A066938 -id:A066938 - cp's OEIS frontend

A067432 Number of ways to represent the n-th prime in form p*q+p+q, where p and q are primes (see A066938).

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

Offset: 1

Reinhard Zumkeller, Feb 15 2002

nonn

a(A049084(A066938(n))) > 0; a(A049084(A198273(n))) = 0; a(A049084(A198277(n))) = n and a(A049084(m)) <> n for m < A198277(n). [Reinhard Zumkeller, Oct 23 2011]

a(n) < A072670(n).

			a(15) = 2 as A000040(15) = 47 = 3*11+3+11 = 5*7+5+7.

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

Cf. A000196, A010051.

a067432 n = length [p | let prime_n = a000040 n,
   p <- takeWhile (< a000196 prime_n) a000040_list,
   let (q,m) = divMod (prime_n - p) (p + 1),
   m == 0, a010051 q == 1]
a067432_list = map a067432 [1..]
-- Reinhard Zumkeller, Oct 23 2011

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

Original entry on oeis.org

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

A072668 Numbers one less than composite numbers.

Original entry on oeis.org

3, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95

Offset: 1

Henry Bottomley, Apr 11 2001

Complement of A006093 (primes minus 1).
Numbers which can be written as i*j+i+j, 0A072670(a(n))>0 for n>1.
a(n)! is divisible by a(n)*(a(n)+1)/2, see A060462.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A060462, A066938, A072672.

Cf. A002808, A006093, A079696.

[n-1: n in [2..120] | not IsPrime(n)]; // Vincenzo Librandi, Jun 09 2015

Select[Range[4, 96], CompositeQ] - 1 (* Michael De Vlieger, Dec 10 2020 *)

for(n=2,100,if(!isprime(n),print1(n-1,", "))) \\ Derek Orr, Jun 08 2015

from sympy import composite
def A072668(n): return composite(n)-1 # Chai Wah Wu, Aug 02 2024

a(n) = A002808(n) - 1.
a(n) = 2*A002808(n) - A079696(n). - Juri-Stepan Gerasimov, Oct 22 2009
a(n) = A060462(n).

A072673 Primes of form prime(n)prime(2n)+prime(n)+prime(2*n).

Original entry on oeis.org

11, 31, 83, 359, 1487, 4283, 4751, 5471, 7127, 12527, 41183, 66863, 71339, 85247, 186119, 274223, 290987, 338687, 373859, 386219, 400679, 465299, 490643, 663407, 720791, 827147, 883739, 1096127, 1124603

Offset: 1

Reinhard Zumkeller, Jun 30 2002

nonn

			A000040(236) = 1487 = 1403+23+61 = 23*61+23+61 = A000040(9)*A000040(18)+A000040(9)+A000040(18), therefore 1487 is a term.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A072672, A066938.

p[n_]:=Module[{p1=Prime[n],p2=Prime[2n]},p1*p2+p1+p2]; Select[ Array[ p,200],PrimeQ] (* Harvey P. Dale, Jul 14 2014 *)

A091301 Primes of the form p*q + p - q, where p and q are distinct primes.

Original entry on oeis.org

5, 7, 13, 17, 19, 29, 31, 37, 41, 43, 61, 67, 73, 89, 97, 103, 109, 113, 127, 137, 139, 149, 151, 157, 181, 193, 197, 199, 211, 229, 233, 241, 257, 271, 277, 281, 283, 307, 313, 317, 337, 349, 353, 373, 379, 389, 397, 401, 409, 421, 433, 449, 457, 461, 463, 487

Offset: 1

Zak Seidov, Feb 21 2004

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

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

Cf. A091310.

nn=100;Take[Select[Union[Flatten[{First[#]*Last[#]+First[#]-Last[#], First[#]*Last[#]- First[#]+Last[#]}&/@Subsets[Prime[Range[nn]],{2}]]], PrimeQ],nn] (* Harvey P. Dale, Jul 12 2014 *)

list(lim)=my(v=List(),t);forprime(q=2,lim,forprime(p=2,(lim+q)\(q+1),if(ispseudoprime(t=p*q+p-q),listput(v,t))));vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Feb 15 2011

Definition clarified by Harvey P. Dale, Jul 12 2014

A091305 Primes of the form p*q - p - q, where p and q are primes.

Original entry on oeis.org

3, 5, 7, 11, 17, 19, 23, 29, 31, 41, 43, 47, 59, 71, 79, 83, 101, 103, 107, 131, 137, 139, 149, 163, 167, 179, 191, 197, 199, 211, 223, 227, 239, 251, 263, 269, 271, 281, 311, 331, 347, 359, 379, 383, 419, 431, 443, 461, 463, 467, 479, 491, 499, 503, 521, 523

Offset: 1

Zak Seidov, Feb 21 2004

Some primes have more than one representation (besides of symmetry in p,q!), e.g. 11 with (p,q)=(2,13) and (3,7).

