A084920 -id:A084920 - cp's OEIS frontend

A006093 a(n) = prime(n) - 1.

1, 2, 4, 6, 10, 12, 16, 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, 256, 262, 268, 270

Offset: 1

N. J. A. Sloane

These are also the numbers that cannot be written as i*j + i + j (i,j >= 1). - Rainer Rosenthal, Jun 24 2001; Henry Bottomley, Jul 06 2002
The values of k for which Sum_{j=0..n} (-1)^j*binomial(k, j)*binomial(k-1-j, n-j)/(j+1) produces an integer for all n such that n < k. Setting k=10 yields [0, 1, 4, 11, 19, 23, 19, 11, 4, 1, 0] for n = [-1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], so 10 is in the sequence. Setting k=3 yields [0, 1, 1/2, 1/2] for n = [-1, 0, 1, 2], so 3 is not in the sequence. - Dug Eichelberger (dug(AT)mit.edu), May 14 2001
n such that x^n + x^(n-1) + x^(n-2) + ... + x + 1 is irreducible. - Robert G. Wilson v, Jun 22 2002
Records for Euler totient function phi.
Together with 0, n such that (n+1) divides (n!+1). - Benoit Cloitre, Aug 20 2002; corrected by Charles R Greathouse IV, Apr 20 2010
n such that phi(n^2) = phi(n^2 + n). - Jon Perry, Feb 19 2004
Numbers having only the trivial perfect partition consisting of a(n) 1's. - Lekraj Beedassy, Jul 23 2006
Numbers n such that the sequence {binomial coefficient C(k,n), k >= n } contains exactly one prime. - Artur Jasinski, Dec 02 2007
Record values of A143201: a(n) = A143201(A001747(n+1)) for n > 1. - Reinhard Zumkeller, Aug 12 2008
From Reinhard Zumkeller, Jul 10 2009: (Start)
The first N terms can be generated by the following sieving process:
  start with {1, 2, 3, 4, ..., N - 1, N};
  for i := 1 until SQRT(N) do
  (if (i is not striked out) then
  (for j := 2 * i + 1 step i + 1 until N do
  (strike j from the list)));
  remaining numbers = {a(n): a(n) <= N}. (End)
a(n) = partial sums of A075526(n-1) = Sum_{1..n} A075526(n-1) = Sum_{1..n} (A008578(n+1) - A008578(n)) = Sum_{1..n} (A158611(n+2) - A158611(n+1)) for n >= 1. - Jaroslav Krizek, Aug 04 2009
A006093 U A072668 = A000027. - Juri-Stepan Gerasimov, Oct 22 2009
A171400(a(n)) = 1 for n <> 2: subsequence of A171401, except for a(2) = 2. - Reinhard Zumkeller, Dec 08 2009
Numerator of (1 - 1/prime(n)). - Juri-Stepan Gerasimov, Jun 05 2010
Numbers n such that A002322(n+1) = n. This statement is stronger than repeating the property of the entries in A002322, because it also says in reciprocity that this sequence here contains no numbers beyond the Carmichael numbers with that property. - Michel Lagneau, Dec 12 2010
a(n) = A192134(A095874(A000040(n))); subsequence of A192133. - Reinhard Zumkeller, Jun 26 2011
prime(a(n)) + prime(k) < prime(a(k) + k) for at least one k <= a(n): A212210(a(n),k) < 0. - Reinhard Zumkeller, May 05 2012
Except for the first term, numbers n such that the sum of first n natural numbers does not divide the product of first n natural numbers; that is, n*(n + 1)/2 does not divide n!. - Jayanta Basu, Apr 24 2013
BigOmega(a(n)) equals BigOmega(a(n)*(a(n) + 1)/2), where BigOmega = A001222. Rationale: BigOmega of the product on the right hand side factorizes as BigOmega(a/2) + Bigomega(a+1) = BigOmega(a/2) + 1 because a/2 and a + 1 are coprime, because BigOmega is additive, and because a + 1 is prime. Furthermore Bigomega(a/2) = Bigomega(a) - 1 because essentially all 'a' are even. - Irina Gerasimova, Jun 06 2013
Record values of A060681. - Omar E. Pol, Oct 26 2013
Deficiency of n-th prime. - Omar E. Pol, Jan 30 2014
Conjecture: All the sums Sum_{k=s..t} 1/a(k) with 1 <= s <= t are pairwise distinct. In general, for any integers d >= -1 and m > 0, if Sum_{k=i..j} 1/(prime(k)+d)^m = Sum_{k=s..t} 1/(prime(k)+d)^m with 0 < i <= j and 0 < s <= t then we must have (i,j) = (s,t), unless d = m = 1 and {(i,j),(s,t)} = {(4,4),(8,10)} or {(4,7),(5,10)}. (Note that 1/(prime(8)+1)+1/(prime(9)+1)+1/(prime(10)+1) = 1/(prime(4)+1) and Sum_{k=5..10} 1/(prime(k)+1) = 1/(prime(4)+1) + Sum_{k=5..7} 1/(prime(k)+1).) - Zhi-Wei Sun, Sep 09 2015
Numbers n such that (prime(i)^n + n) is divisible by (n+1), for all i >= 1, except when prime(i) = n+1. - Richard R. Forberg, Aug 11 2016
a(n) is the period of Fubini numbers (A000670) over the n-th prime. - Federico Provvedi, Nov 28 2020

