A101300 -id:A101300 - cp's OEIS frontend

A045966 a(1)=3; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+2}^e_i.

3, 5, 7, 25, 11, 35, 13, 125, 49, 55, 17, 175, 19, 65, 77, 625, 23, 245, 29, 275, 91, 85, 31, 875, 121, 95, 343, 325, 37, 385, 41, 3125, 119, 115, 143, 1225, 43, 145, 133, 1375, 47, 455, 53, 425, 539, 155, 59, 4375, 169, 605, 161, 475, 61, 1715, 187, 1625, 203, 185, 67

Offset: 1

N. J. A. Sloane

If we had a(1) = 1 (instead of 3), then this would be fully multiplicative with a(prime(k)) = prime(k+2) (see A357852). - Antti Karttunen, Jan 10 2020

From a puzzle proposed by Marc LeBrun.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

See A027748, A124010 for factorization data for n.
Cf. A000040, A003961, A101300.
Sequences with similar definitions: A045967, A045968, A045970, A126272.
A059896 is used to express relationship between terms of this sequence.
A357852 is a slightly better version. - N. J. A. Sloane, Oct 29 2022

a045966 1 = 3
a045966 n = product $ zipWith (^)
            (map a101300 $ a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Jun 03 2013, Dec 23 2011

a[1] = 3; a[n_] := With[{f = FactorInteger[n]}, Times @@ (Prime[PrimePi[f[[All, 1]]]+2]^f[[All, 2]])]; Array[a, 60] (* Jean-François Alcover, Jun 19 2015 *)

A045966(n) = if(1==n,3,my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(1+nextprime(1+f[i, 1]))); factorback(f)); \\ Antti Karttunen, Jan 10 2020

From Peter Munn, Dec 27 2019, for n >= 2, k >= 2: (Start)
a(n) = A003961^2(n).
a(n^k) = a(n)^k.
a(A003961(n)) = A003961(a(n)).
a(A059896(n,k)) = A059896(a(n), a(k)).
(End)

More terms from David W. Wilson

A030661 Product of next 2 primes after n.

Original entry on oeis.org

6, 15, 35, 35, 77, 77, 143, 143, 143, 143, 221, 221, 323, 323, 323, 323, 437, 437, 667, 667, 667, 667, 899, 899, 899, 899, 899, 899, 1147, 1147, 1517, 1517, 1517, 1517, 1517, 1517, 1763, 1763, 1763, 1763

Offset: 1

N. J. A. Sloane

nonn

Cf. A000040.

A030661 := n->nextprime(n) * nextprime(nextprime(n));

a(n) = A151800(n) * A101300(n) . - R. J. Mathar, Aug 09 2019

A105161 Difference between n and the second-smallest prime larger than n.

Original entry on oeis.org

3, 2, 3, 4, 3, 6, 5, 6, 5, 4, 3, 6, 5, 6, 5, 4, 3, 6, 5, 10, 9, 8, 7, 8, 7, 6, 5, 4, 3, 8, 7, 10, 9, 8, 7, 6, 5, 6, 5, 4, 3, 6, 5, 10, 9, 8, 7, 12, 11, 10, 9, 8, 7, 8, 7, 6, 5, 4, 3, 8, 7, 10, 9, 8, 7, 6, 5, 6, 5, 4, 3, 8, 7, 10, 9, 8, 7, 6, 5, 10, 9, 8, 7, 14, 13, 12, 11, 10, 9, 12, 11, 10, 9, 8, 7, 6, 5

Offset: 0

Zak Seidov, Apr 29 2005

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A101300.

with(numtheory); A105161:=n->ithprime(pi(n) + 2); seq(A105161(n), n=0..100); # Wesley Ivan Hurt, Feb 26 2014

Table[Prime[PrimePi[n] + 2], {n, 0, 100}] (* Wesley Ivan Hurt, Feb 26 2014 *)

a(n) = prime(primepi(n)+2) - n; \\ Michel Marcus, Oct 09 2013

a(n)=nextprime(nextprime(n+1)+1)-n \\ Charles R Greathouse IV, Oct 09 2013

from sympy import nextprime
def a(n): return nextprime(nextprime(n)) - n
print([a(n) for n in range(97)]) # Michael S. Branicky, Mar 02 2021

a(n) = prime(pi(n)+2) - n.

A378885 Numbers that are divisible by at least three different primes and the smallest three of them are consecutive primes.

Original entry on oeis.org

30, 60, 90, 105, 120, 150, 180, 210, 240, 270, 300, 315, 330, 360, 385, 390, 420, 450, 480, 510, 525, 540, 570, 600, 630, 660, 690, 720, 735, 750, 780, 810, 840, 870, 900, 930, 945, 960, 990, 1001, 1020, 1050, 1080, 1110, 1140, 1155, 1170, 1200, 1230, 1260, 1290

Offset: 1

Amiram Eldar, Dec 09 2024

All the positive multiples of 30 (A249674 \ {0}) are terms.
Numbers k such that A151800(A020639(k)) | k and also A101300(A020639(k)) | k.
The asymptotic density of this sequence is Sum_{k>=1} (Product_{j=1..k-1} (1-1/prime(j))) / (prime(k)*prime(k+1)*prime(k+2)) = 0.03943839735407432193784... .