If (r,r+2) is a twin prime pair then r is in this sequence (with q=2, p=r+2). - Emmanuel Vantieghem, Jun 02 2025

			31 is a member with p=3, q=17.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A066938 (p*q + p + q), A091301 (p*q + p - q).

Cf. A001359 (subsequence).

N:= 1000: # for terms <= N
P:= select(isprime, [2,seq(i,i=3..N/2,2)]):
S:= {}:
for i from 1 to nops(P) do
  for j from 1 to i do
    x:= P[i]*P[j]-P[i]-P[j];
    if x > N then break fi;
    if isprime(x) then S:= S union {x} fi
od od:
sort(convert(S,list)); # Robert Israel, Jun 05 2025

mp[{p_,q_}]:=p*q-p-q; Take[Union[Select[mp/@Subsets[Prime[Range[100]],{2}], PrimeQ]],60] (* Harvey P. Dale, Nov 27 2011 *)

isA091305(p)=fordiv(p++,d,if(isprime(d+1)&isprime(p/d+1), return(isprime(p-1)))) \\ Charles R Greathouse IV, Feb 15 2011

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

A363638 Primes p such that p+1 can be written as a product of smaller numbers that are also of the form prime+1.

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, 223, 227, 239, 251, 263, 269, 271, 293, 311, 359, 383, 419, 431, 439, 443, 449, 479, 491, 503, 521, 557, 587, 593, 599, 607, 631, 647, 659, 683, 701, 719, 727, 743, 773

Offset: 1

Pontus von Brömssen, Jun 19 2023

nonn

			11 is a term because 11 is prime, 11+1 = 3*4 = (2+1)*(3+1), and 2 and 3 are prime.
223 is a term because 223 is prime, 223+1 = 4*4*14 = (3+1)^2*(13+1), and 3 and 13 are prime. (This is the first term that requires more than two factors, i.e., it is not a term of A066938.)

Cf. A008864, A066938 (subsequence), A363636, A363750.

A256074 Squares representable as k*m + k + m, where k >= m > 1 are squares.

Original entry on oeis.org

49, 169, 324, 441, 961, 1849, 2209, 3249, 5329, 8281, 12321, 15129, 17424, 17689, 24649, 33489, 44521, 58081, 58564, 64009, 65025, 74529, 94249, 103684, 117649, 145161, 177241, 191844, 214369, 237169, 257049, 305809, 361201, 423801, 480249, 494209, 573049, 660969, 700569

Offset: 1

Alex Ratushnyak, Mar 14 2015

nonn

A subsequence of A254671.
The sequence of square roots of a(n) begins: 7, 13, 18, 21, 31, 43, 47, 57, 73, 91, 111, 123, 132, 133, 157, 183, 211, 241, 242, 253, 255, 273, 307, 322, 343.
This sequence is infinite via x = m^2 and y = (m + 1)^2 so then x*y + x + y = m^2 * (m + 1)^2 + m^2 + (m + 1)^2 = m^4 + 2*m^3 + 3*m^2 + 2*m + 1 = (m^2 + m + 1)^2. - David A. Corneth, Oct 19 2024

			a(1) = 49 = 4*9 + 4 + 9.
a(2) = 169 = 9*16 + 9 + 16.

David A. Corneth, Table of n, a(n) for n = 1..10657 (terms <= 10^16)

Cf. A000290, A058031, A066938, A254671.

v=[];for(m=2,100,for(k=m,10^3,if(issquare(s=(k*m)^2+k^2+m^2),v=concat(v,s))));vecsort(v) \\ Derek Orr, Mar 21 2015

More terms from David A. Corneth, Oct 19 2024

A249444 Primes representable as p^q + p + q, where p and q are primes.

Original entry on oeis.org

13, 137, 251, 353, 2213, 4933, 24421, 78137, 148933, 205441, 371311, 493121, 524309, 571873, 912773, 1225153, 1594339, 4330913, 7189253, 13652161, 18191713, 21254213, 28629187, 31855333, 42508901, 49431233, 73560481, 81183173, 99253313, 178454113, 184220581, 192100613

Offset: 1

Alex Ratushnyak, Jan 12 2015

nonn

Cf. A053810, A065017, A066938, A096342, A253776.

cp's OEIS Frontend

A067432 Number of ways to represent the n-th prime in form p*q+p+q, where p and q are primes (see A066938).

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

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A072668 Numbers one less than composite numbers.

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A072673 Primes of form prime(n)*prime(2*n)+prime(n)+prime(2*n).

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A091301 Primes of the form p*q + p - q, where p and q are distinct primes.

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A091305 Primes of the form p*q - p - q, where p and q are primes.

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

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A072673 Primes of form prime(n)prime(2n)+prime(n)+prime(2*n).