A005935 -id:A005935 - cp's OEIS frontend

A244065 Pseudoprimes to base 3 that are not squarefree.

121, 3751, 4961, 7381, 11011, 29161, 32791, 142901, 228811, 239701, 341341, 551881, 566401, 595441, 671671, 784201, 856801, 1016521, 1237951, 1335961, 1433971, 1804231

Offset: 1

Felix Fröhlich, Jun 19 2014

nonn

Must be divisible by the square of a Mirimanoff prime, A014127. - Charles R Greathouse IV, Jun 21 2014

Felix Fröhlich and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 196 terms from Fröhlich)

Cf. A005935, A158358, A014127.

for(n=2, 10^9, if(!isprime(n) && Mod(3, n)^(n-1)==1 && !issquarefree(n), print1(n, ", ")))

list(lim)=my(M=[11,1006003],v=List(),p2);for(i=1,#M,p2=M[i]^2;forstep(n=p2,lim,p2,if(Mod(3,n)^(n-1)==1,listput(v,n))));Set(v) \\ Good for lim <= 9.4 * 10^29; Charles R Greathouse IV, Jun 21 2014

A114245 Number of Fermat pseudoprimes to base 3 less than 10^n.

Original entry on oeis.org

0, 1, 6, 23, 78, 246, 760, 2155, 5804, 15472, 39871, 102839, 264461

Offset: 1

Eric W. Weisstein, Nov 18 2005

Michal Mikuš, Computing the Pseudoprimes up to 10^13, Cybernetics and Informatics, International Conference SSKI, February 10-13, 2010.
Eric Weisstein's World of Mathematics, Fermat Pseudoprime.
Index entries for sequences related to pseudoprimes.

Cf. A005935, A114247, A114249.

Table[Count[Select[Range[2, 10^6], ! PrimeQ[#] && PowerMod[3, # - 1, #] == 1 &], x_ /; x < 10^n], {n, 6}]  (* Robert Price, Jun 09 2019 *)

isFps(n,b)= { if(isprime(n), return(0) ) ; if( (b^(n-1)) % n == 1, return(1), return(0) ) ; } { a=0 ; e=1 ; for(n=1,10^12, if( n == 10^e, e++ ; print(a) ; ) ; a += isFps(n,3) ; ) ; } \\ R. J. Mathar, Feb 10 2007

A005935(a(n)) < 10^n; A005935(a(n)+1) > 10^n. - R. J. Mathar, Feb 10 2007

a(9)-a(10) from Washington Bomfim, Mar 02 2012
a(11)-a(12) from Hiroaki Yamanouchi, Sep 25 2015
a(13) from Mikuš (2010) added by Amiram Eldar, Mar 31 2024

A258189 Pseudoprimes to base 3, written in base 3.

Original entry on oeis.org

10101, 11111, 101121, 220212, 222001, 1022011, 1111221, 2010002, 2101001, 2121001, 10101022, 10122201, 10201001, 10212111, 11111112, 11121201, 12010221, 20210202, 21121121, 100001111, 101010101, 102112011, 110020001, 112112001, 120002211, 122000101, 201201201, 202212002

Offset: 1

Abdul Gaffar Khan, Sep 11 2015

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

Cf. A005935 (pseudoprimes to base 3), A007089 (numbers in base 3).

Cf. A262101, A262102, A262103, A262104, A262105, A262154.

base = 3; t = {}; n = 1;
While[Length[t] < 40, n++;
If[! PrimeQ[n] && PowerMod[base, n - 1, n] == 1,
  AppendTo[t, FromDigits@IntegerDigits[n, 3]]]]; t

lista(nn, b=3) = {for (n=1, nn, if (Mod(b, n)^(n-1)==1 && !ispseudoprime(n) && n>1, print1(subst(Pol(digits(n,b), x), x, 10), ", ");););} \\ Michel Marcus, Sep 30 2015

a(n) = A007089(A005935(n)). - Michel Marcus, Sep 11 2015

a(25) corrected by Georg Fischer, Dec 18 2020

A083739 Pseudoprimes to bases 2, 3, 5 and 7.

Original entry on oeis.org

29341, 46657, 75361, 115921, 162401, 252601, 294409, 314821, 334153, 340561, 399001, 410041, 488881, 512461, 530881, 552721, 658801, 721801, 852841, 1024651, 1152271, 1193221, 1461241, 1569457, 1615681, 1857241, 1909001, 2100901

Offset: 1

Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), May 05 2003

nonn

			a(1)=29341 since it is the first number such that 2^(k-1) = 1 (mod k), 3^(k-1) = 1 (mod k), 5^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).

Amiram Eldar, Table of n, a(n) for n = 1..7469 (terms below 10^12; terms 1..114 from R. J. Mathar)
Jens Bernheiden, Pseudoprimes (in German).
Fred Richman, Primality testing with Fermat's little theorem.
Index entries for sequences related to pseudoprimes.

