A098990 -id:A098990 - cp's OEIS frontend

A048656 a(n) is the number of unitary (and also of squarefree) divisors of n!.

1, 2, 4, 4, 8, 8, 16, 16, 16, 16, 32, 32, 64, 64, 64, 64, 128, 128, 256, 256, 256, 256, 512, 512, 512, 512, 512, 512, 1024, 1024, 2048, 2048, 2048, 2048, 2048, 2048, 4096, 4096, 4096, 4096, 8192, 8192, 16384, 16384, 16384, 16384, 32768, 32768, 32768, 32768

Offset: 1

Labos Elemer

nonn

Let K(n) be the field that is generated over the rationals Q by adjoining the square roots of the numbers 1,2,3,...,n, i.e., K(n) = Q(sqrt(1),sqrt(2),...,sqrt(n)); a(n) is the degree of this field over Q. - Avi Peretz (njk(AT)netvision.net.il), Mar 20 2001
For n>1, a(n) is the number of ways n! can be expressed as the product of two coprime integers p and q such that 0 < p/q < 1, if negative integers are considered as well. This is the answer to the 2nd problem of the International Mathematical Olympiad 2001. Example, for n = 3, the a(3) = 4 products are 3! = (-2)*(-3) = (-1)*(-6) = 1*6 = 2*3. - Bernard Schott, Jan 21 2021
a(n) = number of subsets S of {1,2,...,n} such that every number in S is a prime. - Clark Kimberling, Sep 17 2022

			For n = 7, n! = 5040 = 16*9*5*7 with 4 distinct prime factors, so a(7) = A034444(7!) = 16.
The subsets S of {1, 2, 3, 4} such that every number in S is a prime are these: {}, {2}, {3}, {2, 3}; thus, a(4) = 4. - _Clark Kimberling_, Sep 17 2022

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
International Mathematical Olympiad 2001, Hong Kong Preliminary Selection Contest, Problem 2.
Index to divisibility sequences.
Index to sequences related to Olympiads.

Cf. A000720, A001221, A034444, A357214, A357215.

Cf. A098882, A098990.

Table[2^PrimePi[n], {n, 1, 70}] (* Clark Kimberling, Sep 17 2022 *)

a(n)=2^primepi(n) \\ Charles R Greathouse IV, Apr 07 2012

A001221(n!) = A000720(n) so a(n) = A034444(n!) = 2^A000720(n).

Sum_{n>=1} 1/a(n) = A098882 + 1 = A098990 - 1. - Amiram Eldar, Mar 13 2025

A053760 Smallest positive quadratic nonresidue modulo p, where p is the n-th prime.

Original entry on oeis.org

2, 2, 2, 3, 2, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 2, 2, 2, 7, 5, 3, 2, 3, 5, 2, 3, 2, 2, 3, 3, 2, 3, 2, 2, 3, 2, 2, 5, 2, 2, 2, 7, 5, 2, 3, 2, 3, 2, 2, 3, 7, 7, 2, 3, 5, 2, 3, 2, 3, 2, 2, 2, 11, 5, 2, 2, 5, 2, 2, 3, 7, 3, 2, 2, 5, 2, 2, 3, 7, 2, 2, 7, 5, 3, 2, 3, 5, 2, 3, 2, 13, 3, 2, 2, 5, 2, 3, 2, 2, 2, 2, 2

Offset: 1

Steven Finch, Apr 05 2000

nonn

Assuming the Generalized Riemann Hypothesis, Montgomery proved a(n) << (log p(n))^2, meaning that there is a constant c such that |a(n)| <= c*(log p(n))^2. - Jonathan Vos Post, Jan 06 2007
a(n) < 1 + sqrt(p), where p is the n-th prime (Theorem 3.9 in Niven, Zuckerman, and Montgomery). - Jonathan Sondow, May 13 2010
Treviño proves that a(n) < 1.1 p^(1/4) log p for n > 2 where p is the n-th prime. - Charles R Greathouse IV, Dec 06 2012
a(n) is always a prime, because if x*y is a nonresidue, then x or y must also be a nonresidue. - Jonathan Sondow, May 02 2013
a(n) is the smallest prime q such that the congruence x^2 == q (mod p) has no solution 0 < x < p, where p = prime(n). For n > 1, a(n) is the smallest base b such that b^((p-1)/2) == -1 (mod p), where odd p = prime(n). - Thomas Ordowski, Apr 24 2019

			The 5th prime is 11, and the positive quadratic residues mod 11 are 1^2 = 1, 2^2 = 4, 3^2 = 9, 4^2 = 5 and 5^2 = 3. Since 2 is missing, a(5) = 2.
The only positive quadratic redidue mod 2 is 1, so a(1)=2.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 94-98.
Hugh L. Montgomery, Topics in Multiplicative Number Theory, 3rd ed., Lecture Notes in Mathematics, Vol. 227 (1971), MR 49:2616.
Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991, p. 147.
Paulo Ribenboim, The New Book of Prime Number Records, 3rd ed., Springer-Verlag 1996; Math. Rev. 96k:11112.

