A135590 -id:A135590 - cp's OEIS frontend

A247220 Numbers k such that k^2 + 1 divides 2^k + 1.

0, 2, 4, 386, 20136, 59140, 373164544

Offset: 1

Juri-Stepan Gerasimov, Nov 26 2014

a(8) > 10^12. - Giovanni Resta, May 05 2020
All terms of the sequence are even. a(5), a(6) and a(7) are of the form 2*p + 2 where p is a prime and p mod 14 = 1. - Farideh Firoozbakht, Dec 07 2014
From Jianing Song, Jan 13 2019: (Start)
Among the known terms only a(3) and a(4) are of the form 2*p where p is a prime.
a(n)^2 + 1 is prime for 2 <= n <= 7. Among these primes, the multiplicative order of 2 modulo a(n)^2 + 1 is 2*a(n) except for n = 5, in which case it is 2*a(n)/3. (End)
If a(n)^2 + 1 is composite, then a(n) is also a term of A135590. - Max Alekseyev, Apr 25 2024

			0 is in this sequence because 0^2 + 1 = 1 divides 2^0 + 1 = 2.

Cf. A135590, A247165.

for(n=0,10^5,if(Mod(2,n^2+1)^n==-1,print1(n,", "))); \\ Joerg Arndt, Nov 30 2014

from gmpy2 import powmod
A247220_list = [i for i in range(10**7) if powmod(2,i,i*i+1) == i*i]
# Chai Wah Wu, Dec 03 2014

a(7) from Hiroaki Yamanouchi, Nov 29 2014

A247165 Numbers m such that m^2 + 1 divides 2^m - 1.

Original entry on oeis.org

0, 16, 256, 8208, 65536, 649800, 1382400, 4294967296

Offset: 1

Juri-Stepan Gerasimov, Nov 30 2014

Contains 2^(2^k) = A001146(k) for k >= 2. - Robert Israel, Dec 02 2014
a(9) > 10^12. - Hiroaki Yamanouchi, Sep 16 2018
For each n, a(n)^2 + 1 belongs to A176997, and thus a(n) belongs to either A005574 or A135590. - Max Alekseyev, Feb 08 2024

			0 is in this sequence because 0^2 + 1 = 1 divides 2^0 - 1 = 1.

Cf. A247219 (n^2 - 1 divides 2^n - 1), A247220 (n^2 + 1 divides 2^n + 1).

Cf. A005574, A135590, A176997.

[n: n in [1..100000] | Denominator((2^n-1)/(n^2+1)) eq 1];

select(n -> (2 &^ n - 1) mod (n^2 + 1) = 0, [$1..10^6]); # Robert Israel, Dec 02 2014

a247165[n_Integer] := Select[Range[0, n], Divisible[2^# - 1, #^2 + 1] &]; a247165[1500000] (* Michael De Vlieger, Nov 30 2014 *)

for(n=0,10^9,if(Mod(2,n^2+1)^n==+1,print1(n,", "))); \\ Joerg Arndt, Nov 30 2014

A247165_list = [n for n in range(10**6) if n == 0 or pow(2,n,n*n+1) == 1]
# Chai Wah Wu, Dec 04 2014

a(8) from Chai Wah Wu, Dec 04 2014

Edited by Jon E. Schoenfield, Dec 06 2014

A265285 Carmichael numbers (A002997) k such that k-1 is a square.

Original entry on oeis.org

46657, 2433601, 67371265, 351596817937, 422240040001, 18677955240001, 458631349862401, 286245437364810001, 20717489165917230086401

Offset: 1

Altug Alkan, Dec 06 2015

This sequence contains all Carmichael numbers n such that for all primes p dividing n, p-1 divides n-1 and furthermore, n-1 is a square.

Numbers sqrt(a(n)-1) form a subsequence of A135590. - Max Alekseyev, Apr 25 2024

			46657 is a term because 46657 - 1 = 46656 = 216^2.
2433601 is a term because 2433601 - 1 = 2433600 = 1560^2.

