A001567 -id:A001567 - cp's OEIS frontend

A371760 a(n) is the smallest number k such that the k-th n-gonal number is a Fermat pseudoprime to base 2 (A001567), or -1 if no such number exists.

33, 1093, 73, 17, 97, 11, 193, 17, 89, 11, 193, 73, 673, 13, 257, 33, 41, 15, 97, 65, 1009, 13, 97, 149, 190, 23, 401, 41, 281, 31, 133, 17, 1033, 31, 89, 13, 6, 59, 241, 157, 1217, 91, 145, 37, 937, 29, 1289, 73, 97, 41, 617, 19, 137, 151, 34, 103, 8641, 47, 82

Offset: 3

Amiram Eldar, Apr 05 2024

nonn

The corresponding pseudoprimes are in A371759.

Amiram Eldar, Table of n, a(n) for n = 3..10000
Eric Weisstein's World of Mathematics, Polygonal Number.
Eric Weisstein's World of Mathematics, Poulet Number.
Wikipedia, Polygonal number.
Wikipedia, Pseudoprime.
Index entries for sequences related to pseudoprimes.

Cf. A001567, A293622, A293623, A322130, A321868, A321870, A322123, A322160, A371759.

p[k_, n_] := ((n - 2)*k^2 - (n - 4)*k)/2; pspQ[n_] := CompositeQ[n] && PowerMod[2, n - 1, n] == 1; a[n_] := Module[{k = 2}, While[! pspQ[p[k, n]], k++]; k]; Array[a, 100, 3]

p(k, n) = ((n-2)*k^2 - (n-4)*k)/2;
ispsp(n) = !isprime(n) && Mod(2, n)^(n-1) == 1;
a(n) = {my(k = 2); while(!ispsp(p(k, n)), k++); k;}

A172255 Partial sums of the Fermat pseudoprimes to base 2, A001567.

Original entry on oeis.org

341, 902, 1547, 2652, 4039, 5768, 7673, 9720, 12185, 14886, 17707, 20984, 25017, 29386, 33757, 38438, 43899, 50500, 58457, 66778, 75259, 84170, 94431, 105016, 116321, 129122, 142863, 156610, 170591, 185082, 200791, 216632, 233337, 252042

Offset: 1

Jonathan Vos Post, Jan 29 2010

The subsequence of pseudoprimes in this sequence begins 341; the next term exceeds a(10000) if it exists. - Charles R Greathouse IV, Aug 22 2012

The subsequence of primes in the sequence begins 7673, 17707, 33757, 270763, 484621.

			a(15) = 341 + 561 + 645 + 1105 + 1387 + 1729 + 1905 + 2047 + 2465 + 2701 + 2821 + 3277 + 4033 + 4369 + 4371 = 33757 is prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A001567.

sums(v)=my(s); vector(#v,i, s+=v[i])
sums(select(n->Mod(2, n)^n==2 & !isprime(n), vector(10^5,n,2*n+1))) \\ Charles R Greathouse IV, Jul 09 2015

a(n) = Sum_{i=1..n, odd composite numbers n such that 2^(n-1) == 1 mod n}.

A216404 a(n) = lcm((d1 + 1), (d2 + 1), ..., (dk + 1)), where d1, d2, ..., dk are the prime factors of the n-th Fermat pseudoprime to base 2, A001567(n).

Original entry on oeis.org

96, 36, 132, 126, 740, 280, 384, 360, 90, 1406, 224, 570, 2090, 774, 96, 608, 1408, 168, 4070, 4266, 516, 680, 2656, 1110, 360, 252, 1064, 2340, 672, 7436, 1368, 1184, 1806, 660, 9506, 3384, 252, 11858, 4448, 8246, 4648, 720, 5310, 16058, 8008, 5676, 3630, 17910