T. D. Noe, Table of n, a(n) for n = 1..10000
Robert Baillie and Samuel S. Wagstaff, Lucas pseudoprimes, Mathematics of Computation, Vol. 35, No. 152 (1980), pp. 1391-1417, Math. Rev. 81j:10005, alternative link.
Paul Erdős, Remarks on number theory. I., Mat. Lapok, Vol. 12 (1961), pp. 10-17; Math. Rev. 26 #2410.
Steven R. Finch, Quadratic Residues [Broken link]
Steven R. Finch, Quadratic Residues [From the Wayback machine]
Keith Matthews, Finding n(p), the least quadratic non-residue (mod p)
Enrique Treviño, The least k-th power non-residue, Journal of Number Theory, Vol. 149 (2015),pp. 201-224, alternative link.
Eric Weisstein's World of Mathematics, Quadratic Nonresidue.

Cf. A000229, A020649, A098990.

Table[ p = Prime[n]; First[ Select[ Range[p], JacobiSymbol[#, p] != 1 &]], {n, 1, 100}] (* Jonathan Sondow, Mar 03 2013 *)

residue(n,m)={local(r);r=0;for(i=0,floor(m/2),if(i^2%m==n,r=1));r}
A053760(n)={local(r,m);r=0;m=0;while(r==0,m=m+1;if(!residue(m,prime(n)),r=1));m} \\ Michael B. Porter, May 02 2010

qnr(p)=my(m);while(1,if(!issquare(Mod(m++,p)),return(m)))
a(n)=if(n>1,qnr(prime(n)),2) \\ Charles R Greathouse IV, Feb 27 2013

a(n) = A020649(prime(n)) for n > 1. - Thomas Ordowski, Apr 24 2019

Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = A098990 (Erdős, 1961). - Amiram Eldar, Oct 29 2020

More terms from James Sellers, Apr 08 2000

A249270 Decimal expansion of lim_{n->oo} (1/n)*Sum_{k=1..n} smallest prime not dividing k.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 24 2014

The old definition was "Decimal expansion of the mean value over all positive integers of the least prime not dividing a given integer."

The integer parts of the sequence having this constant as starting value and thereafter x[n+1] = (frac(x[n])+1)*floor(x[n]), where floor and frac are integer and fractional part, are exactly the sequence of the prime numbers: see the Grime-Haran Numberphile video for details. - M. F. Hasler, Nov 28 2020

			2.9200509773161347120925629171120194680027278993214267...

Steven R. Finch, Meissel-Mertens constants: Quadratic residues, Mathematical Constants, Cambridge Univ. Press, 2003, pp. 96—98.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Steven R. Finch, Average least nonresidues, December 4, 2013. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 171.
Dylan Fridman, Juli Garbulsky, Bruno Glecer, James Grime and Massi Tron Florentin, A Prime-Representing Constant, The American Mathematical Monthly, Vol. 126, No. 1 (2019), pp. 70-73, ResearchGate link.
James Grime and Brady Haran, 2.920050977316, Numberphile video, Nov 26 2020.
Paul Pollack, The average least quadratic nonresidue modulo m and other variations on a theme of Erdős, J. Number Theory, Vol. 132, No. 6 (2012), pp. 1185-1202.
Juan L. Varona, A Couple of Transcendental Prime-Representing Constants, arXiv:2012.11750 [math.NT], 2020.
I. A. Weinstein, Family of prime-representing constants: use of the ceiling function, arXiv:2101.00094 [math.GM], 2021.

Cf. A000040, A001223, A002110, A034386, A053669, A098990, A232927.

digits = 103; Clear[s]; s[m_] := s[m] = Sum[(Prime[k] - 1)/Product[Prime[j], {j, 1, k - 1}] // N[#, digits + 100]&, {k, 1, m}]; s[10]; s[m = 20]; While[RealDigits[s[m]] != RealDigits[s[m/2]], m = 2*m]; RealDigits[s[m], 10, digits] // First

def sharp_primorial(n): return sloane.A002110(prime_pi(n));
@CachedFunction
def spv(n):
    b = 0
    for i in (0..n):
        b += 1 / sharp_primorial(i)
    return b
N(spv(300), digits=108) # Jani Melik, Jul 22 2015