G. Tarry, I. Franel, A. Korselt, and G. Vacca, Problème chinois, L'intermédiaire des mathématiciens 6 (1899), pp. 142-144.
Eric Weisstein's World of Mathematics, Carmichael Number.
Index entries for sequences related to Carmichael numbers.

Subsequence of A265237 and of A265328.

Cf. A002997, A135590, A265237, A303791.

isA002997:= proc(n) local F,p;
         if n::even or isprime(n)  then return false fi;
         F:= ifactors(n)[2];
         if max(seq(f[2],f=F)) > 1 then return false fi;
         andmap(f -> (n-1) mod (f[1]-1) = 0,  F)
end proc:
select(isA002997, [seq(4*i^2+1,i=1..10^6)]); # Robert Israel, Dec 08 2015

is_c(n) = { my(f); bittest(n, 0) && !for(i=1, #f=factor(n)~, (f[2, i]==1 && n%(f[1, i]-1)==1)||return) && #f>1 }
for(n=1, 1e10, if(is_c(n) && issquare(n-1), print1(n, ", ")))

lista(kmax) = {my(m); for(k = 2, kmax, m = k^2 + 1; if(!isprime(m), f = factor(k); for(i = 1, #f~, f[i, 2] *= 2); fordiv(f, d, if(!(m % (d+1)) && isprime(d+1), m /= (d+1))); if(m == 1, print1(k^2 + 1, ", ")))); } \\ Amiram Eldar, May 02 2024

a(4)-a(5), using A002997 b-file, from Michel Marcus, Dec 07 2015
a(6) and a(7) from Robert Israel, Dec 08 2015
a(8) from Max Alekseyev, Apr 30 2018
a(9) from Daniel Suteu confirmed by Max Alekseyev, Apr 25 2024

A265328 Carmichael numbers (A002997) k such that k-1 is a perfect power (A001597).

Original entry on oeis.org

1729, 46657, 2433601, 2628073, 19683001, 67371265, 110592000001, 351596817937, 422240040001, 432081216001, 2116874304001, 3176523000001, 18677955240001, 458631349862401, 286245437364810001, 312328165704192001, 12062716067698821000001, 20717489165917230086401, 211215936967181638848001, 411354705193473163968001

Offset: 1

Altug Alkan, Dec 07 2015

From Antti Karttunen, Dec 08 2015: (Start)
The prime factorizations of the first six terms are:
7*13*19, 13*37*97, 17*37*53*73, 7*37*73*139, 13*37*151*271, 5*13*37*109*257
and the prime factorizations of the corresponding perfect powers (numbers one smaller) are:
(2^6 * 3^3), (2^6 * 3^6), (2^6 * 3^2 * 5^2 * 13^2), (2^3 * 3^3 * 23^3), (2^3 * 3^9 * 5^3), (2^8 * 3^6 * 19^2).
(End)
For each k in {22934100, 59553720, 74371320, 242699310, 3190927740, 9214178820, 84855997590}, which is a subset of A270840, k^3+1 is a Carmichael number. - Daniel Suteu, Aug 24 2019
Wagstaff (2024) found that there are no Carmichael numbers k below 10^21 such that k+1 is a perfect power. - Amiram Eldar, Dec 29 2024

			1729 = 7*13*19 is a term because 1729 - 1 = 1728 = 12^3, and 7-1 = 6, 13-1 = 12 and 19-1 = 18 can be all constructed from the primes available in 1728 = (2^6 * 3^3).
2433601 = 17*37*53*73 is a term because 2433601 - 1 = 2433600 = 1560^2, and 16, 36, 52 and 72 can be all constructed from the primes available in 2433600 = (2^6 * 3^2 * 5^2 * 13^2).
67371265 = 5*13*37*109*257 is a term because 67371264 = 8208^2, and 4 (= 2*2), 12 (= 2*2*3), 36 (= 2*2*3*3), 108 (= 2*2*3*3*3) and 256 (= 2^8) can be all constructed from the primes available in 67371264 = (2^8 * 3^6 * 19^2).

