A006093 -id:A006093 - cp's OEIS frontend

A080101 Number of prime powers in all composite numbers between n-th prime and next prime.

0, 1, 0, 2, 0, 1, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Jan 28 2003

nonn

The maximum value of terms in the sequence, through the (10^5)th term, is 2. - Harvey P. Dale, Aug 24 2014

This is conjectured to be the maximum, see also A366833. - Gus Wiseman, Nov 06 2024

			There are two prime powers between 2179 = A000040(327) and 2203 = A000040(328): 2187 = 3^7 and 2197 = 13^3, therefore a(327) = 2, A080102(327) = 2187 and A080103(327) = 2197.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A080102, A080103, A025475, A000961.
For powers of 2 instead of primes we have A244508, see also A013597, A014210, A014234, A304521.
Adding one gives A366833.
For non-prime-powers instead of prime-powers we have A368748.
Positions of positive terms are A377057, primes A053607.
Positions of 0 are A377286.
Positions of 1 are A377287.
Positions of 2 are A377288, primes A053706.
For perfect-powers (instead of prime-powers) we have A377432.
A000015 gives the least prime-power >= n, difference A377282.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820, seconds A376596.
A031218 gives the greatest prime-power <= n, difference A276781.
A046933(n) counts the interval from A008864(n) to A006093(n+1).
A065514 gives the greatest prime-power < prime(n), difference A377289.
A246655 lists the prime-powers not including 1, complement A361102.
A345531 gives the least prime-power > prime(n), difference A377281.
Cf. A001597, A002808, A024619, A065890, A182908, A224363, A376597, A377051, A377054, A377436.

a := proc(n) local c, k, p: c, p := 0, ithprime(n): for k from p+1 to nextprime(p)-1 do if nops(numtheory:-factorset(k)) = 1 then c := c+1: fi: od: c: end:
seq(a(n), n = 1 .. 105); # Lorenzo Sauras Altuzarra, Jul 08 2022

prpwQ[n_]:=Module[{fi=FactorInteger[n]},Length[fi]==1&&fi[[1,2]]>1]; nn=600;With[{pwrs=Table[If[prpwQ[n],1,0],{n,nn}]},Table[Total[ Take[ pwrs,{Prime[n],Prime[n+1]}]],{n,PrimePi[nn]-1}]] (* Harvey P. Dale, Aug 24 2014 *)
Table[Length[Select[Range[Prime[n]+1,Prime[n+1]-1],PrimePowerQ]],{n,30}] (* Gus Wiseman, Nov 06 2024 *)

a(n) = A366833(n) - 1. - Gus Wiseman, Nov 06 2024

A075526 a(n) = A008578(n+2) - A008578(n+1).

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4, 12, 8, 4, 8, 4, 6

Offset: 0

Reinhard Zumkeller, Sep 22 2002

nonn

n appears this number of times in A000720. - Lekraj Beedassy, Jun 19 2006
a(0) = 1, for n >= 1: a(n) = differences between consecutive primes (A001223(n)) = A158611(n+2) - A158611(n+1). Partial sums give A006093 (shifted). - Jaroslav Krizek, Aug 04 2009
First differences of noncomposite numbers. - Juri-Stepan Gerasimov, Feb 17 2010
This is 1 together with A001223. A054541 is 2 together with A001223. A125266 is 3 together with A001223. - Omar E. Pol, Nov 01 2013

Cf. A000040, A001223, A075527, A158611.

Prime[Range[100]] // Differences // Prepend[#, 1]& (* Jean-François Alcover, Dec 11 2021 *)

a(n) = A001223(n) for n>0.