Cf. A001567, A005935, A005936, A005938.

Proper subset of A083737.

a001567 := [] : f := fopen("b001567.txt",READ) : bfil := readline(f) : while StringTools[WordCount](bfil) > 0 do if StringTools[FirstFromLeft]("#",bfil ) <> 0 then ; else bfil := sscanf(bfil,"%d %d") ; a001567 := [op(a001567), op(2,bfil) ] ; fi ; bfil := readline(f) ; od: fclose(f) : isPsp := proc(n,b) if n>3 and not isprime(n) and b^(n-1) mod n = 1 then true; else false; fi; end: isA001567 := proc(n) isPsp(n,2) ; end: isA005935 := proc(n) isPsp(n,3) ; end: isA005936 := proc(n) isPsp(n,5) ; end: isA005938 := proc(n) isPsp(n,7) ; end: isA083739 := proc(n) if isA001567(n) and isA005935(n) and isA005936(n) and isA005938(n) then true ; else false ; fi ; end: n := 1: for psp2 from 1 do i := op(psp2,a001567) ; if isA083739(i) then printf("%d %d ",n,i) ; n :=n+1 ; fi ; od: # R. J. Mathar, Feb 07 2008

Select[ Range[2113920], !PrimeQ[ # ] && PowerMod[2, # - 1, # ] == 1 && PowerMod[3, 1 - 1, # ] == 1 && PowerMod[5, # - 1, # ] == 1 && PowerMod[7, 1 - 1, # ] == 1 & ]

is(n)=!isprime(n)&&Mod(2,n)^(n-1)==1&&Mod(3,n)^(n-1)==1&&Mod(5,n)^(n-1)==1&&Mod(7,n)^(n-1)==1 \\ Charles R Greathouse IV, Apr 12 2012

a(n) = n-th positive integer k(>1) such that 2^(k-1) = 1 (mod k), 3^(k-1) = 1 (mod k), 5^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).

A005938 INTERSECT A083737. - R. J. Mathar, Feb 07 2008

Edited by Robert G. Wilson v, May 06 2003

A210461 Cipolla pseudoprimes to base 3: (9^p-1)/8 for any odd prime p.

Original entry on oeis.org

91, 7381, 597871, 3922632451, 317733228541, 2084647712458321, 168856464709124011, 1107867264956562636991, 588766087155780604365200461, 47690053059618228953581237351, 25344449488056571213320166359119221, 166284933091139163730593611482181209801

Offset: 1

Bruno Berselli, Jan 22 2013 - proposed by Umberto Cerruti (Department of Mathematics "Giuseppe Peano", University of Turin, Italy)

nonn

This is the case a=3 of Theorem 1 in the paper of Hamahata and Kokubun (see Links section).

			91 is in the sequence because 91=((3^3-1)/2)*((3^3+1)/4), even if p=3 divides 3*(3^2-1), and 3^90 = (91*8+1)^15 == 1 (mod 91).
7381 is in the sequence because 7381=((3^5-1)/2)*((3^5+1)/4) and 3^7380 = (7381*472400+1)^369 == 1 (mod 7381).

Michele Cipolla, Sui numeri composti P che verificano la congruenza di Fermat a^(P-1) = 1 (mod P), Annali di Matematica 9 (1904), p. 139-160.

Bruno Berselli, Table of n, a(n) for n = 1..50
Umberto Cerruti, Pseudoprimi di Fermat e numeri di Carmichael (in Italian), 2013.
Y. Hamahata and Y. Kokubun, Cipolla Pseudoprimes, Journal of Integer Sequences, Vol. 10 (2007).

Cf. A005935, A002452, A065091, A210454, A217853.

a210461 = (`div` 8) . (subtract 1) . (9 ^) . a065091
-- Reinhard Zumkeller, Jan 22 2013

Magma
```
[(9^NthPrime(n)-1)/8: n in [2..12]];
```

P:=proc(q)local n;
for n from 2 to q do print((9^ithprime(n)-1)/8);
od; end: P(100); # Paolo P. Lava, Oct 11 2013

Mathematica
```
(9^# - 1)/8 & /@ Prime[Range[2, 12]]
```

Prime(n) := block(if n = 1 then return(2), return(next_prime(Prime(n-1))))$
makelist((9^Prime(n)-1)/8, n, 2, 12);

A328662 Super pseudoprimes (or superpseudoprimes) to base 3: Fermat pseudoprimes to base 3 all of whose divisors that are larger than 1 are either primes or Fermat pseudoprimes to base 3.

