A159611 -id:A159611 - cp's OEIS frontend

A019434 Fermat primes: primes of the form 2^(2^k) + 1, for some k >= 0.

3, 5, 17, 257, 65537

Offset: 1

It is conjectured that there are only 5 terms. Currently it has been shown that 2^(2^k) + 1 is composite for 5 <= k <= 32 (see Eric Weisstein's Fermat Primes link). - Dmitry Kamenetsky, Sep 28 2008
No Fermat prime is a Brazilian number. So Fermat primes belong to A220627. For proof see Proposition 3 page 36 in "Les nombres brésiliens" in Links. - Bernard Schott, Dec 29 2012
This sequence and A001220 are disjoint (see "Other theorems about Fermat numbers" in Wikipedia link). - Felix Fröhlich, Sep 07 2014
Numbers n > 1 such that n * 2^(n-2) divides (n-1)! + 2^(n-1). - Thomas Ordowski, Jan 15 2015
From Jaroslav Krizek, Mar 17 2016: (Start)
Primes p such that phi(p) = 2*phi(p-1); primes from A171271.
Primes p such that sigma(p-1) = 2p - 3.
Primes p such that sigma(p-1) = 2*sigma(p) - 5.
For n > 1, a(n) = primes p such that p = 4 * phi((p-1) / 2) + 1.
Subsequence of A256444 and A256439.
Conjectures:
1) primes p such that phi(p) = 2*phi(p-2).
2) primes p such that phi(p) = 2*phi(p-1) = 2*phi(p-2).
3) primes p such that p = sigma(phi(p-2)) + 2.
4) primes p such that phi(p-1) + 1 divides p + 1.
5) numbers n such that sigma(n-1) = 2*sigma(n) - 5. (End)
Odd primes p such that ratio of the form (the number of nonnegative m < p such that m^q == m (mod p))/(the number of nonnegative m < p such that -m^q == m (mod p)) is a divisor of p for all nonnegative q. - Juri-Stepan Gerasimov, Oct 13 2020
Numbers n such that tau(n)*(number of distinct ratio (the number of nonnegative m < n such that m^q == m (mod n))/(the number of nonnegative m < n such that -m^q == m (mod n))) for nonnegative q is equal to 4. - Juri-Stepan Gerasimov, Oct 22 2020
The numbers of primitive roots for the five known terms are 1, 2, 8, 128, 32768. - Gary W. Adamson, Jan 13 2022
Prime numbers such that every residue is either a primitive root or a quadratic residue. - Keith Backman, Jul 11 2022
If there are only 5 Fermat primes, then there are only 31 odd order groups which have a 2-group automorphism group. See the Miles Englezou link for a proof. - Miles Englezou, Mar 10 2025

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 137-141, 197.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see Table 1, p. 458.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, pp. 78-79.
Richard K. Guy, Unsolved Problems in Number Theory, A3.
Hardy and Wright, An Introduction to the Theory of Numbers, bottom of page 18 in the sixth edition, gives an heuristic argument that this sequence is finite.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 7, 70.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 136-137.

Anas AbuDaqa, Amjad Abu-Hassan, and Muhammad Imam, Taxonomy and Practical Evaluation of Primality Testing Algorithms, arXiv:2006.08444 [cs.CR], 2020.
Cyril Banderier, Pepin's Criterion For Fermat Numbers (in French)
Kent D. Boklan and John H. Conway, Expect at most one billionth of a new Fermat Prime!, arXiv:1605.01371 [math.NT], 2016.
P. Bruillard, S.-H. Ng, E. Rowell, and Z. Wang, On modular categories, arXiv:1310.7050 [math.QA], 2013-2015.
C. K. Caldwell, The Prime Glossary, Fermat number
Miles Englezou, Proof that 5 Fermat primes implies 31 odd order groups with 2-group automorphism groups
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m
P. Mathonet, M. Rigo, M. Stipulanti and N. Zénaïdi, On digital sequences associated with Pascal's triangle, arXiv:2201.06636 [math.NT], 2022.
Romeo Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv:1202.3670 [math.HO], 2012. - _N. J. A. Sloane_, Jun 13 2012
Romeo Meštrović, Goldbach-type conjectures arising from some arithmetic progressions, University of Montenegro, 2018.
Romeo Meštrović, Goldbach's like conjectures arising from arithmetic progressions whose first two terms are primes, arXiv:1901.07882 [math.NT], 2019.
Salah Eddine Rihane, Chèfiath Awero Adegbindin, and Alain Togbé, Fermat Padovan And Perrin Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.6.2.
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 65.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38. Local copy, included here with permission from the editors of Quadrature.
Yaryna Stelmakh, Homeomorphisms of the space of non-zero integers with the Kirch topology, arXiv:2101.04676 [math.GN], 2020.
Eric Weisstein's World of Mathematics, Fermat Number
Eric Weisstein's World of Mathematics, Fermat Prime
Eric Weisstein's World of Mathematics, Pepin's Test
Wikipedia, Fermat number
Haifeng Xu, The largest cycles consist by the quadratic residues and Fermat primes, arXiv:1601.06509 [math.NT], 2016.