Correction for change of offset in A158611 and A008578 in Aug 2009 Jaroslav Krizek, Jan 27 2010

A008330 phi(p-1), as p runs through the primes.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 8, 6, 10, 12, 8, 12, 16, 12, 22, 24, 28, 16, 20, 24, 24, 24, 40, 40, 32, 40, 32, 52, 36, 48, 36, 48, 64, 44, 72, 40, 48, 54, 82, 84, 88, 48, 72, 64, 84, 60, 48, 72, 112, 72, 112, 96, 64, 100, 128, 130, 132, 72, 88, 96, 92, 144, 96, 120, 96, 156, 80, 96, 172, 112

Offset: 1

N. J. A. Sloane

Number of primitive roots in the field with p elements.
Kátai proves that phi(p-1)/(p-1) has a continuous distribution function. - Charles R Greathouse IV, Jul 15 2013
For odd primes p, phi(p-1)<=(p-1)/2 since p has phi(p-1) primitive roots and (p-1)/2 quadratic residues and no primitive root is a quadratic residue. - Geoffrey Critzer, Apr 18 2015

D. H. Lehmer and Emma Lehmer, "Heuristics Anyone?", in: G. Szegö et al. (eds.), Studies in Mathematical Analysis and Related Topics: Essays in Honor of George Pólya, Stanford University Press, 1962, pp. 202-210.

T. D. Noe, Table of n, a(n) for n = 1..10000
P. Erdős, On the density of some sequences of numbers, III., J. London Math. Soc. 13 (1938), pp. 119-127.
Imre Kátai, On distribution of arithmetical functions on the set prime plus one, Compositio Math. 19 (1968), pp. 278-289.
S. S. Pillai, On the sum function connected with primitive roots, Proceedings of the Indian Academy of Sciences - Section A, Vol. 13. No. 6 (1941), 526-529; alternative link.
I. J. Schoenberg, Über die asymptotische Verteilung reeller Zahlen mod 1, Mathematische Zeitschrift 28:1 (1928), pp. 171-199.
P. J. Stephens, An average result for Artin's conjecture, Mathematika, Vol. 16, No. 2 (1969), pp. 178-188.

Cf. A000010, A005596, A241194, A241195 (fraction phi(p-1)/(p-1)), A338364 (partial products).

[EulerPhi(NthPrime(n)-1): n in [1..80]]; // Vincenzo Librandi, Apr 06 2015

A008330 := proc(n)
    numtheory[phi](ithprime(n)-1) ;
end proc:
seq(A008330(n),n=1..100) ;

Table[ EulerPhi[ Prime@n - 1], {n, 70}] (* Robert G. Wilson v, Dec 17 2005 *)

a(n)=eulerphi(prime(n)-1) \\ Charles R Greathouse IV, Dec 08 2011

a(n) = phi(phi(prime(n))). - Robert G. Wilson v, Dec 26 2015
a(n) = phi(A006093(n)). - Michel Marcus, Dec 27 2015
Sum_{k; prime(k) <= x} a(k)/(prime(k)-1) = A * li(x) + O(x/log(x)^D), where A is Artin's constant (A005596), li(x) is the logarithmic integral, and D > 1 (Pillai, 1941; Lehmer and Lehmer 1962; Stephens, 1969). - Amiram Eldar, Jul 23 2025

A366833 Number of times n appears in A362965 (number of primes <= the n-th prime power).

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Paolo Xausa, Oct 25 2023

nonn

Conjecture: a(n) can be only 1, 2, or 3 (with the first occurrences of 3 appearing at n = 4, 9, 30, 327 and 3512).

One less than the number of prime powers between prime(n) and prime(n+1), inclusive. - Gus Wiseman, Jan 09 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000
Paolo Xausa, 1200 X 1200 raster image of a(n), n = 1..1440000, read left to right, top to bottom, showing a(n) = 1 in blue, a(n) = 2 in white and a(n) = 3 in red.