Sum_{k >= 1} (p_k - 1)/(p_1 p_2 ... p_{k-1}), where p_k is the k-th prime number.
Sum_{k >= 0} 1/A034386(k). - Jani Melik, Jul 22 2015
From Amiram Eldar, Oct 29 2020: (Start)
Equals lim_{n->oo} (1/n) * Sum_{k=1..n} A053669(k).
Equals 2 + Sum_{n>=1} (prime(n+1)-prime(n))/prime(n)# = 2 + Sum_{n>=1} A001223(n)/A002110(n). (End)
prime(n+1) = floor(C*prime(n)# - prime(n)*floor(C*prime(n-1)# - 1)) with prime(1)=2 where C is this constant. - Davide Rotondo, Sep 15 2023

Definition revised by N. J. A. Sloane, Nov 29 2020

A232927 a(n) is the smallest k such that the first k primes generate the multiplicative group modulo n.

Original entry on oeis.org

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

Offset: 3

Steven Finch, Dec 02 2013

nonn

H. Brown and H. Zassenhaus, Some empirical observations on primitive roots, J. Number Theory 3 (1971) 306-309.
S. R. Finch, Average least nonresidues, December 4, 2013. [Cached copy, with permission of the author]
P. Pollack, The average least quadratic nonresidue modulo m and other variations on a theme of Erdos, J. Number Theory 132 (2012) 1185-1202.

Cf. A098990, A249270.

A098882 Decimal expansion of the Sum_{n>0} (A000040(n+1)-A000040(n))/(2^n), where A000040(k) gives the k-th prime number.

Original entry on oeis.org

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

Offset: 1

Joseph Biberstine (jrbibers(AT)indiana.edu), Nov 03 2004

			1.6746439660113287789956763090840294116777975887794373283122052201763...

Eric Weisstein's World of Mathematics, Prime Number.

Cf. A000040, A098990.

g:=N->sum((ithprime(n+1)-ithprime(n))/2^n,n=1..N); evalf[106](g(5000)); evalf[106](g(10000));

suminf(k=1, (prime(k+1)-prime(k))/2^k) \\ Michel Marcus, Jan 13 2016

Equals A098990 - 2. - Amiram Eldar, Nov 17 2020

A346173 Decimal expansion of Sum_{k>=1} prime(k)/2^prime(k).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jul 08 2021

This constant is irrational (Hančl and Tijdeman, 2004).

Jaroslav Hančl and Robert Tijdeman, On the irrationality of Cantor series, Journal für die reine und angewandte Mathematik, Vol. 2004, No. 571 (2004), pp. 145-158.

Cf. A034785, A098990, A100127.

RealDigits[Sum[Prime[n]/2^Prime[n], {n, 1, 100}], 10, 100][[1]]

suminf(k=1, prime(k)/2^prime(k)) \\ Michel Marcus, Jul 09 2021

1.09306425770250716540258595269676368295547596540121...

A162541 Primes p such that a splitting of the cyclic group Zp by the perfect 3-shift code {+-1,+-2,+-3} exists.

Original entry on oeis.org

7, 37, 139, 163, 181, 241, 313, 337, 349, 379, 409, 421, 541, 571, 607, 631, 751, 859, 877, 937, 1033, 1087, 1123, 1171, 1291, 1297, 1447, 1453, 1483, 1693, 1741, 1747, 2011, 2161, 2239, 2311, 2371, 2473, 2539, 2647, 2677, 2707, 2719, 2857, 3169, 3361, 3433, 3511, 3547

Offset: 1

Ctibor O. Zizka, Jul 05 2009

nonn

This list was computed by S. Saidi.
From Travis Scott, Oct 04 2022: (Start)
These are also the p whose (phi/3)-th power residues have minimal bases at {1,2,3} (see under Example). Such covers {1a(n)-> {1,2,3}(n) =   7,  37,  139,  163,  181,  241, ... ~    (9*n)*log(n)
       {1,2,4}(n) =  13,  19,   61,   67,   73,   79, ... ~  (9*n/2)*log(n)
       {1,3,5}(n) =  31, 223,  229,  277,  283,  397, ... ~   (27*n)*log(n)
       {1,3,7}(n) =  43, 433,  457,  691, 1069, 1471, ... ~ (81*n/2)*log(n)
       {1,3,9}(n) = 109, 127,  157,  601,  733,  739, ... ~ (81*n/4)*log(n)
       {1,5,7}(n) = 307, 919, 1093, 2179, 2251, 3181, ... ~   (81*n)*log(n)
Note that the k-th q value takes A054272(k) x values and that a(n) = A040034(n) \ {1,2,4}(n). Following a result of Erdős (cf. A053760, A098990) the asymptotic means for q and x are Sum_{n>=1} prime(n)*2/3^n = 2.69463670741804726229622... and Sum_{n>=1} Sum_{prime(n) < k prime < prime(n)^2 OR k = prime(n)^2} D(prime(n),k)*k = 5.69767191389790422108748...
Subsequence of A040034 (2 is not a cubic residue modulo p) such that 3 is neither a residue nor in the same cubic power class as 2. (End)

