A249270 -id:A249270 - cp's OEIS frontend

A087560 Smallest m > n such that gcd(m, n^2) = n.

2, 6, 6, 12, 10, 30, 14, 24, 18, 30, 22, 60, 26, 42, 30, 48, 34, 90, 38, 60, 42, 66, 46, 120, 50, 78, 54, 84, 58, 210, 62, 96, 66, 102, 70, 180, 74, 114, 78, 120, 82, 210, 86, 132, 90, 138, 94, 240, 98, 150, 102, 156, 106, 270, 110, 168, 114, 174, 118, 420, 122

Offset: 1

Reinhard Zumkeller, Oct 24 2003

nonn

Equals n multiplied by the least nontrivial number coprime to n. - Amarnath Murthy, Nov 20 2005

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000005, A053669, A062011, A119416, A249270.

Table[n*Select[Prime[Range[Log2[n] + 1]], ! Divisible[n, #] &][[1]], {n, 61}] (* Ivan Neretin, May 21 2015 *)

a(n) = forprime(p = 2, , if(n%p, return(n*p))); \\ Amiram Eldar, Feb 01 2025

a(n) = n*A053669(n).
A000005(a(n)) = 2*A000005(n) = A062011(n). - Reinhard Zumkeller, May 17 2006
Sum_{k=1..n} ~ c * n^2 / 2, where c = A249270. - Amiram Eldar, Feb 01 2025

A340469 First constant from family of prime-representing constants h_n (h1 = 1.2148208055...) such that ceiling(h_n) = prime(n).

Original entry on oeis.org

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

Offset: 1

Ilya Weinstein, Jan 08 2021

The family of constants h_n (h1 = 1.2148208055...) for generation of the complete sequence of primes with using of a recursive relation for h_n such that ceiling(h_n) = prime(n). The recursive relation h_n = ceiling(h_{n-1})*(h_{n-1}-ceiling(h_{n-1})+2) generates the complete sequence of prime numbers. Constants h_n are irrational for all n.

			h1 = 1.21482080552433374694513123422377095425915026021227...
h2 = 2.42964161104866749389026246844754190851830052042454...
h3 = 4.28892483314600248167078740534262572555490156127363...
etc.

I. A. Weinstein, Family of prime-representing constants: use of the ceiling function, arXiv:2101.00094 [math.GM], 2021.

Cf. A000040, A249270.

N[Sum[(Prime[k]-2)/Product[Prime[n],{n,1,k-1}],{k,1,150}],50]

suminf(k=1, (prime(k)-2)/prod(i=1, k-1, prime(i))) \\ Michel Marcus, Jan 08 2021

h1 = Sum_{k>=1} (prime(k)-2)/Product_{i=1..k-1} prime(i).

Equals A249270 - A064648 - 1. - Antonio Graciá Llorente, Dec 22 2023

A339204 Decimal expansion of the generating constant for the Fibonacci numbers.

Original entry on oeis.org

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

Offset: 1

A.H.M. Smeets, Nov 27 2020

Inspired by the prime generating constant A249270, but here for the Fibonacci numbers, A000045(n); generating the Fibonacci numbers for n > 2.
The producing function is given by f' = floor(f)*(f-floor(f)+1), starting with this constant, f' denoting the next f, and floor(f) being the terms of the sequence produced by this constant.
The number of correct digits obtained from the first n terms from the series expansion for this constant as given in the formula section is roughly about (n^2)/10 (~ (3/7)*(log(Fib(n))^2)) decimal digits; i.e., for a binary representation, about (n^2)/3 binary digits.

			2.95693889137798804983169009791120927869915823439362...

Dylan Friedman, Juli Garbulsky, Bruno Glecer, James Grime, and Massi Tron Florentin, A Prime-Representing Constant, 2019.

Cf. A000045 (Fibonacci).

Cf. A249270 (for primes), A339203 (for Mersenne prime exponents).

with(combinat, fibonacci): evalf(Sum((fibonacci(n) - 1)/Product(fibonacci(k), k = 2..n-1), n = 3..infinity), 120); # Vaclav Kotesovec, Nov 29 2020

Quiet[First[RealDigits[NSum[(Fibonacci[n] - 1)/Fibonorial[n - 1], {n, 3, Infinity}, Method -> {"WynnEpsilon", "ExtraTerms" -> 25}, NSumTerms -> 25, VerifyConvergence -> False, WorkingPrecision -> 105], 10, 100]], General::intnm] (* Jan Mangaldan, Nov 29 2020 *)

suminf(n=3, (fibonacci(n)-1)/prod(k=2, n-1, fibonacci(k))) \\ Michel Marcus, Nov 27 2020

n, sumn, sumd, termd, f0, f1 = 0, 0, 1, 1, 1, 1
while n < 33: # enough to obtain 100 digits
    n, sumn, sumd, termd, f0, f1 = n+1, sumn*termd+sumd*(f0-1), sumd*termd, termd*f0, f0+f1, f0
pre, sumn, i, d = sumn//sumd, sumn%sumd, 0, ""
while i < 100:
    dig, sumn, i = (10*sumn)//sumd, (10*sumn)%sumd, i+1
    d = d+str(dig)
print(str(pre)+"."+d)

Equals Sum_{n > 2} (A000045(n)-1)/(Product_{k = 2..n-1} A000045(k)).