Archimedeans Problems Drive, Eureka, 40 (1979), 28.
Harvey Dubner, Generalized Fermat primes, J. Recreational Math., 18 (1985): 279-280.
M. Gardner, The Colossal Book of Mathematics, pp. 31, W. W. Norton & Co., NY, 2001.
M. Gardner, Mathematical Circus, pp. 251-2, Alfred A. Knopf, NY, 1979.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
Thomas F. Bloom, Unit fractions with shifted prime denominators, arXiv:2305.02689 [math.NT], 2023.
R. P. Boas & N. J. A. Sloane, Correspondence, 1974
Harvey Dubner, Generalized Fermat primes, J. Recreational Math. 18.4 (1985-1986), 279. (Annotated scanned copy)
Armel Mercier, Problem E 3065, American Mathematical Monthly, 1984, p. 649.
Armel Mercier, S. K. Rangarajan, J. C. Binz and Dan Marcus, Problem E 3065, American Mathematical Monthly, No. 4, 1987, pp. 378.
Poo-Sung Park, Additive uniqueness of PRIMES-1 for multiplicative functions, arXiv:1708.03037 [math.NT], 2017.
J. R. Rickard and J. J. Hitchcock, Problem Drive 4, Archimedeans Problems Drive, Eureka, 40 (1979), 28-29, 40. (Annotated scanned copy)
Index entries for sequences generated by sieves

a(n) = K(n, 1) and A034693(K(n, 1)) = 1 for all n. The subscript n refers to this sequence and K(n, 1) is the index in A034693. - Labos Elemer
Cf. A000040, A034694. Different from A075728.
Complement of A072668 (composite numbers minus 1), A072670(a(n))=0.
Essentially the same as A039915.
Cf. A084920, A006093, A050997, A008864, A060800, A131991, A131992, A131993.
Cf. A101301 (partial sums), A005867 (partial products).
Column 1 of the following arrays/triangles: A087738, A249741, A352707, A378979, A379010.
The last diagonal of A162619, and of A174996, the first diagonal in A131424.
Row lengths of irregular triangles A086145, A124223, A212157.

Filtered([1..280],IsPrime)-1; # Muniru A Asiru, Nov 25 2018

a006093 = (subtract 1) . a000040  -- Reinhard Zumkeller, Mar 06 2012

[NthPrime(n)-1: n in [1..100]]; // Vincenzo Librandi, Nov 17 2015

for n from 2 to 271 do if (n! mod n^2 = n*(n-1) and (n<>4) then print(n-1) fi od; # Gary Detlefs, Sep 10 2010
# alternative
A006093 := proc(n)
    ithprime(n)-1 ;
end proc:
seq(A006093(n),n=1..100) ; # R. J. Mathar, Feb 06 2019