			From _Travis Scott_, Oct 04 2022: (Start)
{1,2,3}^12 (mod 37) == {1,26,10} covers the 12th-power residues on Z/37Z.
{1,2,3}^14 (mod 43) == {1,1,36} misses 6. (End)

S. Saidi, Semicrosses and quadratic forms, European J. Combinatorics, vol.16, pp. 191-196, 1995. [This article has a typesetting error at the last term of this sequence on Table 1, p. 195. - _Travis Scott_, Oct 04 2022]

Subsequence of A040034.

Select[Prime@Range@497,Mod[#,3]==1&&DuplicateFreeQ@PowerMod[{1,2,3},(#-1)/3,#]&] (* Travis Scott, Oct 04 2022 *)

From Travis Scott, Oct 04 2022: (Start)
Primes of quadratic form 7x^2 +- 6xy + 36y^2 [from Saidi].
a(n) ~ 9*n*log(n). (End)

Incorrect term deleted and more terms from Travis Scott, Oct 04 2022

A249271 Decimal expansion of the mean value over all positive integers of a function giving the least quadratic nonresidue modulo a given odd integer (this function is precisely defined in A053761).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 24 2014

			3.147755148502400312516674955879769209272937748793398864...

Steven R. Finch, Mathematical Constants, Cambridge Univ. Press, 2003, Meissel-Mertens constants: Quadratic residues, pp. 96—98.

Robert Baillie and Samuel S. Wagstaff, Lucas pseudoprimes, Mathematics of Computation 35 (1980), pp. 1391-1417.

Cf. A053760, A053761, A098990.

digits = 104; Clear[s]; s[m_] := s[m] = 1 + Sum[(Prime[j] + 1)*2^(-j + 1)* Product[1 - 1/Prime[i], {i, 1, j - 1}] // N[#, digits + 100]&, {j, 2, m}] ; s[10]; s[m = 20]; While[RealDigits[s[m]] != RealDigits[s[m/2]], Print[m]; m = 2*m]; RealDigits[s[m], 10, digits] // First

do(lim)=my(p=2,pr=1.,s=1); forprime(q=3,lim, pr*=(1-1/p)/2; s+=(q+1)*pr; p=q); s \\ Charles R Greathouse IV, Dec 20 2017

1 + sum_{j=2..m} (p_j + 1)*2^(-j+1)*prod_{i=1..j-1} (1 - 1/p_i), where p_j is the j-th prime number.

A369508 Decimal expansion of Sum_{n>=1} prime(n)*(-1/2)^n (negated).

Original entry on oeis.org

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

Offset: 0

Pierluigi Failla, Jan 25 2024

			0.662729784185267965388337694463998...

Cf. A098990.

First[RealDigits[Sum[Prime[n](-2)^-n,{n,1000}],10,85]] (* James C. McMahon, Jan 29 2024 *)

suminf(k=1, prime(k)/(-2)^k) \\ Michel Marcus, Jan 25 2024

from sympy import prime
from decimal import Decimal, getcontext
getcontext().prec = 100
def f(k):
    if k == 1:
        return prime(k) * Decimal(-0.5)
    return f(k-1) + prime(k) * Decimal(-0.5) ** k

Showing 1-9 of 9 results.

cp's OEIS Frontend

A048656 a(n) is the number of unitary (and also of squarefree) divisors of n!.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A053760 Smallest positive quadratic nonresidue modulo p, where p is the n-th prime.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A249270 Decimal expansion of lim_{n->oo} (1/n)*Sum_{k=1..n} smallest prime not dividing k.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Sage

Formula

Extensions

A232927 a(n) is the smallest k such that the first k primes generate the multiplicative group modulo n.

Views

Author

Keywords

Links

Crossrefs

A098882 Decimal expansion of the Sum_{n>0} (A000040(n+1)-A000040(n))/(2^n), where A000040(k) gives the k-th prime number.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

A346173 Decimal expansion of Sum_{k>=1} prime(k)/2^prime(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A162541 Primes p such that a splitting of the cyclic group Zp by the perfect 3-shift code {+-1,+-2,+-3} exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica