A070801 -id:A070801 - cp's OEIS frontend

A070802 a(n)=prevprime[sigma(n)]-nextprime[phi(n)]=A070801(n)-A070800(n).

1, 0, 4, 0, 8, 0, 8, 6, 12, 0, 18, 0, 16, 12, 20, 0, 30, 0, 30, 18, 20, 0, 48, 8, 28, 18, 40, 0, 60, 0, 44, 24, 36, 18, 76, 0, 40, 24, 72, 0, 76, 0, 60, 44, 48, 0, 96, 10, 66, 34, 68, 0, 94, 30, 84, 42, 60, 0, 150, 0, 58, 66, 90, 30, 116, 0, 76, 42, 110, 0, 164, 0, 76, 72, 102, 28

Offset: 2

Labos Elemer, May 08 2002

			n=100:sigma[100]=217,prevprime[217]=211, phi[100]=40,nextprime[40]=41,a(100)=211-41=170. The difference is 0 for odd primes.

Cf. A007917, A007818, A000010, A000203, A000720, A000040, A070800, A070801.

Table[Prime[PrimePi[DivisorSigma[1, w]]]- Prime[1+PrimePi[EulerPhi[w]]], {w, 2, 128}]

a(n)=p[Pi(sigma[n])]-p[1+Pi(phi[n])]

A007917 Version 1 of the "previous prime" function: largest prime <= n.

Original entry on oeis.org

2, 3, 3, 5, 5, 7, 7, 7, 7, 11, 11, 13, 13, 13, 13, 17, 17, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 29, 29, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 41, 41, 43, 43, 43, 43, 47, 47, 47, 47, 47, 47, 53, 53, 53, 53, 53, 53, 59, 59, 61, 61, 61, 61, 61, 61, 67, 67, 67, 67, 71, 71, 73, 73, 73, 73

Offset: 2

R. Muller

Version 2 of the "previous prime" function (see A151799) is "largest prime < n". This produces the same sequence of numerical values, except the offset (or indexing) starts at 3 instead of 2.
Maple's "prevprime" function uses version 2.
Also the largest prime dividing n! or lcm(1,...,n). - Labos Elemer, Jun 22 2000
Also largest prime among terms of (n+1)st row of Pascal's triangle. - Jud McCranie, Jan 17 2000
Also largest integer k such that A000203(k) <= n+1. - Benoit Cloitre, Mar 17 2002. - Corrected by Antti Karttunen, Nov 07 2017
Also largest prime factor of A061355(n) (denominator of Sum_{k=0..n} 1/k!). - Jonathan Sondow, Jan 09 2005
Also prime(pi(x)) where pi(x) is the prime counting function = number of primes <= x. - Cino Hilliard, May 03 2005
Also largest prime factor, occurring to the power p, in denominator of Sum_{k=1..n} 1/k^p, for any positive integer p. - M. F. Hasler, Nov 10 2006
For n > 10, these values are close to the most negative eigenvalues of A191898 (conjecture). - Mats Granvik, Nov 04 2011

K. Atanassov, On the 37th and the 38th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 2, 83-85.
J. Castillo, Other Smarandache Type Functions: Inferior/Superior Smarandache f-part of x, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, 202-204.

N. J. A. Sloane, Table of n, a(n) for n = 2..10000
Marc Deleglise and Jean-Louis Nicolas, Maximal product of primes whose sum is bounded, arXiv preprint arXiv:1207.0603 [math.NT], 2012. See Fig. 1. - From _N. J. A. Sloane_, Dec 17 2012
Hans Gunter, Puzzle 145. The Inferior Smarandache Prime Part and Superior Smarandache Prime Part functions; Solutions by Jean Marie Charrier, Teresinha DaCosta, Rene Blanch, Richard Kelley and Jim Howell. F. Smarandache, Only Problems, Not Solutions!.
Eric Weisstein's World of Mathematics, Previous Prime

Cf. A000040, A000203, A005145, A007918, A070801, A113523, A151799, A179278, A175851, A000720.

a007917 n = if a010051' n == 1 then n else a007917 (n-1)
-- Reinhard Zumkeller, May 01 2013, Jul 26 2012

[NthPrime(#PrimesUpTo(n)): n in [2..100]]; // Vincenzo Librandi, Nov 25 2015

Maple
```
A007917 := n-> prevprime(n+1);
```

Table[Prime[PrimePi[n]], {n, 2, 70}] (* Stefan Steinerberger, Apr 06 2006 *)
NextPrime[Range[3,80],-1] (* Harvey P. Dale, Jan 23 2011 *)

a=precprime \\ In older versions, use a(n)=precprime(n)
\\ Charles R Greathouse IV, Jun 15 2011