Table[Prime[n] - 1, {n, 1, 30}] (* Vladimir Joseph Stephan Orlovsky, Apr 27 2008 *)
a[ n_] := If[ n < 1, 0, -1 + Prime @ n] (* Michael Somos, Jul 17 2011 *)
Prime[Range[60]] - 1 (* Alonso del Arte, Oct 26 2013 *)

isA006093(n) = isprime(n+1) \\ Michael B. Porter, Apr 09 2010

A006093(n) = prime(n)-1 \\ Michael B. Porter, Apr 09 2010

\\ Sieve as described in Rainer Rosenthal's comment.
m=270;s=vector(m);for(i=1,m,for(j=i,m,k=i*j+i+j;if(k<=m,s[k]=1)));for(k=1,m,if(s[k]==0,print1(k,", "))); \\ Hugo Pfoertner, May 14 2019

from sympy import prime
for n in range(1,100): print(prime(n)-1, end=', ') # Stefano Spezia, Nov 30 2018

a(n) = (p-1)! mod p where p is the n-th prime, by Wilson's theorem. - Jonathan Sondow, Jul 13 2010
a(n) = A000010(prime(n)) = A000010(A006005(n)). - Antti Karttunen, Dec 16 2012
a(n) = A005867(n+1)/A005867(n). - Eric Desbiaux, May 07 2013
a(n) = A000040(n) - 1. - Omar E. Pol, Oct 26 2013
a(n) = A033879(A000040(n)). - Omar E. Pol, Jan 30 2014

Correction for change of offset in A158611 and A008578 in Aug 2009 Jaroslav Krizek, Jan 27 2010
Obfuscating comments removed by Joerg Arndt, Mar 11 2010
Edited by Charles R Greathouse IV, Apr 20 2010

A008864 a(n) = prime(n) + 1.

Original entry on oeis.org

3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 128, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272, 278, 282, 284

Offset: 1

N. J. A. Sloane, R. K. Guy

Sum of divisors of prime(n). - Labos Elemer, May 24 2001
For n > 1, there are a(n) more nonnegative Hurwitz quaternions than nonnegative Lipschitz quaternions, which are counted in A239396 and A239394, respectively. - T. D. Noe, Mar 31 2014
These are the numbers which are in A239708 or in A187813, but excluding the first 3 terms of A187813, i.e., a number m is a term if and only if m is a term > 2 of A187813, or m is the sum of two distinct powers of 2 such that m - 1 is prime. This means that a number m is a term if and only if m is a term > 2 such that there is no base b with a base-b digital sum of b, or b = 2 is the only base for which the base-b digital sum of m is b. a(6) is the only term such that a(n) = A187813(n); for n < 6, we have a(n) > A187813(n), and for n > 6, we have a(n) < A187813(n). - Hieronymus Fischer, Apr 10 2014
Does not contain any number of the format 1 + q + ... + q^e, q prime, e >= 2, i.e., no terms of A060800,  A131991, A131992, A131993 etc. [Proof: that requires 1 + p = 1 + q + ... + q^e, or p = q*(1 + ... + q^(e-1)). This is not solvable because the left hand side is prime, the right hand side composite.] - R. J. Mathar, Mar 15 2018
1/a(n) is the asymptotic density of numbers whose prime(n)-adic valuation is odd. - Amiram Eldar, Jan 23 2021

C. W. Trigg, Problem #1210, Series Formation, J. Rec. Math., 15 (1982), 221-222.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. P. Boas and N. J. A. Sloane, Correspondence, 1974
N. J. A. Sloane and Brady Haran, Eureka Sequences, Numberphile video (2021).

Cf. A000040, A060800, A131991, A131992, A131993, A141468.
Cf. A007953, A079696, A187813, A239703, A239708.
Column 1 of A341605, column 2 of A286623 and of A328464.
Partial sums of A125266.

a008864 = (+ 1) . a000040
-- Reinhard Zumkeller, Sep 04 2012, Oct 08 2012

[NthPrime(n)+1: n in [1..70]]; // Vincenzo Librandi, Jul 30 2016

A008864:=n->ithprime(n)+1; seq(A008864(n), n=1..50); # Wesley Ivan Hurt, Apr 11 2014