Cf. A000215, A001146, A159611, A220627, A220570.
Subsequence of A147545 and of A334101. Cf. also A333788, A334092.
Cf. A045544.

[2^(2^n)+1 : n in [0..4] | IsPrime(2^(2^n)+1)]; // Arkadiusz Wesolowski, Jun 09 2011

Select[Table[2^(2^n) + 1, {n, 0, 4}], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

for(i=0,10, isprime(2^2^i+1) && print1(2^2^i+1,", ")) \\ M. F. Hasler, Nov 21 2009

[2^(2^n)+1 for n in (0..4) if is_prime(2^(2^n)+1)] # G. C. Greubel, Mar 07 2019

a(n+1) = A180024(A049084(a(n))). - Reinhard Zumkeller, Aug 08 2010

a(n) = 1 + A001146(n-1), if 1 <= n <= 5. - Omar E. Pol, Jun 08 2018

A014499 Number of 1's in binary representation of n-th prime.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 2, 3, 4, 4, 5, 3, 3, 4, 5, 4, 5, 5, 3, 4, 3, 5, 4, 4, 3, 4, 5, 5, 5, 4, 7, 3, 3, 4, 4, 5, 5, 4, 5, 5, 5, 5, 7, 3, 4, 5, 5, 7, 5, 5, 5, 7, 5, 7, 2, 4, 4, 5, 4, 4, 5, 4, 5, 6, 5, 6, 5, 4, 6, 6, 4, 6, 7, 6, 7, 8, 4, 5, 4, 5, 5, 5, 7, 5, 7, 7, 4, 5, 6, 7, 6, 8, 7, 7, 7, 8, 8, 3, 4

Offset: 1

Ingemar Assarsjo (ingemar(AT)binomen.se)

a(n) is the rank of prime(n) in the base-2 dominance order on the natural numbers. - Tom Edgar, Mar 25 2014

			From _M. F. Hasler_, Mar 03 2023: (Start)
a(n) = 1 only for p(n = 1) = 2, the only prime equal to a power of 2.
a(n) = 2 for n in A159611 = A000720(A019434) = {2, 3, 7, 55, 6543} (probably complete), the Fermat primes F[k] = 2^2^k + 1 with k = 0, 1, 2, 3, 4. (On the graph one can distinctly see a(6543) = 2 corresponding to F[4] = 65537.)
a(n) = 3 for n in A000720(A081091) = (4, 5, 6, 8, 12, 13, 19, 21, 25, 32, 33, 44, 98, 106, 116, 136, 174, 191, 310, 313, 319, 565, 568, ...). (End)

T. D. Noe, Table of n, a(n) for n = 1..10000
Tyler Ball and Daniel Juda, Dominance over N, Rose-Hulman Undergraduate Mathematics Journal, Vol. 13, No. 2, Fall 2013.
Christian Elsholtz, Almost all primes have a multiple of small Hamming weight, arXiv:1602.05974 [math.NT], 2016.
Index entries for sequences related to binary expansion of n

Cf. A035103, A035100, A004676, A090455.
Cf. A027697, A027699
Cf. A180024. - Reinhard Zumkeller, Aug 08 2010
Cf. A072084.
Cf. A159611 (indices of 2s), A000720(A081091) (indices of 3s). - M. F. Hasler, Mar 03 2023

a014499 = a000120 . a000040  -- Reinhard Zumkeller, Feb 10 2013

[&+Intseq(NthPrime(n), 2): n in [1..100] ]; // Vincenzo Librandi, Mar 25 2014

Table[Plus @@ IntegerDigits[Prime[n], 2], {n, 1, 100}] (* Vincenzo Librandi, Mar 25 2014 *)

A014499(n)=hammingweight(prime(n)) \\ M. F. Hasler, Nov 20 2009, updated Mar 03 2023

from sympy import prime
def A014499(n): return prime(n).bit_count() # Chai Wah Wu, Mar 22 2023

[sum(i.digits(base=2)) for i in primes_first_n(200)] # Tom Edgar, Mar 25 2014

a(n) = A000120(A000040(n)).
a(A049084(A061712(n))) = n. - Reinhard Zumkeller, Feb 10 2013
a(n) = [x^prime(n)] (1/(1 - x))*Sum_{k>=0} x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Mar 27 2018

A023503 Greatest prime divisor of prime(n) - 1.