A356093 a(n) = numerator((prime(n)-1)/prime(n)#), where prime(n)# = A002110(n) is the n-th primorial.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 8, 3, 1, 2, 1, 6, 4, 1, 1, 2, 1, 2, 1, 1, 12, 1, 1, 4, 16, 10, 1, 1, 18, 8, 3, 1, 4, 1, 2, 5, 2, 27, 1, 2, 1, 6, 1, 32, 14, 3, 1, 1, 1, 2, 4, 1, 8, 25, 128, 1, 2, 9, 2, 4, 1, 2, 3, 1, 4, 2, 1, 8, 1, 2, 16, 1, 1, 2, 9, 1, 2, 6, 40, 4, 1, 2, 1

Offset: 1

Amiram Eldar, Jul 26 2022

f(n) = a(n)/A356094(n) is the asymptotic density of numbers k such that prime(n) = A053669(k) is the smallest prime not dividing k.

The asymptotic mean of A053669 is 2.9200509773... (A249270) which is the weighted mean of the primes with f(n) as weights. The corresponding asymptotic standard deviation, which can be evaluated from the second raw moment Sum_{n>=1} f(n) * prime(n)^2, is 2.8013781465... .

			Fractions begin with 1/2, 1/3, 2/15, 1/35, 1/231, 2/5005, 8/255255, 3/1616615, 1/10140585, 2/462120945, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002110, A006093, A039787, A053669, A249270, A356094 (denominators).

Similar sequences: A038110, A338559, A340818, A341431, A342450, A342479.

primorial[n_] := Product[Prime[i], {i, 1, n}]; Numerator[Table[(Prime[i] - 1)/primorial[i], {i, 1, 100}]]

a(n) = numerator((prime(n)-1)/factorback(primes(n))); \\ Michel Marcus, Jul 26 2022

from math import gcd
from sympy import prime, primorial
def A356093(n): return (p:=prime(n)-1)//gcd(p,primorial(n)) # Chai Wah Wu, Jul 26 2022

a(n) = 1 iff prime(n) is in A039787.
Let f(n) = a(n)/A356094(n):
f(n) = A006093(n)/A002110(n).
Sum_{n>=1} f(n) = 1.
Sum_{n>=1} f(n) * prime(n) = A249270.

A381031 The second smallest prime not dividing n minus the smallest prime not dividing n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 12 2025

nonn

			For n = 1, the least prime not dividing it is 2, and the second least prime not dividing is 3, thus a(1) = 3-2 = 1.
For n = 3, the least nondividing prime is 2, the second least nondividing prime is 5, thus a(3) = 5-2 = 3.
For n = 6 = 2*3, the least nondividing prime is 5, and the second least nondividing prime is 7, thus a(6) = 7-5 = 2.

Antti Karttunen, Table of n, a(n) for n = 1..30030

Cf. A053669, A249270, A380539, A381113.

a[n_] := Module[{p = 1, q = 1, c = 0}, While[c < 2, p = NextPrime[p]; If[! Divisible[n, p], c++; If[c == 1, q = p]]]; p-q]; Array[a, 105] (* Amiram Eldar, Feb 14 2025 *)

A381031(n) = { my(c=0,e=0); forprime(p=2, , if(n%p, c++; if(1==c, e=p, if(2==c, return(p-e))))); };

a(n) = A380539(n) - A053669(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A381113 - A249270 = 2.239091... . - Amiram Eldar, Feb 14 2025

A381113 Decimal expansion of the asymptotic mean of the second smallest prime not dividing k, where k runs over the positive integers (A380539).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Feb 14 2025

			5.15914285965164203013658097450125817200073072141916...

Cf. A002110, A007504, A249270 (analogous constant with smallest prime), A380539.

primorial(k) = prod(i = 1, k, prime(i));
primesum(k) = sum(i = 1, k, prime(i));
suminf(k = 2, prime(k) * (prime(k)-1) * (primesum(k-1)-k+1) / primorial(k))

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A380539(k).

Equals Sum_{k>=2} prime(k) * (prime(k)-1) * (primesum(k-1)-k+1) / primorial(k), where primesum(k) = A007504(k) and primorial(k) = A002110(k).

A339203 Decimal expansion of the generating constant for the exponents of the Mersenne primes.