Prime[Range[70]]+1 (* Vladimir Joseph Stephan Orlovsky, Apr 27 2008 *)

forprime(p=2,1e3,print1(p+1", ")) \\ Charles R Greathouse IV, Jun 16 2011

A008864(n) = (1+prime(n)); \\ Antti Karttunen, Mar 14 2021

[nth_prime(n) +1 for n in (1..70)] # G. C. Greubel, May 20 2019

a(n) = prime(n) + 1 = A000040(n) + 1.
a(n) = A000005(A034785(n)) = A000203(A000040(n)). - Labos Elemer, May 24 2001
a(n) = A084920(n) / A006093(n). - Reinhard Zumkeller, Aug 06 2007
A239703(a(n)) <= 1. - Hieronymus Fischer, Apr 10 2014
From Ilya Gutkovskiy, Jul 30 2016: (Start)
a(n) ~ n*log(n).
Product_{n>=1} (1 + 2/(a(n)*(a(n) - 2))) = 5/2. (End)

A024702 a(n) = (prime(n)^2 - 1)/24.

Original entry on oeis.org

1, 2, 5, 7, 12, 15, 22, 35, 40, 57, 70, 77, 92, 117, 145, 155, 187, 210, 222, 260, 287, 330, 392, 425, 442, 477, 495, 532, 672, 715, 782, 805, 925, 950, 1027, 1107, 1162, 1247, 1335, 1365, 1520, 1552, 1617, 1650, 1855, 2072, 2147, 2185, 2262, 2380, 2420, 2625, 2752, 2882, 3015

Offset: 3

Clark Kimberling, Dec 11 1999

Note that p^2 - 1 is always divisible by 24 since p == 1 or 2 (mod 3), so p^2 == 1 (mod 3) and p == 1, 3, 5, or 7 (mod 8) so p^2 == 1 (mod 8). - Michael B. Porter, Sep 02 2016
For n > 3 and m > 1, a(n) = A000330(m)/(2*m + 1), where 2*m + 1 = prime(n). For example, for m = 8, 2*m + 1 = 17 = prime(7), A000330(8) = 204, 204/17 = 12 = a(7). - Richard R. Forberg, Aug 20 2013
For primes => 5, a(n) == 0 or 2 (mod 5). - Richard R. Forberg, Aug 28 2013
The only primes in this sequence are 2, 5 and 7 (checked up to n = 10^7). The set of prime factors, however, appears to include all primes. - Richard R. Forberg, Feb 28 2015
Subsequence of generalized pentagonal numbers (cf. A001318): a(n) = k_n*(3*k_n - 1)/2, for k_n in {1, -1, 2, -2, 3, -3, 4, 5, -5, -6, 7, -7, 8, 9, 10, -10, ...} = A024699(n-2)*((A000040(n) mod 6) - 3)/2, n >= 3. - Daniel Forgues, Aug 02 2016
The only primes in this sequence are indeed 2, 5 and 7. For a prime p >= 5, if both p + 1 and p - 1 contains a prime factor > 3, then (p^2 - 1)/24 = (p + 1)*(p - 1)/24 contains at least 2 prime factors, so at least one of p +  1 and p - 1 is 3-smooth. Let's call it s. Also, If (p^2 - 1)/24 is a prime, then A001222(p^2-1) = 5. Since A001222(p+1) and A001222(p-1) are both at least 2, A001222(s) <= 5 - 2 = 3. From these we can see the only possible cases are p = 7, 11 and 13. - Jianing Song, Dec 28 2018

			For n = 6, the 6th prime is 13, so a(6) = (13^2 - 1)/24 = 168/24 = 7.

Charles R Greathouse IV, Table of n, a(n) for n = 3..10000
Brady Haran and Matt Parker, Squaring Primes, Numberphile video (2018).
Carlos Rivera, Puzzle 1060. Can you find more solutions?, The Prime Puzzles and Problems Connection. [Asks for squares in this sequence]

Subsequence of generalized pentagonal numbers A001318.

Cf. A075888.