Run lengths of A362965.
Subtracting one gives A080101.
For non prime powers we have A368748.
Positions of terms > 1 are A377057.
Positions of 1 are A377286.
Positions of 2 are A377287.
For perfect powers we have A377432.
For squarefree we have A373198.
A000015 gives the least prime power >= n, difference A377282.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A024619 and A361102 list the non prime powers, differences A375708 and A375735.
A031218 gives the greatest prime power <= n, difference A276781.
A046933(n) counts the interval from A008864(n) to A006093(n+1).
A246655 lists the prime powers not including 1.
A366835 counts primes between prime powers.
Cf. A053607, A053706, A065514, A068435, A080102, A304521, A345531, A377289.
Cf. A024620, A027883, A067871, A075526, A080769, A151800, A377436, A379157.

With[{upto=1000},Map[Length,Most[Split[PrimePi[Select[Range[upto],PrimePowerQ]]]]]] (* Considers prime powers up to 1000 *)

a(n) = A080101(n) + 1. - Gus Wiseman, Jan 09 2025

A246392 Numbers n such that Phi(10, n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

2, 3, 5, 10, 11, 12, 16, 20, 21, 22, 33, 37, 38, 43, 47, 48, 55, 71, 75, 76, 80, 81, 111, 121, 126, 131, 133, 135, 136, 141, 155, 157, 158, 165, 176, 177, 180, 203, 223, 242, 245, 251, 253, 256, 257, 258, 265, 268, 276, 286, 290, 297, 307, 322, 323, 342, 361, 363, 366, 375, 377, 385, 388, 396, 411

Offset: 1

Eric Chen, Nov 13 2014

nonn

Numbers n such that (n^5+1)/(n+1) is prime, or numbers n such that A060884(n) is prime.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 893 terms from Robert Price)
OEIS Wiki, Cyclotomic Polynomials at x=sigma(n)

Cf. A008864 (1), A006093 (2), A002384 (3), A005574 (4), A049409 (5), A055494 (6), A100330 (7), A000068 (8), A153439 (9), this sequence (10), A162862 (11), A246397 (12), A217070 (13), A006314 (16), A217071 (17), A164989 (18), A217072 (19), A217073 (23), A153440 (27), A217074 (29), A217075 (31), A006313 (32), A097475 (36), A217076 (37), A217077 (41), A217078 (43), A217079 (47), A217080 (53), A217081 (59), A217082 (61), A006315 (64), A217083 (67), A217084 (71), A217085 (73), A217086 (79), A153441 (81), A217087 (83), A217088 (89), A217089 (97), A006316 (128), A153442 (243), A056994 (256), A056995 (512), A057465 (1024), A057002 (2048), A088361 (4096), A088362 (8192), A226528 (16384), A226529 (32768), A226530 (65536).

Cf. A060884, A085398, A259257.

[n: n in [1..500]| IsPrime((n^5+1) div (n+1))]; // Vincenzo Librandi, Nov 14 2014

A246392:=n->`if`(isprime((n^5+1)/(n+1)),n,NULL): seq(A246392(n), n=1..500); # Wesley Ivan Hurt, Nov 15 2014

Select[Range[700], PrimeQ[(#^5 + 1) / (# + 1)] &] (* Vincenzo Librandi, Nov 14 2014 *)

for(n=1,10^3,if(isprime(polcyclo(10,n)),print1(n,", "))); \\ Joerg Arndt, Nov 13 2014

A001747 2 together with primes multiplied by 2.

Original entry on oeis.org

2, 4, 6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502

Offset: 1

N. J. A. Sloane

When supplemented with 8, may be considered the "even primes", since these are the even numbers n = 2k which are divisible just by 1, 2, k and 2k. - Louis Zuckerman (louis(AT)trapezoid.com), Sep 12 2000
Sequence gives solutions of sigma(n) - phi(n) = n + tau(n) where tau(n) is the number of divisors of n.
Numbers n such that sigma(n) = 3*(n - phi(n)).
Except for 2, orders of non-cyclic groups k (in A060679(n)) such that x^k==1 (mod k) has only 1 solution 2<=x<=k. - Benoit Cloitre, May 10 2002
Numbers n such that A092673(n) = 2. - Jon Perry, Mar 02 2004
Except for initial terms, this sequence = A073582 = A074845 = A077017. Starting with the term 10, they are identical. - Robert G. Wilson v, Jun 15 2004
Together with 8 and 16, even numbers n such that n^2 does not divide (n/2)!. - Arkadiusz Wesolowski, Jul 16 2011
Twice noncomposite numbers. - Omar E. Pol, Jan 30 2012

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A060679, A009530, A098764.