Equals A006530(A000142(n)). - Jonathan Sondow, Jan 09 2005
Equals A006530(A056040(n)). - Peter Luschny, Mar 04 2011
a(n) = A000040(A049084(A007918(n)) + 1 - A010051(n)). - Reinhard Zumkeller, Jul 26 2012
From Wesley Ivan Hurt, May 22 2013: (Start)
omega( Product_{i=2..n} a(i) ) = pi(n).
Omega( Product_{i=2..n} a(i) ) = n - 1. (End)
For n >= 2, a(A000203(n)) = A070801(n). - Antti Karttunen, Nov 07 2017
a(n) = n + 1 - Sum_{i=1..n} floor(pi(i)/pi(n)) = n + 1 - A175851(n). - Ridouane Oudra, Jun 24 2024
Conjecture: a(n) = floor(log(Sum_{k=2..n} exp(A000010(k)+1))). - Joseph M. Shunia, Aug 09 2024
a(n) = A000040(A000720(n)). - Ridouane Oudra, Oct 04 2024

Edited by N. J. A. Sloane, Apr 06 2008

A070804 Number of primes not exceeding phi(n).

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 3, 2, 3, 2, 4, 2, 5, 3, 4, 4, 6, 3, 7, 4, 5, 4, 8, 4, 8, 5, 7, 5, 9, 4, 10, 6, 8, 6, 9, 5, 11, 7, 9, 6, 12, 5, 13, 8, 9, 8, 14, 6, 13, 8, 11, 9, 15, 7, 12, 9, 11, 9, 16, 6, 17, 10, 11, 11, 15, 8, 18, 11, 14, 9, 19, 9, 20, 11, 12, 11, 17, 9, 21, 11, 16, 12, 22, 9, 18, 13, 16

Offset: 1

Labos Elemer, May 08 2002

			n=50: phi[50]=20,Pi[20]=8=a(50)

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Cf. A007818, A000010, A000720, A070800, A070801, A070802, A070803.

[#PrimesUpTo(EulerPhi(n)): n in [1..90]]; // Vincenzo Librandi, Mar 26 2017

Table[PrimePi[EulerPhi[n]], {n, 1, 256}]

a(n) = A000720(A000010(n)) = pi(phi(n)).

A070800 Smallest prime greater than phi(n): a(n) = nextprime(phi(n)).

Original entry on oeis.org

2, 3, 3, 5, 3, 7, 5, 7, 5, 11, 5, 13, 7, 11, 11, 17, 7, 19, 11, 13, 11, 23, 11, 23, 13, 19, 13, 29, 11, 31, 17, 23, 17, 29, 13, 37, 19, 29, 17, 41, 13, 43, 23, 29, 23, 47, 17, 43, 23, 37, 29, 53, 19, 41, 29, 37, 29, 59, 17, 61, 31, 37, 37, 53, 23, 67, 37, 47, 29, 71, 29, 73, 37

Offset: 2

Labos Elemer, May 08 2002

Antti Karttunen, Table of n, a(n) for n = 2..16384

Cf. A000010, A070801, A151800, A078773.

Table[Prime[1+PrimePi[EulerPhi[w]]], {w, 2, 128}] (* or *)
NextPrime[EulerPhi[#]] & /@ Range[2,80] (* Harvey P. Dale, Jan 30 2011 *)

a(n)=nextprime(eulerphi(n)+1) \\ Charles R Greathouse IV, Oct 31 2011

a(n) = prime(1 + primepi(phi(n))) = A151800(A000010(n)).

A085379 Greatest prime as sum of distinct divisors of n.

Original entry on oeis.org

3, 3, 7, 5, 11, 7, 13, 13, 17, 11, 23, 13, 23, 23, 31, 17, 37, 19, 41, 31, 23, 23, 59, 31, 41, 37, 53, 29, 71, 31, 61, 47, 53, 47, 89, 37, 59, 53, 89, 41, 89, 43, 83, 73, 71, 47, 113, 7, 83, 71, 97, 53, 113, 71, 113, 79, 89, 59, 167, 61, 31, 103, 127, 83, 139, 67

Offset: 2

Reinhard Zumkeller, Jun 26 2003

nonn

			The divisors of n = 50 are {1,2,5,10,25,50}, the sums of distinct divisors that are prime: 2, 3 = 2+1, 5, 7 = 5+2, 11 = 10+1, 13 = 10+2+1, 17 = 10+5+2, 31 = 25+5+1, 37 = 25+10+2, 41 = 25+10+5+1, 43 = 25+10+5+2+1, 53 = 50+2+1, 61 = 50+10+1, 67 = 50+10+5+2 and 83 = 50+25+5+2+1. Therefore a(50) = 83 < 89 = A070801(50) and A085381(3) = 50.

Amiram Eldar, Table of n, a(n) for n = 2..10001
Eric Weisstein's World of Mathematics, Divisor Function.

Cf. A000203, A023194, A062700, A070801, A085380, A085381.

a[n_] := Module[{d = Divisors[n], c, x}, c = Rest@CoefficientList[Series[Product[1 + x^d[[i]], {i, 1, Length[d]}], {x, 0, Total[d]}], x]; Max[Select[Position[c, ?(# > 0 &)] // Flatten, PrimeQ]]]; Array[a, 100, 2] (* _Amiram Eldar, Mar 01 2024 *)

a(n) <= A070801(n) <= A000203(n).
a(A085380(n)) = A070801(A085380(n)).
a(A085381(n)) < A070801(A085381(n)).
a(A023194(n)) = A000203(A023194(n)) = A062700(n).