A024702:=n->(ithprime(n)^2-1)/24: seq(A024702(n), n=3..70); # Wesley Ivan Hurt, Mar 01 2015

(Prime[Range[3,100]]^2-1)/24 (* Vladimir Joseph Stephan Orlovsky, Mar 15 2011 *)

a(n)=prime(n)^2\24 \\ Charles R Greathouse IV, May 30 2013

is(n)=my(k);issquare(24*n+1,&k)&&isprime(k) \\ Charles R Greathouse IV, May 31 2013

a(n) = (A000040(n)^2 - 1)/24 = (A001248(n) - 1)/24. - Omar E. Pol, Dec 07 2011
a(n) = A005097(n-1)*A006254(n-1)/6. - Bruno Berselli, Dec 08 2011
a(n) = A084920(n)/24. - R. J. Mathar, Aug 23 2013
a(n) = A127922(n)/A000040(n) for n >= 3. - César Aguilera, Nov 01 2019

A154945 Decimal expansion of Sum_{p} 1/(p^2-1), summed over the primes p = A000040.

Original entry on oeis.org

5, 5, 1, 6, 9, 3, 2, 9, 7, 6, 5, 6, 9, 9, 9, 1, 8, 4, 4, 3, 9, 7, 3, 1, 0, 2, 3, 9, 7, 1, 3, 4, 3, 5, 7, 8, 1, 3, 1, 5, 0, 0, 3, 7, 7, 7, 7, 8, 6, 2, 8, 2, 5, 2, 2, 3, 0, 6, 1, 7, 3, 3, 4, 0, 5, 9, 5, 6, 5, 5, 9, 7, 6, 4, 1, 0, 7, 0, 6, 7, 1, 0, 7, 7, 7, 5, 0, 9, 8, 3, 1, 6, 8, 2, 7, 7, 9, 6, 0, 7, 2, 5, 0, 5, 8

Offset: 0

R. J. Mathar, Jan 17 2009

By geometric series expansion, the same as the sum over the prime zeta function at even arguments, P(2i), i=1,2,....

(Pi^2/6)*density of A190641, the numbers divisible by exactly one prime with exponent greater than 1. - Charles R Greathouse IV, Aug 02 2016

			0.551693297656999184439731023971343578131500377778628252230...

Jacques Grah, Comportement moyen du cardinal de certains ensembles de facteurs premiers, Monatsh. Math., Vol. 118 (1994), pp. 91-109, Corollary 6.1.
Carl Pomerance and Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory, Vol. 1, Iss. 1 (2011), pp. 52-66. See p. 61.
Index to constants which are prime zeta sums {0,1,1}

Cf. A000040, A000961, A056798, A084920, A085548, A085964, A085966, A085968, A152447, A154932, A190641.

digits = 105; m0 = 2 digits; Clear[rd]; rd[m_] := rd[m] = RealDigits[delta1 = Sum[PrimeZetaP[2n], {n, 1, m}] , 10, digits][[1]]; rd[m0]; rd[m = 2m0];
While[rd[m] != rd[m-m0], Print[m]; m = m+m0]; Print[N[delta1, digits]]; rd[m] (* Jean-François Alcover, Sep 11 2015, updated Mar 16 2019 *)

eps()=2.>>bitprecision(1.)
primezeta(s)=my(t=s*log(2)); sum(k=1, lambertw(t/eps())\t, moebius(k)/k*log(abs(zeta(k*s))))
sumpos(n=1,primezeta(2*n)) \\ Charles R Greathouse IV, Aug 02 2016

sumeulerrat(1/(p^2-1)) \\ Amiram Eldar, Mar 18 2021

Equals Sum_{k>=1} 1/A084920(k) = Sum_{i>=1} P(2i) = A085548+A085964+A085966+A085968+... = A152447+A085548-A154932.
Equals Sum_{k>=2} 1/A000961(k)^2 = Sum_{k>=2} 1/A056798(k). - Amiram Eldar, Sep 21 2020
Equals (A136141 + A179119)/2. - Artur Jasinski, Mar 31 2025

More digits from Jean-François Alcover, Sep 11 2015

A046970 Dirichlet inverse of the Jordan function J_2 (A007434).

Original entry on oeis.org

1, -3, -8, -3, -24, 24, -48, -3, -8, 72, -120, 24, -168, 144, 192, -3, -288, 24, -360, 72, 384, 360, -528, 24, -24, 504, -8, 144, -840, -576, -960, -3, 960, 864, 1152, 24, -1368, 1080, 1344, 72, -1680, -1152, -1848, 360, 192, 1584, -2208, 24, -48, 72, 2304, 504, -2808, 24, 2880, 144, 2880, 2520, -3480, -576

Offset: 1

Douglas Stoll, dougstoll(AT)email.msn.com

B(n+2) = -B(n)*((n+2)*(n+1)/(4*Pi^2))*z(n+2)/z(n) = -B(n)*((n+2)*(n+1)/(4*Pi^2)) * Sum_{j>=1} a(j)/j^(n+2).