Original entry on oeis.org

2, 2, 3, 5, 3, 2, 3, 11, 7, 5, 3, 5, 7, 23, 13, 29, 5, 11, 7, 3, 13, 41, 11, 3, 5, 17, 53, 3, 7, 7, 13, 17, 23, 37, 5, 13, 3, 83, 43, 89, 5, 19, 3, 7, 11, 7, 37, 113, 19, 29, 17, 5, 5, 2, 131, 67, 5, 23, 7, 47, 73, 17, 31, 13, 79, 11, 7, 173, 29, 11, 179, 61, 31, 7, 191

Offset: 2

Clark Kimberling

nonn

Baker & Harman (1998) show that there are infinitely many n such that a(n) > prime(n)^0.677. This improves on earlier work of Goldfeld, Hooley, Fouvry, Deshouillers, Iwaniec, Motohashi, et al.
Fouvry shows that a(n) > prime(n)^0.6683 for a positive proportion of members of this sequence. See Fouvry and also Baker & Harman (1996) which corrected an error in the former work.
The record values are the Sophie Germain primes A005384. - Daniel Suteu, May 09 2017
Conjecture: every prime is in the sequence. Cf. A035095 (see my comment). - Thomas Ordowski, Aug 06 2017
a(n) is 2 for n in A159611, and is at most 3 for n in A174099. Conjecture: liminf a(n) = 3. - Jeppe Stig Nielsen, Jul 04 2020

T. D. Noe, Table of n, a(n) for n = 2..10000
R. C. Baker and G. Harman, The Brun-Titchmarsh theorem on average, Analytic Number Theory (Proceedings in honor of Heini Halberstam), Birkhäuser, Boston, 1996, pp. 39-103.
R. Baker and G. Harman, Shifted primes without large prime factors, Acta Arithmetica 83 (1998), pp. 331-361.
Étienne Fouvry, Théorème de Brun-Titchmarsh; application au théorème de Fermat, Invent. Math 79 (1985), 383-407.
D. M. Goldfeld, On the number of primes p for which p + a has a large prime factor, Mathematika 16 (1969), pp. 23-27.
R. R. Hall, Some properties of the sequence {p-1}, Acta Arith. 28 (1975/76), 101-105.
G. Harman, On the greatest prime factor of p-1 with effective constants

Cf. A006093, A006530.

Cf. A005384, A035095, A159611, A174099.

A023503 := proc(n)
    A006530(ithprime(n)-1) ;
end proc:
seq( A023503(n),n=2..80) ; # R. J. Mathar, Sep 07 2016

Table[FactorInteger[Prime[n] - 1][[-1, 1]], {n, 2, 100}] (* T. D. Noe, Jun 08 2011 *)

a(n) = vecmax(factor(prime(n)-1)[,1]); \\ Michel Marcus, Aug 15 2015

a(n) = A006530(A006093(n)). - Michel Marcus, Aug 15 2015

Comments, references, and links from Charles R Greathouse IV, Mar 04 2011

A098006 (p-1)/2 - phi(p-1) as p runs through the odd primes.

Original entry on oeis.org

0, 0, 1, 1, 2, 0, 3, 1, 2, 7, 6, 4, 9, 1, 2, 1, 14, 13, 11, 12, 15, 1, 4, 16, 10, 19, 1, 18, 8, 27, 17, 4, 25, 2, 35, 30, 27, 1, 2, 1, 42, 23, 32, 14, 39, 57, 39, 1, 42, 4, 23, 56, 25, 0, 1, 2, 63, 50, 44, 49, 2, 57, 35, 60, 2, 85, 72, 1, 62, 16, 1, 63, 66, 81, 1, 2, 78, 40, 76, 29, 114, 47

Offset: 2

N. J. A. Sloane, Sep 08 2004

In the Luca-Walsh paper it is shown that there are infinitely many numbers not in this sequence. See A098047.

a(n)=0 for Fermat primes (A019434). a(n)=1 for safe primes (A005385). a(n)=2 for A090866. The least prime p for which (p-1)/2-phi(p-1)=n or 0 if there is no such prime is given by A134765(n). Sequence A134854(k) gives the least prime for which a(n)=2^(k-1). For k not a power of 2, it can be shown that if k is in this sequence, then it appears for a prime p <= 1+k^2. - T. D. Noe, Nov 13 2007

J. Browkin and A. Schinzel, On integers not of the form n-phi(n), Colloq. Math., 68 (1995), 55-58.
F. Luca and P. G. Walsh, On the number of nonquadratic residues which are not primitive roots, Colloq. Math., 100 (2004), 91-93.

