A067432 -id:A067432 - cp's OEIS frontend

A072670 Number of ways to write n as i*j + i + j, 0 < i <= j.

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 2, 0, 2, 0, 2, 1, 1, 0, 3, 1, 1, 1, 2, 0, 3, 0, 2, 1, 1, 1, 4, 0, 1, 1, 3, 0, 3, 0, 2, 2, 1, 0, 4, 1, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 5, 0, 1, 2, 3, 1, 3, 0, 2, 1, 3, 0, 5, 0, 1, 2, 2, 1, 3, 0, 4, 2, 1, 0, 5, 1, 1, 1, 3, 0, 5, 1, 2, 1, 1, 1, 5, 0, 2, 2, 4, 0, 3, 0, 3, 3

Offset: 0

Reinhard Zumkeller, Jun 30 2002

nonn

a(n) is the number of partitions of n+1 with summands in arithmetic progression having common difference 2. For example a(29)=3 because there are 3 partitions of 30 that are in arithmetic progressions: 2+4+6+8+10, 8+10+12 and 14+16. - N-E. Fahssi, Feb 01 2008
From Daniel Forgues, Sep 20 2011: (Start)
a(n) is the number of nontrivial factorizations of n+1, in two factors.
a(n) is the number of ways to write n+1 as i*j + i + j + 1 = (i+1)(j+1), 0 < i <= j. (End)
a(n) is the number of ways to write n+1 as i*j, 1 < i <= j. - Arkadiusz Wesolowski, Nov 18 2012
For a generalization, see comment in A260804. - Vladimir Shevelev, Aug 04 2015
Number of partitions of n into 3 parts whose largest part is equal to the product of the other two. - Wesley Ivan Hurt, Jan 04 2022

			a(11)=2: 11 = 1*5 + 1 + 5 = 2*3 + 2 + 3.
From _Daniel Forgues_, Sep 20 2011 (Start)
Number of nontrivial factorizations of n+1 in two factors:
  0 for the unit 1 and prime numbers
  1 for a square: n^2 = n*n
  1 for 6 (2*3), 10 (2*5), 14 (2*7), 15 (3*5)
  1 for a cube: n^3 = n*n^2
  2 for 12 (2*6, 3*4), for 18 (2*9, 3*6) (End)

Robert Israel, Table of n, a(n) for n = 0..10000
Joseph W. Andrushkiw, Roman I. Andrushkiw and Clifton E. Corzatt, Representations of Positive Integers as Sums of Arithmetic Progressions, Mathematics Magazine, Vol. 49, No. 5 (Nov., 1976), pp. 245-248.
M. A. Nyblom and C. Evans, On the enumeration of partitions with summands in arithmetic progression, Australian Journal of Combinatorics, Vol. 28 (2003), pp. 149-159.
Vladimir Shevelev, Representation of positive integers by the form x1...xk+x1+...+xk, arXiv:1508.03970 [math.NT], 2015.

Cf. A001620, A067432, A066938, A072668, A006093, A072671, A161840, A260803, A260804.

0, seq(ceil(numtheory:-tau(n+1)/2)-1, n=1..100); # Robert Israel, Aug 04 2015

p2[n_] := 1/2 (Length[Divisors[n]] - 2 + ((-1)^(Length[Divisors[n]] + 1) + 1)/2); Table[p2[n + 1], {n, 0, 104}] (* N-E. Fahssi, Feb 01 2008 *)
Table[Ceiling[DivisorSigma[0, n + 1]/2] - 1, {n, 0, 104}] (* Arkadiusz Wesolowski, Nov 18 2012 *)

is_ok(k,i,j)=0=i&&k===i*j+i+j;
first(m)=my(v=vector(m,z,0));for(l=1,m,for(j=1,l,for(i=1,j,if(is_ok(l,i,j),v[l]++))));concat([0],v); /* Anders Hellström, Aug 04 2015 */

a(n)=(numdiv(n+1)+issquare(n+1))/2-1 \\ Charles R Greathouse IV, Jul 14 2017

a(n) = A038548(n+1) - 1.
From N-E. Fahssi, Feb 01 2008: (Start)
a(n) = p2(n+1), where p2(n) = (1/2)*(d(n) - 2 + ((-1)^(d(n)+1)+1)/2); d(n) is the number of divisors of n: A000005.
G.f.: Sum_{n>=1} a(n) x^n = 1/x Sum_{k>=2} x^(k^2)/(1-x^k). (End)
lim_{n->infinity} a(A002110(n)-1) = infinity. - Vladimir Shevelev, Aug 04 2015
a(n) = A161840(n+1)/2. - Omar E. Pol, Feb 27 2019
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 3) / 2, where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 14 2024

A066938 Primes of the form p*q+p+q, where p and q are primes.