Apart from signs also Sum_{d|n} core(d)^2*mu(n/d) where core(x) is the squarefree part of x. - Benoit Cloitre, May 31 2002

			a(3) = -8 because the divisors of 3 are {1, 3} and mu(1)*1^2 + mu(3)*3^2 = -8.
a(4) = -3 because the divisors of 4 are {1, 2, 4} and mu(1)*1^2 + mu(2)*2^2 + mu(4)*4^2 = -3.
E.g., a(15) = (3^2 - 1) * (5^2 - 1) = 8*24 = 192. - _Jon Perry_, Aug 24 2010
G.f. = x - 3*x^2 - 8*x^3 - 3*x^4 - 24*x^5 + 24*x^6 - 48*x^7 - 3*x^8 - 8*x^9 + ...

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1965, pp. 805-811.
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, p. 48.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
P. G. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408.
Paul W. Oxby, A Function Based on Chebyshev Polynomials as an Alternative to the Sinc Function in FIR Filter Design, arXiv:2011.10546 [eess.SP], 2020.
David M. Smith, Multiple-Precision Evaluation of Bernoulli Numbers, Loyola Marymount Univ., 2024. See p. 3.
Wikipedia, Riemann zeta function.

Dirichlet inverse of Jordan totient function J_r(n): A023900 (r = 1), A063453(r = 3), A189922 (r = 4).

Cf. A007434, A027641, A027642, A027748, A046692, A076479, A084920, A322360.

a046970 = product . map ((1 -) . (^ 2)) . a027748_row
-- Reinhard Zumkeller, Jan 19 2012

Jinvk := proc(n,k) local a,f,p ; a := 1 ; for f in ifactors(n)[2] do p := op(1,f) ; a := a*(1-p^k) ; end do: a ; end proc:
A046970 := proc(n) Jinvk(n,2) ; end proc: # R. J. Mathar, Jul 04 2011

muDD[d_] := MoebiusMu[d]*d^2; Table[Plus @@ muDD[Divisors[n]], {n, 60}] (Lopez)
Flatten[Table[{ x = FactorInteger[n]; p = 1; For[i = 1, i <= Length[x], i++, p = p*(1 - x[[i]][[1]]^2)]; p}, {n, 1, 50, 1}]] (* Jon Perry, Aug 24 2010 *)
a[ n_] := If[ n < 1, 0, Sum[ d^2 MoebiusMu[ d], {d, Divisors @ n}]]; (* Michael Somos, Jan 11 2014 *)
a[ n_] := If[ n < 2, Boole[ n == 1], Times @@ (1 - #[[1]]^2 & /@ FactorInteger @ n)]; (* Michael Somos, Jan 11 2014 *)

A046970(n)=sumdiv(n,d,d^2*moebius(d)) \\ Benoit Cloitre

{a(n) = if( n<1, 0, direuler( p=2, n, (1 - X*p^2) / (1 - X))[n])}; /* Michael Somos, Jan 11 2014 */

from math import prod
from sympy import primefactors
def A046970(n): return prod(1-p**2 for p in primefactors(n)) # Chai Wah Wu, Sep 08 2023

Multiplicative with a(p^e) = 1 - p^2.
a(n) = Sum_{d|n} mu(d)*d^2.
abs(a(n)) = Product_{p prime divides n} (p^2 - 1). - Jon Perry, Aug 24 2010
From Wolfdieter Lang, Jun 16 2011: (Start)
Dirichlet g.f.: zeta(s)/zeta(s-2).
a(n) = J_{-2}(n)*n^2, with the Jordan function J_k(n), with J_k(1):=1. See the Apostol reference, p. 48. exercise 17. (End)
a(prime(n)) = -A084920(n). - R. J. Mathar, Aug 28 2011
G.f.: Sum_{k>=1} mu(k)*k^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 15 2017
a(n) = Sum_{d divides n} d * (sigma_1(d))^(-1) * sigma_1(n/d), where (sigma_1(n))^(-1) = A046692(n) denotes the Dirichlet inverse of sigma_1(n). - Peter Bala, Jan 26 2024
a(n) = A076479(n) * A322360(n). - Amiram Eldar, Feb 02 2024