Original entry on oeis.org

91, 121, 671, 703, 949, 1541, 1891, 2701, 3281, 7381, 8401, 12403, 14383, 15203, 16531, 18721, 23521, 24727, 28009, 30857, 31621, 31697, 38503, 44287, 46999, 47197, 49051, 49141, 55261, 55969, 63139, 72041, 74593, 79003, 82513, 83333, 88573, 88831, 90751, 96139

Offset: 1

Amiram Eldar, Oct 24 2019

nonn

The super pseudoprimes to base 2 are the super-Poulet numbers (A050217).
Includes all the semiprimes in A005935. The first terms that are not semiprimes are 7381, 512461, 532171, 1018601, ... (A328663).
Subsequence of A271116. - Bill McEachen, Nov 06 2020

			91 is in the sequence since it is a Fermat pseudoprime to base 3, and its proper divisors that are larger than 1 are the primes 7 and 13.
7381 is in the sequence since it is a Fermat pseudoprime to base 3, and its proper divisors that are larger than 1 are the primes 11 and 61, and the composite numbers 121 and 671 that are Fermat pseudoprimes to base 3.

Michal Krížek, Florian Luca, and Lawrence Somer, 17 Lectures on Fermat Numbers: From Number Theory to Geometry, Springer-Verlag, New York, 2001, chapter 12, Fermat's Little Theorem, Pseudoprimes, and Superpseudoprimes, pp. 130-146.

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. Fehér and P. Kiss, Note on super pseudoprime numbers, Ann. Univ. Sci. Budapest, Eötvös Sect. Math., Vol. 26 (1983), pp. 157-159, entire volume.
B. M. Phong, On super pseudoprimes which are products of three primes, Ann. Univ. Sci. Budapest. Eótvós Sect. Math., Vol. 30 (1987), pp. 125-129, entire volume.
Andrzej Rotkiewicz, Solved and unsolved problems on pseudoprime numbers and their generalizations, Applications of Fibonacci numbers, Springer, Dordrecht, 1999, pp. 293-306.
Lawrence Somer, On superpseudoprimes, Mathematica Slovaca, Vol. 54, No. 5 (2004), pp. 443-451.

Subsequence of A005935.

Cf. A050217.

aQ[n_]:=  CompositeQ[n] && AllTrue[Rest[Divisors[n]], PowerMod[3, #-1, #] == 1 &]; Select[Range[10^5], aQ]

A328663 Super pseudoprimes to base 3 (A328662) with more than two prime factors (counted with multiplicity).

Original entry on oeis.org

7381, 512461, 532171, 1018601, 2044657, 3882139, 5934391, 8624851, 10802017, 14396449, 19383673, 25708453, 32285041, 35728129, 35807461, 38316961, 43040161, 53369149, 58546753, 59162891, 64464919, 71386849, 75397891, 79511671, 81276859, 83083001, 84890737, 85636609

Offset: 1

Amiram Eldar, Oct 24 2019

nonn

Super pseudoprimes to base 3 are Fermat pseudoprimes to base 3 all of whose composite divisors are also Fermat pseudoprimes to base 3. Therefore all the Fermat pseudoprimes to base 3 that are semiprimes are super pseudoprimes. This sequence contains the nontrivial terms of A328662, i.e. terms with at least one composite proper divisor.
Fehér and Kiss proved that there are infinitely many terms with 3 distinct prime factors (their proof was for all bases a > 1 that are not divisible by 4. Phong proved it for all bases a > 1).
The first term, 7381, is not squarefree. What is the next such term?

			512461 is in the sequence since it is a Fermat pseudoprime to base 3, 3^512460 == 1 (mod 512461), and all of its divisors that are larger than 1 are either primes (31, 61, and 271), or Fermat pseudoprimes to base 3 (1891, 8401, 16531, 512461).

Michal Krížek, Florian Luca, and Lawrence Somer, 17 Lectures on Fermat Numbers: From Number Theory to Geometry, Springer-Verlag, New York, 2001, chapter 12, Fermat's Little Theorem, Pseudoprimes, and Superpseudoprimes, pp. 130-146.

Amiram Eldar, Table of n, a(n) for n = 1..400
J. Fehér and P. Kiss, Note on super pseudoprime numbers, Ann. Univ. Sci. Budapest, Eötvös Sect. Math., Vol. 26 (1983), pp. 157-159, entire volume.
B. M. Phong, On super pseudoprimes which are products of three primes, Ann. Univ. Sci. Budapest. Eótvós Sect. Math., Vol. 30 (1987), pp. 125-129, entire volume.
Andrzej Rotkiewicz, Solved and unsolved problems on pseudoprime numbers and their generalizations, Applications of Fibonacci numbers, Springer, Dordrecht, 1999, pp. 293-306.
Lawrence Somer, On superpseudoprimes, Mathematica Slovaca, Vol. 54, No. 5 (2004), pp. 443-451.