Offset: 1

Marius Coman, Sep 06 2012

nonn

It is notable how many primes are obtained if we add or subtract 1 from these numbers.
Primes obtained by adding 1 and the corresponding Fermat pseudoprime in the brackets: 97(341), 37(561), 127(1105), 281(1729), 571(3277), 97(4371), 1409(5461), 2657(10261), 2341(13747), 673(13981), 661(18705), 46499(30121), 8009(31621), 3631(34945), 17911(35333).
Primes obtained by subtracting 1 and the corresponding Fermat pseudoprime in the brackets: 131(645),  739(1387), 383(1905), 359(2047), 89(2465), 223(2821), 569(3277), 2089(4033), 773(4369), 607(4681), 167(6601), 1109(10585), 359(11305), 251(12801), 1063(13741), 2339(13747), 1367(15709), 659(18705), 251(23001), 4447(25761), 719(30889), 5309(31417), 160479(31609), 17909(35333).
Numbers from sequence which do not lead to a prime number adding or subtracting 1 (and the corresponding Fermat pseudoprimes to base 2 in the bracketts): 1406(2701), 4070(7957), 4266(8321), 516(8481), 680(8911), 7436(14491), 1184(15841), 1806(16705), 9506(18721), 3384(19951), 11858(23377), 8246(29341), 5676(33153). Interesting analogies can be found between these "exceptions": subtracting 1 from the ones of the form 10*k + 6 often yields semiprimes, etc.
There are probably many other interesting utilities for the function from the sequence above as for the function a(n) = lcm(d1-1, d2-1, ..., dk-1), where d1, d2, ..., dk are the prime factors of the n-th Fermat pseudoprime to base 2 A001567(n).

Eric Weisstein's World of Mathematics, Poulet Number

Cf. A001567.

A242276 Irregular array of factors of n-th Poulet number read by rows, where row n corresponds to A001567(n).

Original entry on oeis.org

11, 31, 3, 11, 17, 3, 5, 43, 5, 13, 17, 19, 73, 7, 13, 19, 3, 5, 127, 23, 89, 5, 17, 29, 37, 73, 7, 13, 31, 29, 113, 37, 109, 17, 257, 3, 31, 47, 31, 151, 43, 127, 7, 23, 41, 73, 109, 53, 157, 3, 11, 257, 7, 19, 67, 31, 331, 5, 29, 73, 5, 7, 17, 19, 3, 17, 251, 7, 13, 151, 59, 233, 11, 31, 41, 43, 337, 23, 683, 7, 31, 73, 5, 13

Offset: 1

Felix Fröhlich, Aug 16 2014

			The first three Poulet numbers (2-pseudoprimes) are 341 = 11*31, 561 = 3*11*17, and 645 = 3*5*43, so the sequence begins:
11, 31;
3, 11, 17;
3, 5, 43;
etc.

Felix Fröhlich, Table of n, a(n) for n = 1..2165 (all terms up to 10^7)

Cf. A001567, A007011, A084653, A084654, A084655.

forcomposite(n=1, 1e4, if(Mod(2, n)^(n-1)==1, f=factor(n)[, 1]; for(i=1, #f, print1(f[i], ", "))))

A265653 Integers k such that (k-1)^3 + 1 is a Fermat pseudoprime to base 2 (A001567).

Original entry on oeis.org

13, 37, 139, 271, 547, 4801, 7561, 12841, 14701, 358201, 678481, 16139971, 22934101, 55058581, 59553721, 74371321, 113068381, 116605861, 242699311, 997521211, 1592680321, 1652749201, 3190927741, 5088964801, 6974736757, 9214178821

Offset: 1

Altug Alkan, Dec 12 2015

nonn

Corresponding Fermat pseudoprimes to base 2 are 1729, 46657, 2628073, 19683001, 162771337, 110592000001, 432081216001, ...

There is only one composite term up to 10^10: 14701. It also appears in A265628 (see comments). Can we say that if there is a Fermat pseudoprime to base 2 of the form (k-1)^3 + 1, k is a prime number most of the time? Are there other composite terms like 14701?

			13 is a term because (13-1)^3 + 1 = 1729, which is a Fermat pseudoprime to base 2.
37 is a term because (37-1)^3 + 1 = 46657, which is a Fermat pseudoprime to base 2.

Cf. A000040, A001567, A265628, A270840.

Select[Range[10^6], ! PrimeQ@ # && PowerMod[2, (# - 1), #] == 1 &@((# - 1)^3 + 1) &] (* Michael De Vlieger, Dec 12 2015, after Farideh Firoozbakht at A001567 *)

is(n) = {Mod(2, n)^n==2 & !isprime(n) & n>1};
for(n=1, 1e10, if(is((n-1)^3+1), print1(n, ", ")));

a(n) = A270840(n) + 1.

A265684 Sarrus numbers (A001567) that are the average of two consecutive primes.