G. Tarry, I. Franel, A. Korselt, and G. Vacca, Problème chinois, L'intermédiaire des mathématiciens 6 (1899), pp. 142-144.
Samuel S. Wagstaff, Ramanujan's taxicab number and its ilk, The Ramanujan Journal, Vol. 64, No. 3 (2024), pp. 761-764; ResearchGate link, author's copy.
Eric Weisstein's World of Mathematics, Carmichael Number.
Index entries for sequences related to Carmichael numbers.

Contains A265285 as a subsequence.

Cf. A001597, A002997, A135590, A270840.

Select[Cases[Range[1, 10^7, 2], n_ /; Mod[n, CarmichaelLambda@ n] == 1 && ! PrimeQ@ n], GCD @@ FactorInteger[# - 1][[All, 2]] > 1 &] (* Michael De Vlieger, Dec 14 2015, after Ant King at A001597 and Artur Jasinski at A002997 *)

is_c(n)={my(f); bittest(n, 0) && !for(i=1, #f=factor(n)~, (f[2, i]==1 && n%(f[1, i]-1)==1)||return) && #f>1}
for(n=1, 1e10, if(is_c(n) && ispower(n-1), print1(n, ", ")))

use ntheory ":all"; foroddcomposites { say if is_power($-1) && is_carmichael($) } 1e8; # Dana Jacobsen, May 05 2017

More terms from Dana Jacobsen, May 05 2017

a(17) from Daniel Suteu confirmed, a(18)-a(20) added by Max Alekseyev, Apr 25 2024

A270840 Numbers n such that n^3 + 1 is a 2-pseudoprime (or Sarrus number).

Original entry on oeis.org

12, 36, 138, 270, 546, 4800, 7560, 12840, 14700, 358200, 678480, 16139970, 22934100, 55058580, 59553720, 74371320, 113068380, 116605860, 242699310, 997521210, 1592680320, 1652749200, 3190927740, 5088964800, 6974736756, 9214178820

Offset: 1

Jeppe Stig Nielsen, Mar 23 2016

nonn

Since n^3 + 1 is clearly composite (n>1), this is n such that 2^(n^3) == 1 (mod n^3+1).

There is only one n such that n + 1 is a composite number up to 10^10: 14700. - Altug Alkan, Mar 27 2016

Cf. A001567, A135590, A265653.

PARI
```
is(n)=Mod(2,n^3+1)^(n^3)==1
```

a(n) = A265653(n) - 1. - Altug Alkan, Mar 24 2016

More terms from Altug Alkan, Mar 24 2016

A333316 Numbers k such that k^2 + 1 is a Fermat pseudoprime to base 3.

Original entry on oeis.org

216, 660, 1484, 1560, 8208, 52164, 544320, 592956, 649800, 4321800, 5103210, 6182220, 10621380, 21415680, 24471720, 135307008, 359624088, 535019100, 1071782250, 1113233520, 1227427740, 1527496740, 9462748008, 143935711920

Offset: 1

Amiram Eldar, Mar 14 2020

a(24) > 7*10^10.
The corresponding pseudoprimes a(n)^2 + 1 are 46657, 435601, 2202257, 2433601, 67371265, ...
a(25) > 7.5*10^11. - Giovanni Resta, Mar 15 2020

			216 is a term since 216^2 + 1 = 46657 is a Fermat pseudoprime to base 3.

Cf. A002522, A005574, A005935, A135590.

Select[Range[10^3], CompositeQ[#^2 + 1] && PowerMod[3, #^2, #^2 + 1] == 1 &]

a(24) from Giovanni Resta, Mar 15 2020

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A247220 Numbers k such that k^2 + 1 divides 2^k + 1.

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A247165 Numbers m such that m^2 + 1 divides 2^m - 1.

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A265285 Carmichael numbers (A002997) k such that k-1 is a square.

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A265328 Carmichael numbers (A002997) k such that k-1 is a perfect power (A001597).

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A270840 Numbers n such that n^3 + 1 is a 2-pseudoprime (or Sarrus number).

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A333316 Numbers k such that k^2 + 1 is a Fermat pseudoprime to base 3.

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