A085380 Numbers k such that (greatest prime as sum of distinct divisors of k) = (greatest prime not greater sum of divisors of k).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 70, 71, 72, 73, 74, 76, 77

Offset: 1

Reinhard Zumkeller, Jun 26 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..6990

Complement of A085381.

Cf. A085379, A070801.

A085379(a(n)) = A070801(a(n)).

A085381 Numbers k such that (greatest prime as sum of distinct divisors of k) < (greatest prime not greater sum of divisors of k).

Original entry on oeis.org

22, 49, 50, 62, 68, 69, 75, 81, 82, 87, 91, 94, 95, 98, 99, 106, 116, 119, 121, 122, 133, 134, 142, 146, 154, 159, 169, 182, 183, 187, 202, 213, 214, 217, 218, 225, 226, 243, 247, 249, 253, 259, 262, 265, 268, 273, 274, 279, 287, 292, 299, 301, 302, 303

Offset: 1

Reinhard Zumkeller, Jun 26 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..3010

Complement of A085380.

Cf. A085379, A070801.

A085379(a(n)) < A070801(a(n)).

A202111 a(n) = sigma(n) - p, where p is the largest prime < sigma(n).

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 5, 1, 1, 1, 2, 1, 2, 1, 1, 1, 5, 1, 1, 2, 1, 3, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 7, 1, 1, 5, 1, 1, 11, 4, 4, 1, 1, 1, 7, 1, 7, 1, 1, 1, 1, 1, 7, 1, 14, 1, 5, 1, 13, 7, 5, 1, 2, 1, 1, 11, 1, 7, 1, 1, 5, 8, 13, 1, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 5, 7, 1, 1, 4, 5, 6, 1, 5, 1, 11, 1, 5

Offset: 2

Michel Lagneau, Dec 11 2011

nonn

			a(12) = sigma(12) - 23 = 28 - 23 = 5.

Antti Karttunen, Table of n, a(n) for n = 2..16385

Cf. A000203, A049711, A070801, A072918.

A202111 := proc(n)
    A049711(numtheory[sigma](n) );
end proc:
seq(A202111(n),n=2..80) ; # R. J. Mathar, Dec 13 2011

f[n_]:=Module[{nf=DivisorSigma[1,n]},nf-NextPrime[nf,-1]];f/@Range[3,90]

A049711(n) = (n-precprime(n-1)); \\ This function from M. F. Hasler, Sep 09 2015
A202111(n) = A049711(sigma(n)); \\ Antti Karttunen, Nov 07 2017

a(n) = A049711(A000203(n)). - R. J. Mathar, Dec 13 2011

More terms from Antti Karttunen, Nov 07 2017

A067792 a(n) is the least prime >= sigma(n).

Original entry on oeis.org

2, 3, 5, 7, 7, 13, 11, 17, 13, 19, 13, 29, 17, 29, 29, 31, 19, 41, 23, 43, 37, 37, 29, 61, 31, 43, 41, 59, 31, 73, 37, 67, 53, 59, 53, 97, 41, 61, 59, 97, 43, 97, 47, 89, 79, 73, 53, 127, 59, 97, 73, 101, 59, 127, 73, 127, 83, 97, 61, 173, 67, 97, 107, 127, 89, 149, 71, 127

Offset: 1

Benoit Cloitre, Feb 07 2002

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sigma(n)

Cf. A000203, A007918, A070801, A074495.

Array[If[PrimeQ@ #, #, NextPrime@ #] &@ DivisorSigma[1, #] &, 68] (* Michael De Vlieger, Nov 16 2017 *)

A067792(n) = nextprime(sigma(n)); \\ Antti Karttunen, Nov 16 2017

From Antti Karttunen, Nov 17 2017: (Start)
a(n) = A007918(A000203(n)).
a(n) <= A074495(n).
(End)

Definition clarified by Antti Karttunen, Nov 16 2017

cp's OEIS Frontend

A070802 a(n)=prevprime[sigma(n)]-nextprime[phi(n)]=A070801(n)-A070800(n).

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A007917 Version 1 of the "previous prime" function: largest prime <= n.

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A070804 Number of primes not exceeding phi(n).

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Magma

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A070800 Smallest prime greater than phi(n): a(n) = nextprime(phi(n)).

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A085379 Greatest prime as sum of distinct divisors of n.

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A085380 Numbers k such that (greatest prime as sum of distinct divisors of k) = (greatest prime not greater sum of divisors of k).

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A085381 Numbers k such that (greatest prime as sum of distinct divisors of k) < (greatest prime not greater sum of divisors of k).

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A202111 a(n) = sigma(n) - p, where p is the largest prime < sigma(n).

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Maple