Original entry on oeis.org

645, 7957, 11305, 15841, 25761, 35333, 126217, 194221, 212421, 332949, 464185, 635401, 656601, 741751, 934021, 1193221, 1357441, 1459927, 1620385, 1690501, 1969417, 2704801, 3911197, 4154161, 4209661, 5095177, 5284333, 5351537, 5758273, 6189121, 6212361, 7820201, 8134561, 8209657

Offset: 1

Altug Alkan, Dec 13 2015

nonn

Inspired by A265669.
Motivation was the form of differences between consecutive primes that generate this sequence. It seems that 12*k appears in differences most of the time. For the first 175 term of this sequence, the relevant proportion is 161/175.
Differences between corresponding consecutive primes are 4, 12, 12, 36, 4, 12, 12, 36, 4, 4, 24, 24, 4, 60, 24, 24, 24, 12, 12, 36, 12, 24, 12, 24, 36, 12, 12, 12, 12, 24, 4, 60, 24, 48, 36, 12, 24, 36, 24, 20, 12, 84, 36, 12, 24, 24, 12, 24, 36, 12, 12, 36, ...

			645 is a term because it is a Sarrus number and the average of the consecutive primes 643 and 647.
7957 is a term because it is a Sarrus number and the average of the consecutive primes 7951 and 7963.

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

Intersection of A001567 and A024675.

Cf. A265669.

Select[Range[200000], CompositeQ[#] && PowerMod[2, (# - 1), #] == 1 && NextPrime[#] - # == # - NextPrime[#, -1] &] (* Amiram Eldar, Jun 28 2019 *)

is(n)={Mod(2, n)^n==2 && !isprime(n)}
forcomposite(n=2, 1e7, if(is(n) && (nextprime(n)-n)==(n-precprime(n)), print1(n, ", ")))

A270639 Fermat pseudoprimes (A001567) that are the sum of three consecutive primes.

Original entry on oeis.org

13741, 16705, 150851, 208465, 249841, 252601, 258511, 410041, 486737, 635401, 1052503, 1082401, 1457773, 1507963, 1579249, 1615681, 2113921, 2184571, 3090091, 3375487, 3726541, 4682833, 4895065, 5044033, 5133201, 6233977, 6255341, 6350941, 6474691, 6912079, 7259161

Offset: 1

Altug Alkan, Mar 20 2016

nonn

In other words, Fermat pseudoprimes to base 2 of the form p + q + r where p, q and r are consecutive primes.
If a Fermat pseudoprime is the sum of n consecutive primes, it is so obvious that the minimum value of n is 3.
Intersection of A001567 and A034961.

			4567, 4583 and 4591 are consecutive primes and their sum is 13741, a Fermat pseudoprime.
84191, 84199 and 84211 are consecutive primes and their sum is 252601, a Fermat pseudoprime.

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

Cf. A001567, A034961, A270267.

isA001567(n) = {Mod(2, n)^n==2 && !ispseudoprime(n) && n > 1}
a034961(n) = my(p=prime(n), q=nextprime(p+1)); p+q+nextprime(q+1);
for(n=1, 200000, if(isA001567(a034961(n)), print1(a034961(n), ", ")));

A273471 Primes p such that at least one of 1093p or 1093p^2 is a Poulet number, i.e., a term of A001567.

Original entry on oeis.org

1093, 4733, 21841, 503413, 1948129, 112901153, 23140471537, 467811806281, 4093204977277417, 8861085190774909, 556338525912325157, 86977595801949844993, 275700717951546566946854497, 3194753987813988499397428643895659569

Offset: 1

Felix Fröhlich, May 23 2016

The prime factors of 2^1092-1 that are congruent to 1 modulo 364 (the multiplicative order of 2 modulo 1093). - Max Alekseyev, Aug 30 2016

G. P. Michon, Wieferich primes and some of their Poulet multiples
Factordb, Factorization of 2^1092-1

Cf. A001220, A001567, A273472.

forprime(p=1, , if(Mod(2, 1093*p)^(1093*p-1)==1 || Mod(2, 1093*p^2)^(1093*p^2-1)==1, print1(p, ", ")))

A273472 Primes p such that at least one of 3511p or 3511p^2 is a Poulet number, i.e., a term of A001567.