Corrected and extended by Vladeta Jovovic, Jul 25 2001

Additional comments from Wilfredo Lopez (chakotay147138274(AT)yahoo.com), Jul 01 2005

A084921 a(n) = lcm(p-1, p+1) where p is the n-th prime.

Original entry on oeis.org

3, 4, 12, 24, 60, 84, 144, 180, 264, 420, 480, 684, 840, 924, 1104, 1404, 1740, 1860, 2244, 2520, 2664, 3120, 3444, 3960, 4704, 5100, 5304, 5724, 5940, 6384, 8064, 8580, 9384, 9660, 11100, 11400, 12324, 13284, 13944, 14964, 16020, 16380

Offset: 1

Reinhard Zumkeller, Jun 11 2003

nonn

This sequence consists of terms of sequences A055523 and A055527 for prime n > 2. - Toni Lassila (tlassila(AT)cc.hut.fi), Feb 02 2004

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Christian Krause, LODA program for A084921
Index entries for sequences related to lcm's

Cf. A000040, A006093, A008864, A009286, A024700, A055523, A055527, A084920, A084922.

a084921 n = lcm (p - 1) (p + 1)  where p = a000040 n
-- Reinhard Zumkeller, Jun 01 2013

[3] cat [(p^2-1)/2: p in PrimesInInterval(3,300)]; // G. C. Greubel, May 03 2024

LCM[#-1,#+1]&/@Prime[Range[50]] (* Harvey P. Dale, Oct 09 2018 *)

a(n)=if(n<2,3,(prime(n)^2-1)/2) \\ Charles R Greathouse IV, May 15 2013

[3]+[(n^2-1)/2 for n in prime_range(3,301)] # G. C. Greubel, May 03 2024

a(n) = A084920(n)/2 for n > 1.
a(n) = 3*A084922(n) for n > 2.
a(n) = A009286(A000040(n)). - Enrique Pérez Herrero, May 17 2012
a(n) ~ 0.5 n^2 log^2 n. - Charles R Greathouse IV, May 15 2013
Product_{n>=1} (1 + 1/a(n)) = 2. - Amiram Eldar, Jan 23 2021
a(n) = (A000040(n)^2 - 1) / 2 for n > 1. - Christian Krause, Mar 27 2021
a(n) = (3/2)*A024700(n-2), for n > 1. - G. C. Greubel, May 03 2024

A127917 Product of three numbers: n-th prime, previous number, and following number.

Original entry on oeis.org

6, 24, 120, 336, 1320, 2184, 4896, 6840, 12144, 24360, 29760, 50616, 68880, 79464, 103776, 148824, 205320, 226920, 300696, 357840, 388944, 492960, 571704, 704880, 912576, 1030200, 1092624, 1224936, 1294920, 1442784, 2048256, 2247960, 2571216, 2685480, 3307800

Offset: 1

Artur Jasinski, Feb 06 2007

a(n) is the order of the matrix group SL(2,prime(n)). - Tom Edgar, Sep 28 2015

G. C. Greubel, Table of n, a(n) for n = 1..10000
J. B. Marshall, On the extension of Fermat's theorem to matrices of order n, Proceedings of the Edinburgh Mathematical Society 6 (1939) 85-91. See (11) page 90-91 when p=2.
Index to sequences related to prime powers.

Cf. A036689, A034953, A084920, A117762.

Cf. A065470, A065487.

