A072491 -id:A072491 - cp's OEIS frontend

A072492 Values of n for which A072491(n)=3.

27, 35, 51, 57, 65, 77, 87, 93, 95, 117, 119, 121, 122, 123, 125, 135, 143, 145, 147, 148, 155, 161, 171, 177, 185, 187, 189, 190, 203, 205, 207, 208, 209, 215, 217, 219, 220, 221, 237, 245, 247, 249, 250, 255, 261, 267, 275, 287, 289, 291, 292, 297, 299

Offset: 1

Amarnath Murthy, Jul 14 2002

Define f(1) = 0. For n>=2, let f(n) = n - p where p is the largest prime <= n. A072491(n) = number of iterations of f to reach 0, starting from n.

p+4 is a term if p is a prime but p+2 and p+4 are both composite.

			27 is a term as f(27)=27-23=4, f(4)=4-3=1 and f(1) = 0. (3 steps.)

Robert Price, Table of n, a(n) for n = 1..10000

Cf. A072491.

f[1]=0; f[n_] := n-Prime[PrimePi[n]]; a72491[n_] := Module[{k, x}, For[k=0; x=n, x>0, k++; x=f[x], Null]; k]; Select[Range[300], a72491[ # ]==3&]

Edited by Dean Hickerson, Nov 26 2002

A051034 Minimal number of primes needed to sum to n.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 3, 2, 1, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 3, 2, 3, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2

Offset: 2

Eric W. Weisstein

			a(2) = 1 because 2 is already prime.
a(4) = 2 because 4 = 2+2 is a partition of 4 into 2 prime parts and there is no such partition with fewer terms.
a(27) = 3 because 27 = 3+5+19 is a partition of 27 into 3 prime parts and there is no such partition with fewer terms.

T. D. Noe, Table of n, a(n) for n = 2..10000
Olivier Ramaré, On Šnirel'man's constant, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, 22:4 (1995), pp. 645-706.
Yannick Saouter, Vinogradov's theorem is true up to 10^20
Terence Tao, Every odd number greater than 1 is the sum of at most five primes, arXiv:1201.6656 [math.NT], 2012 preprint, to appear in Mathematics of Computation.
Eric Weisstein's World of Mathematics, Prime Partition
Index entries for sequences related to Goldbach conjecture

Cf. A025583, A004526, A007944, A007962, A000607, A051036, A010051, A061358, A068307, A103765.

Different from A072491.

(* Assuming Goldbach's conjecture *) a[p_?PrimeQ] = 1; a[n_] := If[ Reduce[ n == x + y, {x, y}, Primes] === False, 3, 2]; Table[a[n], {n, 2, 112}] (* Jean-François Alcover, Apr 03 2012 *)

issum(n,k)=if(k==1,isprime(n),k--;forprime(p=2,n,if(issum(n-p,k),return(1)));0)
a(n)=my(k);while(!issum(n,k++),);k \\ Charles R Greathouse IV, Jun 01 2011

a(n) = 1 iff n is prime. a(2n) = 2 (for n > 1) if Goldbach's conjecture is true. a(2n+1) = 2 (for n >= 1) if 2n+1 is not prime, but 2n-1 is. a(2n+1) >= 3 (for n >= 1) if both 2n+1 and 2n-1 are not primes (for sufficiently large n, a(2n+1) = 3 by Vinogradov's theorem, 1937). - Franz Vrabec, Nov 30 2004
a(n) <= 3 for all n, assuming the Goldbach conjecture. - N. J. A. Sloane, Jan 20 2007
a(2n+1) <= 5, see Tao 2012. - Charles R Greathouse IV, Jul 09 2012
Assuming Goldbach's conjecture, a(n) <= 3. In particular, a(p)=1; a(2*n)=2 for n>1; a(p+2)=2 provided p+2 is not prime; otherwise a(n)=3. - Sean A. Irvine, Jul 29 2019
a(2n+1) <= 3 by Helfgott's proof of Goldbach's ternary conjecture, and hence a(n) <= 4 in general. - Charles R Greathouse IV, Oct 24 2022

More terms from Naohiro Nomoto, Mar 16 2001

A066352 Pillai sequence: a(n) is the smallest term in A007924 requiring n primes.

Original entry on oeis.org

0, 1, 4, 27, 1354, 401429925999155061

Offset: 0

Copied from www.primepuzzles.net by Frank Ellermann, Dec 19 2001

a(5) computed independently in 2007 by R. J. Mathar and Luca & Thangadurai, both using Thomas Nicely's tables.
On Cramer's conjecture, the number of primes required is O(log* n), where log* is the iterated logarithm, so the rate of growth of a(n) is tetrational in n. - Charles R Greathouse IV, Aug 28 2010
The next term likely has hundreds of millions of digits. - Charles R Greathouse IV, Jun 29 2015

			The greatest prime <= 27 is 23; the greatest prime <= 27-23 is 3; 27-23-3 = 1, so the Pillai representation of 27 is 23+3+1, which uses more terms than all preceding numbers.

S. S. Pillai, "An arithmetical function concerning primes", Annamalai University Journal (1930), pp. 159-167.

Florian Luca and Ravindranathan Thangadurai, On an arithmetic function considered by Pillai, Journal de théorie des nombres de Bordeaux 21:3 (2009), pp. 695-701.
M. T. Marcos, Smarandache Prime Base representation, prime puzzle 141.
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
S. S. Pillai, On an Arithmetic Function concerning Primes, Journal Of The Annamalai University, Vol-1 (1932), pp. 159-167 (pages 204-212 in pdf).

Cf. A007924.

A072491(n)=if(n<4,n>0,1+A072491(n-precprime(n)))
print1(r=0);for(n=1,1e7,if(A072491(n)>r,r=a(n);print1(", "n)))
\\ Charles R Greathouse IV, Feb 04 2013

a(n) = 2*p(m) - p(m-1) with minimal m = pi(a(n)) so that p(m) = a(n-1) + p(m-1), where p(n) is A008578(n).

Edited by Charles R Greathouse IV, Oct 28 2009

Entry rewritten by Charles R Greathouse IV, Aug 28 2010

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A072492 Values of n for which A072491(n)=3.

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A051034 Minimal number of primes needed to sum to n.

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A066352 Pillai sequence: a(n) is the smallest term in A007924 requiring n primes.

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