Equals {2} UNION {A100484}.

Concatenation([2], List([1..60], n-> 2*Primes[n])); # G. C. Greubel, May 18 2019

[2] cat [2*NthPrime(n): n in [1..60]]; // G. C. Greubel, May 18 2019

Join[{2},2*Prime[Range[60]]] (* Harvey P. Dale, Jul 23 2013 *)

print1(2);forprime(p=2,97,print1(", "2*p)) \\ Charles R Greathouse IV, Jan 31 2012

[2]+[2*nth_prime(n) for n in (1..60)] # G. C. Greubel, May 18 2019

a(n) = A001043(n) - A001223(n+1), except for initial term.
a(n) = A116366(n-2,n-2) for n>2. - Reinhard Zumkeller, Feb 06 2006
A006093(n) = A143201(a(n+1)) for n>1. - Reinhard Zumkeller, Aug 12 2008
a(n) = 2*A008578(n). - Omar E. Pol, Jan 30 2012, and Reinhard Zumkeller, Feb 16 2012

A056993 a(n) is the smallest k >= 2 such that k^(2^n)+1 is prime, or -1 if no such k exists.

Original entry on oeis.org

2, 2, 2, 2, 2, 30, 102, 120, 278, 46, 824, 150, 1534, 30406, 67234, 70906, 48594, 62722, 24518, 75898, 919444

Offset: 0

Robert G. Wilson v, Sep 06 2000

Smallest base value yielding generalized Fermat primes. - Hugo Pfoertner, Jul 01 2003
The first 5 terms correspond with the known (ordinary) Fermat primes. A probable candidate for the next entry is 62722^131072+1, discovered by Michael Angel in 2003. It has 628808 decimal digits. - Hugo Pfoertner, Jul 01 2003
For any n, a(n+1) >= sqrt(a(n)), because k^(2^(n+1))+1 = (k^2)^(2^n)+1. - Jeppe Stig Nielsen, Sep 16 2015
Does the sequence contain any perfect squares? If a(n) is a perfect square, then a(n+1) = sqrt(a(n)). - Jeppe Stig Nielsen, Sep 16 2015
If for a particular n, a(n) exists, then a(i) exist for all i=0,1,2,...,n. No proof is known that this sequence is infinite. Such a result would clearly imply the infinitude of A002496. - Jeppe Stig Nielsen, Sep 18 2015
919444 is a candidate for a(20). See Zimmermann link. -  Serge Batalov, Sep 02 2017
Now PrimeGrid has tested and double checked all b^(2^20) + 1 with b < 919444, so we have proof that a(20) = 919444. - Jeppe Stig Nielsen, Dec 30 2017

			The primes are 2^(2^0) + 1 = 3, 2^(2^1) + 1 = 5, 2^(2^2) + 1 = 17, 2^(2^3) + 1 = 257, 2^(2^4) + 1 = 65537, 30^(2^5) + 1, 102^(2^6) + 1, ....

Yves Gallot, Generalized Fermat Prime Search
Lucile and Yves Gallot, Generalized Fermat Prime Search
Michael Goetz, id=103235 of Top 5000 Primes
Luke Harmon, Gaetan Delavignette, Arnab Roy, and David Silva, PIE: p-adic Encoding for High-Precision Arithmetic in Homomorphic Encryption, Cryptology ePrint Archive 2023/700.
Stephen Scott, id=84401 of Top 5000 Primes
Sylvanus A. Zimmerman, PrimeGrid’s Generalized Fermat Prime Search

Cf. A006093, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.