[6] cat [NthPrime(n)*(NthPrime(n)^2-1): n in [2..40]]; // Vincenzo Librandi, Sep 29 2015

Table[(Prime[n] + 1) Prime[n](Prime[n] - 1), {n, 1, 100}]
Table[p(p^2-1),{p,Prime[Range[40]]}] (* Harvey P. Dale, Apr 26 2025 *)

forprime(p=2,1e3,print1(6*binomial(p+1,3)", ")) \\ Charles R Greathouse IV, Jun 16 2011

a(n) = prime(n)*(prime(n)^2-1);
vector(40, n, a(n)) \\ Altug Alkan, Sep 28 2015

a(n) = prime(n)*(prime(n)^2-1). - Tom Edgar, Sep 28 2015
a(n) = 2 * A117762(n), for n > 1. - Altug Alkan, Sep 28 2015
From Amiram Eldar, Nov 22 2022: (Start)
Product_{n>=1} (1 + 1/a(n)) = A065487.
Product_{n>=1} (1 - 1/a(n)) = A065470. (End)

A049001 a(n) = prime(n)^2 - 2.

Original entry on oeis.org

2, 7, 23, 47, 119, 167, 287, 359, 527, 839, 959, 1367, 1679, 1847, 2207, 2807, 3479, 3719, 4487, 5039, 5327, 6239, 6887, 7919, 9407, 10199, 10607, 11447, 11879, 12767, 16127, 17159, 18767, 19319, 22199, 22799, 24647, 26567, 27887

Offset: 1

N. J. A. Sloane

Smallest numbers k such that k*prime(n)^2 + 1 is a square. - Bruno Berselli, Apr 19 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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.

Cf. A001248, A049002, A065481.

Cf. A084920, A166010, A182200; A182174.

a049001 = subtract 2 . a001248  -- Reinhard Zumkeller, Jul 30 2015

A049001:=n->ithprime(n)^2-2; seq(A049001(k), k=1..50); # Wesley Ivan Hurt, Oct 11 2013

Table[Prime[n]^2-2, {n, 50}] (* Wesley Ivan Hurt, Oct 11 2013 *)

a(n) = prime(n)^2 - 2; \\ Amiram Eldar, Nov 07 2022

a(n) = A001248(n) - 2.
a(n) = A182200(n) + 1. - Wesley Ivan Hurt, Oct 11 2013
Product_{n>=1} (1 - 1/a(n)) = A065481. - Amiram Eldar, Nov 07 2022

A117762 a(1) = 6; for n>1, a(n) = prime(n)*(prime(n)^2 - 1)/2.

Original entry on oeis.org

6, 12, 60, 168, 660, 1092, 2448, 3420, 6072, 12180, 14880, 25308, 34440, 39732, 51888, 74412, 102660, 113460, 150348, 178920, 194472, 246480, 285852, 352440, 456288, 515100, 546312, 612468, 647460, 721392, 1024128, 1123980, 1285608, 1342740, 1653900

Offset: 1

Roger L. Bagula, Apr 14 2006

a(n) is the order of the matrix group PSL(2,prime(n)). - corrected by Tom Edgar, Sep 28 2015

Blyth and Robertson, Essential Student Algebra, Volume 5: Groups,Chapman and Hall, New York, page 14

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A084920, A127918.

[6] cat [NthPrime(n)*(NthPrime(n)^2-1)/2: n in [2..40]]; // Vincenzo Librandi, Sep 29 2015

a[n_]= If[n==1, 6, Prime[n]*(Prime[n]^2 -1)/2];
Table[a[n], {n,40}]
Join[{6}, Table[Prime[n] (Prime[n]^2 - 1)/2, {n, 2, 40}]] (* Vincenzo Librandi, Sep 29 2015 *)

a(n) = prime(n)*(prime(n)^2-1)/2;
vector(40, n, a(n+1)) \\ Altug Alkan, Sep 28 2015

def A117762(n): return nth_prime(n)*(nth_prime(n)^2-1)/2 + 3*int(n==1)
[A117762(n) for n in range(1,41)] # G. C. Greubel, Jul 21 2023

a(n) = A127918(n), n>1.

a(n) = A000040(n)*A084921(n). - R. J. Mathar, Jan 29 2024

cp's OEIS Frontend

A006093 a(n) = prime(n) - 1.

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A008864 a(n) = prime(n) + 1.

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A024702 a(n) = (prime(n)^2 - 1)/24.

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A154945 Decimal expansion of Sum_{p} 1/(p^2-1), summed over the primes p = A000040.

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A046970 Dirichlet inverse of the Jordan function J_2 (A007434).

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