Original entry on oeis.org

3511, 10531, 1024921, 1969111, 4633201, 409251961, 21497866557571, 194900834792501371, 4242734772486358591, 85488365519409100951, 255375215316698521591, 1439538040790707946401, 5302306226370307681801, 2728334536034592865339299805712535332071, 1514527568177848809210967221069334182785475908756709327091, 559791068131697034376217936561708451475280017605178661418575551, 656640320787712008058581244241126148909602076298405712103045387152988908318802087128873347971063698441918022286945981375193401, 25006596829256741460214169653933852849128490077459890197421900490545433220443136638425782879171530372521984642165350019685875922245867185516694881

Offset: 1

Felix Fröhlich, May 23 2016

The prime factors of 2^3510-1 that are congruent to 1 modulo 1755 (the multiplicative order of 2 modulo 3511). - Max Alekseyev, Aug 30 2016

G. P. Michon, Wieferich primes and some of their Poulet multiples
Factordb, Factorization of 2^3510-1

Cf. A001220, A001567, A273471.

forprime(p=1, , if(Mod(2, 3511*p)^(3511*p-1)==1 || Mod(2, 3511*p^2)^(3511*p^2-1)==1, print1(p, ", ")))

Terms a(8) onward from Max Alekseyev, Aug 30 2016

A291617 Numbers p_1p_2...p_k such that (2^p_1-1)(2^p_2-1)...(2^p_k-1) is a Poulet number (A001567), where p_i are primes and k >= 2.

Original entry on oeis.org

230, 341, 1387, 2047, 2701, 3277, 4033, 4369, 4681, 5461, 7957, 8321, 10261, 13747, 14491, 15709, 18721, 19951, 23377, 31323, 31417, 31609, 31621, 35333, 38193, 42799, 49141, 49981, 60701, 60787, 65077, 65281, 80581, 83333, 85489

Offset: 1

Max Alekseyev and Thomas Ordowski, Aug 28 2017

nonn

Rotkiewicz (1965) proved that (2^p-1)*(2^q-1) is a Poulet number if and only if p*q is a Poulet number, where p,q are distinct primes. It follows that this sequence contains all nonsquare terms in A214305.
Generally, the sequence includes all squarefree super-Poulet numbers.
The terms n = 230, 31323, 38193, ... are not in A050217. Are there infinitely many such terms?

			The number n = 341 = 11*31 is a term, because m = (2^11-1)*(2^31-1) = 4395899025409 is a Poulet number.

Max Alekseyev, Table of n, a(n) for n = 1..66
A. Rotkiewicz, Sur les nombres pseudopremiers de la forme M_p M_q, Elemente der Mathematik 20 (1965): 108-109. (in French)

Cf. A001567, A050217, A214305.

Select[Select[Range[10^4], CompositeQ@ # && SquareFreeQ@ # &], ! PrimeQ[#] && PowerMod[2, (# - 1), #] == 1 &@ Apply[Times, Map[2^# - 1 &, FactorInteger[#][[All, 1]] ]] &] (* Michael De Vlieger, Aug 30 2017 *)

{ is_A291617(n) = my(p,m); if(isprime(n),return(0)); p=factor(n); m=prod(i=1,matsize(p)[1], (2^p[i,1]-1)^p[i,2]); Mod(2,m)^m==2; }

cp's OEIS Frontend

A371760 a(n) is the smallest number k such that the k-th n-gonal number is a Fermat pseudoprime to base 2 (A001567), or -1 if no such number exists.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A172255 Partial sums of the Fermat pseudoprimes to base 2, A001567.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A216404 a(n) = lcm((d1 + 1), (d2 + 1), ..., (dk + 1)), where d1, d2, ..., dk are the prime factors of the n-th Fermat pseudoprime to base 2, A001567(n).

Views

Author

Keywords

Comments

Links

Crossrefs

A242276 Irregular array of factors of n-th Poulet number read by rows, where row n corresponds to A001567(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A265653 Integers k such that (k-1)^3 + 1 is a Fermat pseudoprime to base 2 (A001567).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A265684 Sarrus numbers (A001567) that are the average of two consecutive primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A270639 Fermat pseudoprimes (A001567) that are the sum of three consecutive primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A273471 Primes p such that at least one of 1093*p or 1093*p^2 is a Poulet number, i.e., a term of A001567.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A273472 Primes p such that at least one of 3511*p or 3511*p^2 is a Poulet number, i.e., a term of A001567.

A273471 Primes p such that at least one of 1093p or 1093p^2 is a Poulet number, i.e., a term of A001567.

A273472 Primes p such that at least one of 3511p or 3511p^2 is a Poulet number, i.e., a term of A001567.

A291617 Numbers p_1p_2...p_k such that (2^p_1-1)(2^p_2-1)...(2^p_k-1) is a Poulet number (A001567), where p_i are primes and k >= 2.