Cf. A019434 (Fermat primes).

f[n_] := (p = 2^n; k = 2; While[cp = k^p + 1; !PrimeQ@cp, k++ ]; k); Do[ Print[{n, f@n}], {n, 0, 17}] (* Lei Zhou, Feb 21 2005 *)

a(n)=my(k=2);while(!isprime(k^(2^n)+1),k++);k \\ Anders Hellström, Sep 16 2015

a(n) = A085398(2^(n+1)). - Jianing Song, Jun 13 2022

from Robert G. Wilson v, Oct 30 2000
from Jeppe Stig Nielsen, Aug 07 2005
and 75898 from Lei Zhou, Feb 01 2012
919444 from Jeppe Stig Nielsen, Dec 30 2017

A377432 Number of perfect-powers x in the range prime(n) < x < prime(n+1).

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 0, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 1

Gus Wiseman, Oct 31 2024

nonn

Perfect-powers (A001597) are numbers with a proper integer root, complement A007916.

			Between prime(4) = 7 and prime(5) = 11 we have perfect-powers 8 and 9, so a(4) = 2.

For prime-powers instead of perfect-powers we have A080101.
Non-perfect-powers in the same range are counted by A377433.
Positions of 1 are A377434.
Positions of 0 are A377436.
Positions of terms > 1 are A377466.
For powers of 2 instead of primes we have A377467, for prime-powers A244508.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect-powers, differences A053289.
A007916 lists the non-perfect-powers, differences A375706.
A046933 counts the interval from A008864(n) to A006093(n+1).
A081676 gives the greatest perfect-power <= n.
A246655 lists the prime-powers not including 1, complement A361102.
A366833 counts prime-powers between primes, see A053706, A053607, A304521, A377286.
A377468 gives the least perfect-power > n.
Cf. A000015, A002808, A024619, A031218, A053707, A064113, A065514, A065890, A080769, A377051, A377282.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[Length[Select[Range[Prime[n]+1, Prime[n+1]-1],perpowQ]],{n,100}]

a(n) + A377433(n) = A046933(n) = prime(n+1) - prime(n) - 1.

A057002 Numbers n such that n^1024 + 1 is prime (a generalized Fermat prime).

Original entry on oeis.org

1, 824, 1476, 1632, 2462, 2484, 2520, 3064, 3402, 3820, 4026, 6640, 7026, 7158, 9070, 12202, 12548, 12994, 13042, 15358, 17646, 17670, 18336, 19564, 20624, 22500, 24126, 26132, 26188, 26240, 29074, 29658, 30778, 31126, 32244, 33044, 34016

Offset: 1

Robert G. Wilson v, Sep 09 2000

nonn

This sequence is infinite under Bunyakovsky's conjecture. - Charles R Greathouse IV, Apr 26 2012

T. D. Noe, Table of n, a(n) for n = 1..1000 (from Yves Gallot)
Yves Gallot, Generalized Fermat Prime Search
Jeppe Stig Nielsen, Generalized Fermat Primes sorted by base.
Eric Weisstein's World of Mathematics, Generalized Fermat Number

Other sequences of numbers n such that n^(2^k)+1 is prime for fixed k: A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.

Cf. A006093.

Do[ k = 1; While[ PowerMod[ n, 1024, 2*k*1024 + 1 ] != 2*k*1024 && k < 2*10^6, k++ ]; If[ k == 2*10^6 && PrimeQ[ n^1024 + 1 ], Print[ n ] ], {n, 2, 13954, 2} ]
Do[If[PrimeQ[n^1024 + 1], Print[n], ## &[]], {n, 1, 100}] (* Includes first term and runs faster, Daniel Jolly, Nov 04 2014 *)

isA057002(n) = isprime(n^1024+1) \\ Michael B. Porter, Apr 03 2010

More terms from Jeppe Stig Nielsen, Sep 27 2003

Edited at the suggestion of T. D. Noe by N. J. A. Sloane, May 14 2008

A084920 a(n) = (prime(n)-1)*(prime(n)+1).