T. D. Noe, Table of n, a(n) for n = 2..10000
T. D. Noe, Finding primes p for which (p-1)/2 - phi(p-1) = k

Cf. A000010, A051953, A098047, A176095 (p runs through the odd numbers).

a098006 n = a005097 (n-1) - a000010 (a006093 n)
-- Reinhard Zumkeller, Mar 26 2013

[(NthPrime(n)-1)/2 - EulerPhi(NthPrime(n)-1): n in [2..100]]; // Vincenzo Librandi, Jan 10 2017

A098006 := proc(n)
    local p;
    p := ithprime(n+1) ;
    (p-1)/2-numtheory[phi](p-1) ;
end proc:
seq(A098006(n),n=1..30) ; # R. J. Mathar, Jan 09 2017

Table[(Prime[n] - 1)/2 - EulerPhi[Prime[n] - 1], {n, 2, 85}] (* Robert G. Wilson v, Sep 09 2004 *)
Table[(n-1)/2-EulerPhi[n-1],{n,Prime[Range[2,100]]}] (* Harvey P. Dale, Oct 23 2016 *)

forprime(p=3,1e3,print1(p\2-eulerphi(p-1)", ")) \\ Charles R Greathouse IV, Feb 04 2013

a(n) = A005097(n-1) - A000010(A006093(n)); a(A159611(n)) = 0. - Reinhard Zumkeller, Mar 26 2013

A335432 Number of anti-run permutations of the prime indices of Mersenne numbers A000225(n) = 2^n - 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 6, 2, 6, 2, 36, 1, 6, 6, 24, 1, 24, 1, 240, 6, 24, 2, 1800, 6, 6, 6, 720, 6, 1800, 1, 120, 24, 6, 24, 282240, 2, 6, 24, 15120, 2, 5760, 6, 5040, 720, 24, 6, 1451520, 2, 5040, 120, 5040, 6, 1800, 720, 40320, 24, 720, 2, 1117670400, 1, 6, 1800, 5040, 6

Offset: 1

Gus Wiseman, Jul 02 2020

nonn

An anti-run is a sequence with no adjacent equal parts.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(1) = 1 through a(10) = 6 permutations:
  ()  (2)  (4)  (2,3)  (11)  (2,4,2)  (31)  (2,3,7)  (21,4)  (11,2,5)
                (3,2)                       (2,7,3)  (4,21)  (11,5,2)
                                            (3,2,7)          (2,11,5)
                                            (3,7,2)          (2,5,11)
                                            (7,2,3)          (5,11,2)
                                            (7,3,2)          (5,2,11)

Andrew Howroyd, Table of n, a(n) for n = 1..200

The version for factorial numbers is A335407.
Anti-run compositions are A003242.
Anti-run patterns are A005649.
Permutations of prime indices are A008480.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Strict permutations of prime indices are A335489.
Cf. A056239, A106351, A112798, A114938, A278990, A292884, A325535, A335125.
Mersenne numbers: A000225, A046051, A046800, A046801, A049093, A059305, A159611, A325610, A325611, A325612, A325625.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[2^n-1]],!MatchQ[#,{_,x_,x_,_}]&]],{n,0,30}]

\\ See A335452 for count.
a(n) = {count(factor(2^n-1)[,2])} \\ Andrew Howroyd, Feb 03 2021

a(n) = A335452(A000225(n)).

Terms a(51) and beyond from Andrew Howroyd, Feb 03 2021

A378616 Greatest non prime power <= prime(n).

Original entry on oeis.org

1, 1, 1, 6, 10, 12, 15, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 255, 262, 268, 270

Offset: 1

Gus Wiseman, Dec 06 2024

nonn

Conjecture: Equal to A006093(n) = prime(n) - 1 except at terms of A159611.

