1). Thus q is a strong pseudoprime in base a.

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A178530 Numbers k with the property that there exist nonnegative integers a and b such that k = concat(a,b) = a^2+b^2.

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A181782 Odd composite numbers n that are strong pseudoprimes to some base a, 2 <= a <= n-2.

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A181781 Numbers n that are Euler pseudoprimes to some base b, 2 <= b <= n-2.

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A181780 Numbers n which are Fermat pseudoprimes to some base b, 2 <= b <= n-2.

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