Original entry on oeis.org

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

Offset: 1

A.H.M. Smeets, Nov 27 2020

Inspired by the prime generating constant A249270, but here for the exponents of the Mersenne primes, A000043(n).
The producing function is given by f' = floor(f)*(f-floor(f)+1), starting with this constant, f' denoting the next f, and floor(f) being the next term of the sequence being produced by this constant.
Note that this constant is useless in trying to predict the next Mersenne prime exponent. A new known next Mersenne prime exponent will only enable us to calculate this constant more precisely.

			2.93009444726879573667795218699043578505760116717999...

Dylan Friedman, Juli Garbulsky, Bruno Glecer, James Grime, and Massi Tron Florentin, A Prime-Representing Constant, 2019.

Cf. A000043.

Cf. A249270 (for primes), A339204 (for Fibonacci numbers).

Equals Sum_{n > 0} (A000043(n)-1)/(Product_{k = 1..n-1} A000043(k)).

A339412 a(n) = floor(x(n)) where x(n) = (frac(x(n-1))+1)*floor(x(n-1)) and x(1) = Pi.

Original entry on oeis.org

3, 3, 4, 5, 5, 7, 10, 10, 13, 17, 31, 35, 67, 123, 223, 305, 414, 822, 1550, 2224, 3273, 4560, 7804, 14372, 15493, 20080, 40039, 44226, 71916, 130773, 183760, 316165, 613602, 1066559, 1138668, 1202427, 2022144, 2251837, 2477524, 4479491, 7192184, 11256849

Offset: 1

Michael Turniansky, Dec 03 2020

nonn

Inspired by A249270.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
James Grime and Brady Haran, 2.920050977316, Numberphile video, Nov 26 2020.

Cf. A000796, A249270.

b:= proc(n) option remember; `if`(n=1, Pi,
      (f-> (frac(f)+1)*floor(f))(b(n-1)))
    end:
a:= n-> floor(b(n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Dec 03 2020

Block[{a = {Pi}, $MaxExtraPrecision = 10^3}, Do[AppendTo[a, (FractionalPart[#] + 1) Floor[#]] &@ a[[-1]], 41]; Floor /@ a] (* Michael De Vlieger, Dec 04 2020 *)

{(⌊{(⌊⍵)×1+1|⍵}⍣⍵)○1x}¨0,⍳100

lista(nn) = {localprec(500); my(vx = vector(nn)); vx[1] = Pi; for (n=2, nn, vx[n] = (frac(vx[n-1])+1)*floor(vx[n-1]);); apply(floor, vx);} \\ Michel Marcus, Dec 03 2020

A339764 Decimal expansion of Varona's constant = Sum_{k >= 1} prime(k)/2^(k + k!).

Original entry on oeis.org

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

Offset: 0

José María Grau Ribas, Dec 20 2020

Varona's constant v is transcendental and generates the primes via prime(1)=2=floor(4*v) and for n>1 prime(n) = floor(v*2^(n+n!)) - 2^(1+n!-(n-1)!)*floor(v*2^(n-1+(n-1)!)).

			0.69726565107703208923339843750000000025860875718090325175312203818889

Juan L. Varona, A couple of transcendental prime-representing constants, arXiv:2012.11750 [math.NT], 2020.
Index entries for transcendental numbers

Cf. A249270, A016104, A051021.

First@RealDigits@N[Sum[Prime[i]/2^(i + i!), {i, 1, 12}], 300]

suminf(k=1, prime(k)/2^(k + k!)) \\ Michel Marcus, Dec 21 2020

A377010 Decimal expansion of the asymptotic mean of A376928: lim_{m->oo} (1/m) * Sum_{k=1..m} A376928(k).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 12 2024

			1.74466340501740192345730888352581703598570050436409...

Cf. A002110, A034386, A151800, A249270, A376928.

\p 120
f(k) = prod(i = 1, k, prime(i));
1/2 + suminf(k = 1, prime(k) * (1/f(k) - 1/f(k+1)))

Equals 1/2 + Sum_{p prime} p * (1/p# - 1/nextprime(p)#), where nextprime(p) = A151800(p) and p# = A034386(p).

Equals 1/2 + Sum_{k>=1} prime(k) * (1/A002110(k) - 1/A002110(k+1)).

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A087560 Smallest m > n such that gcd(m, n^2) = n.

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A340469 First constant from family of prime-representing constants h_n (h1 = 1.2148208055...) such that ceiling(h_n) = prime(n).

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A339204 Decimal expansion of the generating constant for the Fibonacci numbers.

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A356093 a(n) = numerator((prime(n)-1)/prime(n)#), where prime(n)# = A002110(n) is the n-th primorial.

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A381031 The second smallest prime not dividing n minus the smallest prime not dividing n.

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A381113 Decimal expansion of the asymptotic mean of the second smallest prime not dividing k, where k runs over the positive integers (A380539).

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A339203 Decimal expansion of the generating constant for the exponents of the Mersenne primes.

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