			The first number line below shows the non prime powers. The second shows the primes:
--1-------------6----------10----12----14-15-------18----20-21-22----24--
=====2==3====5=====7==========11====13==========17====19==========23=====

For nonprime instead of non prime power we have A156037.
Restriction of A378367.
Lengths are A378615.
For nonsquarefree: A378032 (diffs A378034), restriction of A378033 (diffs A378036).
A000040 lists the primes, differences A001223
A000961 and A246655 list the prime powers, differences A057820.
A024619 lists the non prime powers, differences A375735, seconds A376599.
A080101 counts prime powers between primes (exclusive), inclusive A366833.
A361102 lists the non powers of primes, differences A375708.
Prime powers between primes:
- A377057 positive
- A377286 zero
- A377287 one
- A377288 two
Cf. A006093, A053607, A143731, A159611, A304521, A343249, A345531, A356068, A368748, A377281, A377289, A377703, A377781.

Table[Max[Select[Range[Prime[n]],Not@*PrimePowerQ]],{n,100}]

A343557 Indices of the prime factors of Fermat numbers in the sequence of primes.

Original entry on oeis.org

2, 3, 7, 55, 116, 6543, 10847, 23974, 27567, 76709, 177975, 457523, 887643, 1625567, 2751966, 3772007, 9385401, 42401669, 61136051, 301137372, 2946723445, 7632981296, 24728168164, 98261951745, 99582868271, 159657063059, 231641062432, 851793186025, 870658222248

Offset: 1

Lorenzo Sauras Altuzarra, Apr 28 2021

nonn

			A000040(a(5)) = A000040(116) = 641 = A023394(5).

Cf. A000040 (primes), A000720 (primepi), A023394 (prime factors of Fermat primes).

Supersequence of A159611.

q:=n->(irem(2^(2^padic:-ordp(ithprime(n)-1, 2))-1, ithprime(n)) = 0):
select(q, [$1..10^5])[]; # Lorenzo Sauras Altuzarra, Feb 20 2023

is_a023394(p)=p>2 && Mod(2,p)^lift(Mod(2,znorder(Mod(2,p)))^p)==1 && isprime(p) \\ after Charles R Greathouse IV in A023394
my(i=1); forprime(p=1, , if(is_a023394(p), print1(i, ", ")); i++) \\ Felix Fröhlich, Apr 30 2021

a(n) = A000720(A023394(n)).

A000040(a(n)) = A023394(n).

More terms from Michel Marcus, Apr 29 2021

More terms from Amiram Eldar, Apr 29 2021

A173995 Continued fraction expansion of sum of reciprocals of Fermat primes.

Original entry on oeis.org

0, 1, 1, 2, 9, 1, 3, 5, 1, 2, 1, 1, 1, 1, 3, 1, 7, 1, 31, 1, 2, 4, 5

Offset: 1

Jonathan Vos Post, Mar 04 2010

If there are only five Fermat primes, a(24) = 2 is the last term of this sequence. Otherwise, a(24) = a(25) = 1 and a(26) is large (billions of digits).

This sequence is finite if and only if A019434 is finite.

			(1/3) + (1/5) + (1/17) + (1/257) + (1/65537) = 2560071829/4294967295 = 0 + 1/1+ 1/1+ 1/2+ 1/9+ 1/1+ 1/3+ 1/5+ 1/1+ 1/2+ 1/1+ 1/1+ 1/1+ 1/1+ 1/3+ 1/1+ 1/7+ 1/1+ 1/31+ 1/1+ 1/2+ 1/4+ 1/5+ 1/2.

S. W. Golomb, Irrationality of the sum of reciprocals of fermat numbers and other functions, NASA Technical Report 19630013175, Accession ID 63N23055, Contract/grant NAS7-100, 4 pp., Jet Propulsion Laboratory, Jan 01 1962.

Cf. A019434, A000215, A159611, A173898 (sum of reciprocals of Mersenne primes), A007400.

(* Assuming 65537 is the largest Fermat prime *) ContinuedFraction[Sum[1/(2^(2^n) + 1), {n, 0, 4}]] (* Alonso del Arte, Apr 21 2013 *)

Continued fraction of Sum_{i >= 1} 1/A019434(i).

Sequence corrected and comments added by Charles R Greathouse IV, Feb 04 2011

cp's OEIS Frontend

A019434 Fermat primes: primes of the form 2^(2^k) + 1, for some k >= 0.

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A014499 Number of 1's in binary representation of n-th prime.

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A023503 Greatest prime divisor of prime(n) - 1.

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A098006 (p-1)/2 - phi(p-1) as p runs through the odd primes.

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A335432 Number of anti-run permutations of the prime indices of Mersenne numbers A000225(n) = 2^n - 1.

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A378616 Greatest non prime power <= prime(n).

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