			60 = 2^2 * 3 * 5 is a term since 2, 3 and 5 are consecutive primes.
770 = 2 * 5 * 7 * 11 is not a term since its smallest prime divisor is 2 and it is not divisible by 3, the prime next to 2.
1365 = 3 * 5 * 7 * 13 is a term since 3, 5 and 7 are consecutive primes.

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

Subsequence of A000977.
Subsequences: A046301, A378884.
Cf. A020639, A101300, A151800, A249674.

q[k_] := Module[{p = FactorInteger[k][[;; , 1]]}, Length[p] > 2 && p[[2]] == NextPrime[p[[1]]] && p[[3]] == NextPrime[p[[2]]]]; Select[Range[1300], q]

is(k) = if(k == 1, 0, my(p = factor(k)[,1]); #p > 2 && p[2] == nextprime(p[1]+1) && p[3] == nextprime(p[2]+1));

A084754 Triangle read by rows: row n lists the first n primes greater than n.

Original entry on oeis.org

2, 3, 5, 5, 7, 11, 5, 7, 11, 13, 7, 11, 13, 17, 19, 7, 11, 13, 17, 19, 23, 11, 13, 17, 19, 23, 29, 31, 11, 13, 17, 19, 23, 29, 31, 37, 11, 13, 17, 19, 23, 29, 31, 37, 41, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59

Offset: 1

Amarnath Murthy and Jason Earls, Jul 12 2003

			Triangle starts:
  2;
  3,  5;
  5,  7, 11;
  5,  7, 11, 13;
  7, 11, 13, 17, 19;
  7, 11, 13, 17, 19, 23;
  ...

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000040, A000720, A101300, A151800.

[NthPrime(#PrimesUpTo(n) + k): k in [1..n], n in [1..16]]; // G. C. Greubel, May 13 2023

Table[Prime[PrimePi[n]+k], {n,16}, {k,n}]//Flatten (* G. C. Greubel, May 13 2023 *)

def A084754(n,k): return nth_prime(prime_pi(n)+k)
flatten([[A084754(n,k) for k in range(1,n+1)] for n in range(1,17)]) # G. C. Greubel, May 13 2023

From G. C. Greubel, May 13 2023: (Start)
T(n, k) = prime(PrimePi(n) + k).
T(n, 1) = A151800(n).
T(n, 2) = A101300(n). (End)

Edited by David Wasserman, Jan 05 2005

A085918 Primes p such that for some k the number of terms > 0 and < 1 in the Farey sequence of order k is p.

Original entry on oeis.org

3, 5, 11, 17, 31, 41, 71, 79, 101, 127, 139, 149, 179, 199, 211, 229, 241, 269, 277, 307, 359, 383, 431, 449, 541, 773, 829, 881, 1259, 1307, 1327, 1493, 1831, 1933, 2141, 2551, 3373, 3947, 4127, 4831, 4957, 5021, 5153, 5323, 5431, 5569, 5813, 6091, 6329

Offset: 1

Cino Hilliard, Aug 16 2003

Or, prime numbers of the form Sum_{j=2..k} phi(j). - Jorge Coveiro, Dec 22 2004. Examples: phi(2)+phi(3) = 3; phi(2)+phi(3)+phi(4) = 5; phi(2)+phi(3)+phi(4)+phi(5)+phi(6) = 11; phi(2)+phi(3)+phi(4)+phi(5)+phi(6)+phi(7) = 17.

Does this sequence have an infinite number of terms?

			The Farey sequence of order 4 is {0, 1/4, 1/3, 1/2, 2/3, 3/4, 1}. The number of terms > 0 and < 1 is 5, which is prime, so 5 is a term.

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

Cf. A078334, A005728, A101300, A000010.

Select[Accumulate[Table[EulerPhi[k], {k, 2, 150}]], PrimeQ] (* Amiram Eldar, Jul 06 2024 *)

/* Farey sequence of order n */ fareycountp(n) = { for(x=2,n, y = farey(x); if(isprime(y),print1(y",")); ) }
farey(n) = { c=1; m=n*(n-2)+2; a=vector(m); for(x=1,n, for(y=x,n, v = x/y; if(v<1, c++; a[c]=v; ) ) ); a = vecsort(a); c=0; for(x=2,m, if(a[x]<>a[x-1] & a[x]<>0, \ print1(a[x]","); c++; ) ); return(c) }

Definition corrected by Jonathan Sondow, Apr 21 2005

cp's OEIS Frontend

A045966 a(1)=3; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+2}^e_i.

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A030661 Product of next 2 primes after n.

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A105161 Difference between n and the second-smallest prime larger than n.

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A378885 Numbers that are divisible by at least three different primes and the smallest three of them are consecutive primes.

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A084754 Triangle read by rows: row n lists the first n primes greater than n.

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A085918 Primes p such that for some k the number of terms > 0 and < 1 in the Farey sequence of order k is p.

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