Cf. A178997

Subsequence of A005935, A328662.

aQ[n_]:=  PrimeOmega[n] > 2 && AllTrue[Rest[Divisors[n]], PowerMod[3, #-1, #] == 1 &]; Select[Range[10^5], aQ]

A000864 Deceptive nonprimes: composite numbers k that divide the repunit R_{k-1}.

Original entry on oeis.org

91, 259, 451, 481, 703, 1729, 2821, 2981, 3367, 4141, 4187, 5461, 6533, 6541, 6601, 7471, 7777, 8149, 8401, 8911, 10001, 11111, 12403, 13981, 14701, 14911, 15211, 15841, 19201, 21931, 22321, 24013, 24661, 27613, 29341, 34133

Offset: 1

Tim Ray (c268scm(AT)semovm.semo.edu)

nonn

Francis and Ray call these numbers "deceptive primes".
Pseudoprimes to base 10, A005939, not divisible by 3. If k is in the sequence, then (10^k-1)/9 is in the sequence, by Steuerwald's theorem; see A005935. - Thomas Ordowski, Apr 10 2016
41041 is the first term that has four prime divisors. - Altug Alkan, Apr 10 2016

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
R. Francis and T. Ray, The deceptive primes to 2.10^7, Missouri J. Math. Sci. 12 (2000), no. 3, 145-158.

Cf. A002275, A005939.

select(t -> not isprime(t) and (10&^(t-1) - 1) mod (9*t) = 0, [seq(t,t=3..10^5,2)]); # Robert Israel, Apr 10 2016

p=5;forprime(q=7,1e5,forstep(n=p+2,q-2,2,if(n%5 && Mod(10,9*n)^(n-1)==1,print1(n", ")));p=q) \\ Charles R Greathouse IV, Jul 31 2011

A083735 Pseudoprimes to bases 3 and 7.

Original entry on oeis.org

703, 1105, 2465, 10585, 18721, 19345, 29341, 38503, 46657, 50881, 75361, 76627, 88831, 104653, 115921, 146611, 162401, 188191, 213265, 226801, 252601, 278545, 286903, 294409, 314821, 334153, 340561, 359341, 385003, 385201, 399001, 410041

Offset: 1

Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), May 05 2003

nonn

			a(1)=703 since it is the first number such that 3^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).

Amiram Eldar, Table of n, a(n) for n = 1..16479 (terms 1..151 from R. J. Mathar)
F. Richman, Primality testing with Fermat's little theorem
Index entries for sequences related to pseudoprimes

Intersection of A005935 and A005938. - R. J. Mathar, Apr 05 2011

Select[Range[420000],!PrimeQ[#]&&PowerMod[3,#-1,#]==1&&PowerMod[7,#-1,#] == 1&] (* Harvey P. Dale, Mar 08 2014 *)

is(n)=!isprime(n)&&Mod(7,n)^(n-1)==1&&Mod(3,n)^(n-1)==1 \\ Charles R Greathouse IV, Apr 12 2012

a(n) = n-th positive integer k(>1) such that 3^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).

A271116 Integers n such that round(3^n/12) is divisible by n.

Original entry on oeis.org

1, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307

Offset: 1

Altug Alkan, Mar 31 2016

In other words, numbers n such that A015518(n-1) is divisible by n.
This sequence generates prime numbers except 2 and 3.
The first few composite terms in this sequence are 91, 121, 671, 703, 949. Note that they are pseudoprimes to base 3.
Also, numbers k such that 3^(k-1) mod (4*k) = 1. In the first million terms only 908 terms are nonprimes. - David A. Corneth, Oct 02 2020

			5 is a term because round(3^5/12) = 20 is divisible by 5.
6 is not a term because round(3^6/12) = 61 which is not divisible by 6. - _David A. Corneth_, Oct 02 2020

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A005935, A015518.

Select[Range@ 308, Divisible[Round[3^#/12], #] &] (* Michael De Vlieger, Mar 31 2016 *)

f(n) = round(3^n/12);
for(n=1, 1e3, if(f(n) % n == 0, print1(n, ", ")));

is(n) = lift(Mod(3, 4*n)^(n-1))==1 \\ David A. Corneth, Oct 02 2020

cp's OEIS Frontend

A244065 Pseudoprimes to base 3 that are not squarefree.

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A114245 Number of Fermat pseudoprimes to base 3 less than 10^n.

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A258189 Pseudoprimes to base 3, written in base 3.

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A083739 Pseudoprimes to bases 2, 3, 5 and 7.

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A210461 Cipolla pseudoprimes to base 3: (9^p-1)/8 for any odd prime p.

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A328662 Super pseudoprimes (or superpseudoprimes) to base 3: Fermat pseudoprimes to base 3 all of whose divisors that are larger than 1 are either primes or Fermat pseudoprimes to base 3.

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A328663 Super pseudoprimes to base 3 (A328662) with more than two prime factors (counted with multiplicity).

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