A158913 -id:A158913 - cp's OEIS frontend

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

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

A206036 Numbers m such that sigma(m) = sigma(k) has solution for distinct numbers m and k.

Original entry on oeis.org

6, 10, 11, 14, 15, 16, 17, 20, 21, 23, 24, 25, 26, 28, 30, 31, 33, 34, 35, 38, 39, 40, 41, 42, 44, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 65, 66, 68, 69, 70, 71, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 92, 93, 94

Offset: 1

Jaroslav Krizek, Feb 03 2012

nonn

			6 and 11 are in the sequence because sigma(6) = sigma(11) = 12.
7 is not on the list because sigma(7) = 8 and there is no other integer for which sigma(n) = 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Complement of A211656.

Cf. A000203, A054973, A066074, A158913.

max = 9000; sigmaList = Table[DivisorSigma[1, n], {n, Prime[PrimePi[max]]}]; Select[Range[Floor[Sqrt[max]]], Count[sigmaList, sigmaList[[#]]] > 1 &] (* Alonso del Arte, Feb 06 2012 *)

is(k) = invsigmaNum(sigma(k)) > 1; \\ Amiram Eldar, Dec 15 2024, using Max Alekseyev's invphi.gp

A206447 Composite numbers n such that sigma(n) = sigma(d) has solution for some other composite number d.

Original entry on oeis.org

14, 15, 16, 20, 24, 25, 26, 28, 30, 33, 35, 38, 39, 40, 42, 44, 46, 48, 51, 54, 55, 56, 58, 60, 62, 65, 66, 68, 69, 70, 75, 77, 78, 80, 82, 84, 87, 88, 90, 92, 94, 95, 96, 99, 102, 104, 105, 108, 110, 112, 114, 115, 116, 118, 119, 120, 122, 123, 124, 125

Offset: 1

Jaroslav Krizek, Feb 07 2012

nonn

			Composite numbers 14 and 15 are in sequence because sigma(14) = sigma(15) = 24.

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

Cf. A206036, A158913, A066073.

N:= 500:
Res:= {}: Q:= {}:
for n from 4 to N do
  if isprime(n) then next fi;
  s:= numtheory:-sigma(n);
  if not assigned(V[s]) then
     V[s]:= n;
     if s > N then Q:= Q union {n} fi;
  else
     Res:= Res union {n,V[s]};
     if s > N then Q:= Q minus {V[s]} fi;
  fi
od:
convert(select(`<`,Res, min(Q)),list); # Robert Israel, Dec 17 2017

t2 = Table[If[PrimeQ[n], 0, DivisorSigma[1, n]], {n, 1000}]; Select[Range[132], ! PrimeQ[#] && Length[Position[t2, t2[[#]]]] > 1 &] (* T. D. Noe, Feb 27 2012 *)

A248793 Sigma(n) - 1 for n such that sigma(n) - 1 is prime.

Original entry on oeis.org

2, 3, 5, 11, 7, 17, 11, 13, 23, 23, 17, 19, 41, 31, 23, 59, 41, 29, 71, 31, 47, 53, 47, 37, 59, 89, 41, 43, 83, 71, 47, 71, 97, 53, 71, 79, 89, 59, 167, 61, 103, 83, 67, 71, 73, 113, 139, 167, 79, 83, 223, 107, 131, 179, 89, 233, 167, 127, 251, 97, 101, 103

Offset: 1

Jaroslav Krizek, Nov 01 2014

a(n) = corresponding values of primes p = sigma(A248792(n)) - 1, where A248792(n) = numbers n such that sigma(n) - 1 is prime.

If there are at least two numbers k, h such that a(k) = a(h) = p, then p is in A158913.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
OEIS Wiki, Cyclotomic Polynomials at x=n, n! and sigma(n)

Cf. A000203, A000040, A066073, A248792.

[a: n in [1..1000] | IsPrime(a) where a is SumOfDivisors(n)-1]

F:= proc(n)
local r;
r:= numtheory:-sigma(n)-1;
if isprime(r) then r else NULL fi
end proc:
seq(F(n),n=1..1000); # Robert Israel, Nov 02 2014

a248793[n_Integer] :=
Cases[DivisorSigma[1, #] - 1 & /@ Range[n], ?PrimeQ]; a248793[104] (* _Michael De Vlieger, Nov 07 2014 *)

for(n=1,10^3,if(isprime(sigma(n)-1),print1(sigma(n)-1,", "))) \\ Derek Orr, Nov 01 2014

a(n) = A000203(A248792(n)) - 1.

If A248792(n) is a prime p, then a(n) = A248792(n) = p.

A261023 Least number k such that prime(n) = sigma(k) - k - 1.

Original entry on oeis.org

4, 9, 6, 10, 121, 22, 289, 34, 529, 841, 58, 1369, 30, 82, 2209, 42, 3481, 118, 4489, 5041, 70, 6241, 6889, 78, 9409, 10201, 202, 60, 214, 102, 16129, 17161, 18769, 84, 138, 298, 24649, 26569, 27889, 29929, 32041, 358, 36481, 238, 186, 394, 44521, 49729, 51529

Offset: 1

Paolo P. Lava, Aug 07 2015

For any prime k <= p^2. In fact if k = p^2 we have that sigma(p) = sigma(p^2) - p^2, that is 1 + p = 1 + p + p^2 - p^2.

			sigma(2) = 3 and 4 is the least number such that sigma(4) - 4 = 7 - 4 = 3.
sigma(13) = 14 and 22 is the least number such that sigma(22) - 22 = 36 - 22 = 14.

Robert Israel and Paolo P. Lava, Table of n, a(n) for n = 1..1229 (first 100 from Paolo P. Lava)

Cf. A008864, A048050, A070015, A158913.

with(numtheory): P:=proc(q) local a,k,n; for n from 1 to q do
if isprime(n) then for k from 1 to q do
if sigma(n)=sigma(k)-k then print(k); break; fi; od;
fi; od; end: P(10^9);

Table[k = 1; While[DivisorSigma[1, Prime@ p] != DivisorSigma[1, k] - k, k++]; k, {p, 60}] (* Michael De Vlieger, Aug 07 2015 *)

a(n) = my(k = 1, p = prime(n)); while(sigma(k)-k-1 != p, k++); k; \\ Michel Marcus, Aug 12 2015

first(m)=my(v=vector(m),k);for(i=1,m,k=1;while(!(prime(i)==sigma(k)-k-1),k++);v[i]=k;);v; \\ Anders Hellström, Aug 14 2015

a(n) = A070015(A008864(n)). - Robert Israel, Aug 14 2015

A158914 Primes p such that there is a composite c with sigma_2(p)=sigma_2(c).

Original entry on oeis.org

7, 47, 157, 3863

Offset: 1

T. D. Noe, Mar 30 2009

No other terms less than 10^8.
The corresponding composite numbers are 6, 40, 136, and 3352.
Is this sequence finite?
See A158913 for the sequence for sigma_1.
Terms 47, 157, 3863 are x values of solutions to Pell-Fermat equation x^2 - 85*y^2 = 84, where y is also prime and c = 8*y. For any other solution (x,y) formed by primes, x is a term. - Max Alekseyev, Jun 14 2025

tp=DivisorSigma[2,Select[Range[4000],PrimeQ]]; tc=DivisorSigma[2,Select[Range[4000],!PrimeQ[ # ]&]]; Sqrt[Intersection[tp,tc]-1]

A206449 Values of sigma(p) of primes p such that sigma(p) = sigma(c) has a solution for some composite number c.

Original entry on oeis.org

12, 18, 24, 32, 42, 48, 54, 60, 72, 80, 84, 90, 98, 104, 108, 114, 128, 132, 140, 152, 168, 180, 182, 192, 224, 228, 234, 240, 252, 264, 270, 272, 294, 308, 312, 360, 384, 390, 420, 432, 434, 440, 444, 450, 468, 480, 492, 504, 522, 558, 570, 572, 588, 594, 600

Offset: 1

Jaroslav Krizek, Feb 07 2012

nonn

Corresponding values of sigma(p) of primes p from A158913.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203, A158913 (primes p such that there is a composite c with sigma(p)=sigma(c)), A206448 (values of sigma(c) of composite numbers c such that sigma(c) = sigma(d) has a solution for any other composite number d).

Union@ Select[ DivisorSigma[1, Select[ Range@ 100, !PrimeQ@# &]], PrimeQ[# - 1] &] (* Robert G. Wilson v, Feb 09 2012 *)

list(lim) = forprime(p = 1, lim, if(invsigmaNum(p+1) > 1, print1(p+1, ", "))); \\ Amiram Eldar, Dec 16 2024, using Max Alekseyev's invphi.gp

a(n) = A158913(n) + 1.

cp's OEIS Frontend

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

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A206036 Numbers m such that sigma(m) = sigma(k) has solution for distinct numbers m and k.

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A206447 Composite numbers n such that sigma(n) = sigma(d) has solution for some other composite number d.

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A248793 Sigma(n) - 1 for n such that sigma(n) - 1 is prime.

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A261023 Least number k such that prime(n) = sigma(k) - k - 1.

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A158914 Primes p such that there is a composite c with sigma_2(p)=sigma_2(c).

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A206449 Values of sigma(p) of primes p such that sigma(p) = sigma(c) has a solution for some composite number c.

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