Original entry on oeis.org

3, 8, 24, 48, 120, 168, 288, 360, 528, 840, 960, 1368, 1680, 1848, 2208, 2808, 3480, 3720, 4488, 5040, 5328, 6240, 6888, 7920, 9408, 10200, 10608, 11448, 11880, 12768, 16128, 17160, 18768, 19320, 22200, 22800, 24648, 26568, 27888, 29928

Offset: 1

Reinhard Zumkeller, Jun 11 2003

Squares of primes minus 1. - Wesley Ivan Hurt, Oct 11 2013

Integers k for which there exist exactly two positive integers b such that (k+1)/(b+1) is an integer. - Benedict W. J. Irwin, Jul 26 2016

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Barry Brent, On the constant terms of certain meromorphic modular forms for Hecke groups, arXiv:2212.12515 [math.NT], 2022.
Barry Brent, On the Constant Terms of Certain Laurent Series, Preprints (2023) 2023061164.
Nik Lygeros and Olivier Rozier, A new solution to the equation tau(p) == 0 (mod p), J. Int. Seq. 13 (2010), Article 10.7.4.
Lực Ta, Enumeration of virtual quandles up to isomorphism, arXiv:2506.16536 [math.GT], 2025. See p. 6.

Cf. A000040, A005563, A049001, A154945, A166010, A182200, A182174.
Cf. A006093, A008864, A084921, A084922.
Cf. A066872, A001248, A024702, A013661, A065469.

a084920 n = (p - 1) * (p + 1) where p = a000040 n
-- Reinhard Zumkeller, Aug 27 2013

[p^2-1: p in PrimesUpTo(200)]; // Vincenzo Librandi, Mar 30 2015

A084920:=n->ithprime(n)^2-1; seq(A084920(k), k=1..50); # Wesley Ivan Hurt, Oct 11 2013

Table[Prime[n]^2 - 1, {n, 50}] (* Wesley Ivan Hurt, Oct 11 2013 *)
Prime[Range[50]]^2-1 (* Harvey P. Dale, Oct 02 2021 *)

a(n) = (prime(n)-1)*(prime(n)+1); \\ Michel Marcus, Jul 28 2016

[(p-1)*(p+1) for p in primes(200)] # Bruno Berselli, Mar 30 2015

a(n) = A006093(n) * A008864(n);
a(n) = A084921(n)*2, for n > 1; a(n) = A084922(n)*6, for n > 2.
Product_{n > 0} a(n)/A066872(n) = 2/5. a(n) = A001248(n) - 1. - R. J. Mathar, Feb 01 2009
a(n) = prime(n)^2 - 1 = A001248(n) - 1. - Vladimir Joseph Stephan Orlovsky, Oct 17 2009
a(n) ~ n^2*log(n)^2. - Ilya Gutkovskiy, Jul 28 2016
a(n) = (1/2) * Sum_{|k|<=2*sqrt(p)} k^2*H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime. - Seiichi Manyama, Dec 31 2017
a(n) = 24 * A024702(n) for n > 2. - Jianing Song, Apr 28 2019
Sum_{n>=1} 1/a(n) = A154945. - Amiram Eldar, Nov 09 2020
From Amiram Eldar, Nov 07 2022: (Start)
Product_{n>=1} (1 + 1/a(n)) = Pi^2/6 (A013661).
Product_{n>=1} (1 - 1/a(n)) = A065469. (End)

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A080101 Number of prime powers in all composite numbers between n-th prime and next prime.

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A075526 a(n) = A008578(n+2) - A008578(n+1).

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A008330 phi(p-1), as p runs through the primes.

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A366833 Number of times n appears in A362965 (number of primes <= the n-th prime power).

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A246392 Numbers n such that Phi(10, n) is prime, where Phi is the cyclotomic polynomial.

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A001747 2 together with primes multiplied by 2.

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A056993 a(n) is the smallest k >= 2 such that k^(2^n)+1 is prime, or -1 if no such k exists.

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