Original entry on oeis.org

11, 17, 23, 31, 41, 47, 53, 59, 71, 79, 83, 89, 107, 113, 127, 131, 151, 167, 179, 191, 227, 239, 251, 263, 269, 271, 293, 311, 359, 383, 419, 431, 439, 443, 449, 479, 491, 503, 521, 587, 593, 599, 607, 631, 647, 659, 683, 701, 719, 727, 743, 773, 809, 827

Offset: 1

Reinhard Zumkeller, Jan 24 2002

nonn

For p not equal to q, either p*q or p+q is odd, so their sum is odd.
The representation is ambiguous, e.g. 2*7+2+7 = 23 = 3*5+3+5.
Complement of A198273 with respect to A000040. - Reinhard Zumkeller, Oct 23 2011
None of these primes are in A158913 since if p*q+p+q is a prime, then sigma(p*q+p+q) = sigma(p*q). - Amiram Eldar, Nov 15 2021

			59 is in the sequence because 59 = 2 * 19 + 2 + 19.

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

Cf. A054973, A067432, A072668, A072673, A088709, A158913, A198273, A198277.

a066938 n = a066938_list !! (n-1)
a066938_list = map a000040 $ filter ((> 0) . a067432) [1..]
-- Reinhard Zumkeller, Oct 23 2011

nn = 1000; n2 = PrimePi[nn/3]; Select[Union[Flatten[Table[(Prime[i] + 1) (Prime[j] + 1) - 1, {i, n2}, {j, n2}]]], # <= nn && PrimeQ[#] &]

is(n)=fordiv(n+1,d,my(p=d-1,q=(n+1)/d-1); if(isprime(p) && isprime(q), return(isprime(n)))); 0 \\ Charles R Greathouse IV, Jul 23 2013

A067432(A049084(a(n))) > 0. - Reinhard Zumkeller, Oct 23 2011

A054973(a(n)+1) >= 2. - Amiram Eldar, Nov 15 2021

Edited by Robert G. Wilson v, Feb 01 2002

A198277 a(n) is the smallest prime such that exactly n prime pairs (p,q) exist with a(n) = p * q + p + q.

Original entry on oeis.org

2, 11, 23, 71, 239, 719, 2879, 5039, 1439, 10079, 37799, 126719, 55439, 110879, 181439, 191519, 166319, 635039, 514079, 665279, 1330559, 907199, 3243239, 831599, 2948399, 6320159, 4989599, 15301439, 14137199, 5266799, 11531519, 8315999, 23284799, 17463599, 45208799, 52390799, 34594559, 111767039, 95633999, 117976319, 70685999, 68468399

Offset: 0

Reinhard Zumkeller, Oct 23 2011

nonn

A067432(A049084(a(n))) = n and A067432(A049084(m)) <> n for m < a(n).

Charles R Greathouse IV, Table of n, a(n) for n = 0..115
Reinhard Zumkeller, Illustration of initial terms

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a198277 n = a000040 . (+ 1) . fromJust $ elemIndex n a067432_list

ct(n)=sumdiv(n+1,d,if(d^2>n,0, isprime(d-1)&&isprime(n\d)))
v=vector(60);forprime(p=2,1e9, t=ct(p);if(t && !v[t], v[t]=p; print(t" "p))); v \\ with 0's for unknown; Charles R Greathouse IV, Jul 24 2013

a(27)-a(33) from Donovan Johnson, Oct 24 2011

a(34)-a(41) from Charles R Greathouse IV, Jul 24 2013

A198273 Primes not of the form p*q + p + q for any primes p and q.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 29, 37, 43, 61, 67, 73, 97, 101, 103, 109, 137, 139, 149, 157, 163, 173, 181, 193, 197, 199, 211, 223, 229, 233, 241, 257, 277, 281, 283, 307, 313, 317, 331, 337, 347, 349, 353, 367, 373, 379, 389, 397, 401, 409, 421, 433, 457, 461, 463

Offset: 1

Reinhard Zumkeller, Oct 23 2011

A067432(A049084(a(n))) = 0; complement of A066938 with respect to A000040.

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

Cf. A000040, A049084, A066938, A067432.

a198273 n = a198273_list !! (n-1)
a198273_list = map a000040 $ filter ((== 0) . a067432) [1..]

nn = 500; n2 = PrimePi[nn/3]; Complement[Prime[Range[PrimePi[nn]]], Select[Union[Flatten[Table[(Prime[i] + 1) (Prime[j] + 1) - 1, {i, n2}, {j, n2}]]], # <= nn && PrimeQ[#] &]] (* T. D. Noe, Nov 22 2011 *)
Reap[For[P=2, P<500, P = NextPrime[P], If[Reduce[P == p*q + p + q, {p, q}, Primes] === False, Print[P]; Sow[P]]]][[2, 1]] (* Jean-François Alcover, Dec 10 2015 *)

do(lim)=my(v=Set(),t);;forprime(p=3,lim,forprime(q=2,p-1,t=p*q+p+q;if(t>lim,break);v=setunion(v,[t])));setminus(primes(primepi(lim)),v) \\ Charles R Greathouse IV, Nov 22 2011

cp's OEIS Frontend

A072670 Number of ways to write n as i*j + i + j, 0 < i <= j.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A066938 Primes of the form p*q+p+q, where p and q are primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A198277 a(n) is the smallest prime such that exactly n prime pairs (p,q) exist with a(n) = p * q + p + q.

Views

Author

Keywords

Comments

Links

Programs

Haskell

PARI

Extensions

A198273 Primes not of the form p*q + p + q for any primes p and q.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI