A058340 -id:A058340 - cp's OEIS frontend

A066073 Composite numbers k such that sigma(k) - 1 is prime.

6, 10, 14, 15, 20, 21, 24, 26, 30, 33, 34, 35, 38, 40, 44, 46, 51, 52, 55, 57, 58, 60, 63, 65, 74, 76, 78, 84, 85, 86, 88, 90, 92, 93, 96, 105, 111, 114, 117, 118, 120, 123, 124, 126, 130, 135, 136, 141, 143, 145, 147, 153, 155, 158, 161, 164, 166, 168, 172, 174

Offset: 1

Labos Elemer, Dec 03 2001

Composite numbers k such that sigma(k) = sigma(p) has a solution in the primes p. - Jaroslav Krizek, Feb 03 2012
Complement of A000040 (primes) with respect to A248792 (numbers n such that sigma(n) - 1 is prime). - Jaroslav Krizek, Nov 13 2014
Numbers k such that sigma(k) - 1 is greater than k and prime. - Giuseppe Coppoletta, Dec 22 2014

			30, 46, 51, and 55 are in the sequence because each is a composite number n such that sigma(n)-1 = 71, which is prime; 71 itself is excluded from the sequence by definition.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000040, A000203, A058340, A066071, A066072, A066073, A066074, A066075, A066076, A066077, A066080, A248792.

filter:= proc(n)
local s;
s:= numtheory:-sigma(n)-1;
  s > n and isprime(s);
end proc:
select(filter, [$2..1000]); # Robert Israel, Dec 22 2014

Do[s=-1+DivisorSigma[1, m]; If[PrimeQ[s]&&!PrimeQ[m], Print[m]], {m, 1, 256}]
Select[Range[200],CompositeQ[#]&&PrimeQ[DivisorSigma[1,#]-1]&] (* Harvey P. Dale, Jan 13 2025 *)

isA066073(n)=!isprime(n)&&isprime(sigma(n)-1) \\ Charles R Greathouse IV, Feb 20 2012

[n for n in (2..174) if (sigma(n)-1).is_prime() and sigma(n)-1>n] # Giuseppe Coppoletta, Dec 22 2014

A066071 Nonprime numbers k such that phi(k) + 1 is prime.

Original entry on oeis.org

1, 4, 6, 8, 9, 10, 12, 14, 18, 21, 22, 26, 27, 28, 32, 34, 36, 38, 40, 42, 46, 48, 49, 54, 55, 57, 58, 60, 62, 63, 74, 75, 76, 77, 82, 86, 88, 91, 93, 94, 95, 98, 99, 100, 106, 108, 110, 111, 114, 115, 117, 118, 119, 122, 124, 125, 126, 132, 133, 134, 135, 142, 145, 146

Offset: 1

Labos Elemer, Dec 03 2001

nonn

A039698 with the primes removed. For every prime p, 2p is in the sequence. - Ray Chandler, May 26 2008

Includes 3*p for p in A005382 and p^2 for p in A065508. - Robert Israel, Dec 29 2017

			Solutions to 1+phi(x)=13 are {13, 21, 26, 28, 36, 42} of which the 5 composites are in the sequence.

Iain Fox, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

Cf. A000010, A000040, A005382, A006093, A039649, A039698, A058339, A058340, A065508, A066072, A066073, A066074, A066075, A066076, A066077, A066080.

[n: n in [1..200] |not IsPrime(n) and IsPrime(EulerPhi(n)+1)]; // Vincenzo Librandi, Jul 02 2016

select(n -> not isprime(n) and isprime(1+numtheory:-phi(n)), [$1..1000]); # Robert Israel, Dec 29 2017

Select[Complement[Range@ #, Prime@ Range@ PrimePi@ #] &@ 150, PrimeQ[EulerPhi@ # + 1] &] (* Michael De Vlieger, Jul 01 2016 *)

isok(k) = { !isprime(k) && isprime(eulerphi(k) + 1) } \\ Harry J. Smith, Nov 10 2009

A066076 Primes p such that there is a unique solution to p = sigma(x) - 1.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 29, 37, 43, 61, 67, 73, 101, 109, 137, 149, 157, 163, 173, 193, 197, 199, 211, 229, 241, 257, 277, 281, 283, 313, 317, 331, 337, 347, 349, 353, 367, 373, 379, 397, 401, 409, 421, 457, 461, 463, 487, 499, 509, 523, 541

Offset: 1

Labos Elemer, Dec 03 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000040, A000203, A058340, A066071, A066072, A066073, A066074, A066075, A066076, A066077, A066080.

With[{s = KeySort@ PositionIndex@ Array[DivisorSigma[1, #] - 1 &, 10^5]}, Take[#, 51] &@ Keys@ KeySelect[s, PrimeQ@ # && Length@ Lookup[s, #] == 1 &]] (* Michael De Vlieger, Jul 16 2017 *)

{ n=0; for (m=1, 10^9, p=prime(m); a=1; for (x=1, p - 1, if (p == (sigma(x) - 1), a++; break)); if (a==1, write("b066076.txt", n++, " ", p); if (n==1000, return)) ) } \\ Harry J. Smith, Nov 10 2009

is(n) = isprime(n) && invsigmaNum(n + 1) == 1; \\ Amiram Eldar, Aug 18 2024, using Max Alekseyev's invphi.gp

If A066075(m) = 1, then prime(m) is a term.

A066074 Primes arising in A066073.

Original entry on oeis.org

11, 17, 23, 23, 41, 31, 59, 41, 71, 47, 53, 47, 59, 89, 83, 71, 71, 97, 71, 79, 89, 167, 103, 83, 113, 139, 167, 223, 107, 131, 179, 233, 167, 127, 251, 191, 151, 239, 181, 179, 359, 167, 223, 311, 251, 239, 269, 191, 167, 179, 227, 233, 191, 239, 191, 293

Offset: 1

Labos Elemer, Dec 03 2001

nonn

			p=71 appears in the sequence at 9th, 16th, 17th and 19th positions as -1+sigma(x) for x=30, 46, 51, 55.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000203, A058340, A066071, A066072, A066073, A066074, A066075, A066076, A066077, A066080.

Do[s=-1+DivisorSigma[1, n]; If[PrimeQ[s]&&!PrimeQ[n], Print[s]], {n, 1, 256}]

{ n=0; for (m=1, 10^9, if (!isprime(m) && isprime(p=sigma(m) - 1), write("b066074.txt", n++, " ",p); if (n==1000, return)) ) } \\ Harry J. Smith, Nov 10 2009

lista(nn) = forcomposite(n=1, nn, if (isprime(p=(sigma(n)-1)), print1(p, ", "))); \\ Michel Marcus, Jan 05 2018

A066075 Number of solutions x to prime(n) = sigma(x) - 1, where prime(n) is the n-th prime.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 3, 2, 3, 1, 1, 5, 1, 2, 3, 3, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 6, 1, 4, 2, 5, 1, 1, 1, 1, 3, 3, 1, 3, 7, 1, 6, 1, 2, 3, 2, 1, 1, 1, 3, 2, 4, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 6, 2, 1, 1, 1, 4, 1, 8, 4, 2, 2, 3, 1, 1, 1, 3, 9, 1, 2, 1, 10, 1, 2, 1, 1

Offset: 1

Labos Elemer, Dec 03 2001

nonn

prime(n) itself is always the largest solution, but often composite solutions also occur.

If a(n) = 1, then the single solution is prime(n).

			For n = 96, prime(96) = 503, 503 = sigma(x)-1 has 10 solutions together with 503: {204, 220, 224, 246, 284, 286, 334, 415, 451, 503}, so a(96) = 10.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Number of solutions to A000040(n) = A000203(x) - 1.

Cf. A000040, A000203, A054973, A058340, A066071, A066072, A066073, A066074, A066075, A066076, A066077, A066080.

{ for (n=1, 1000, a=1; for (x=1, prime(n) - 1, if (prime(n) == (sigma(x) - 1), a++)); write("b066075.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 10 2009

a(n) = invsigmaNum(prime(n)+1); \\ Amiram Eldar, Dec 16 2024, using Max Alekseyev's invphi.gp

a(n) = A054973(prime(n)+1). - Amiram Eldar, Dec 16 2024

A066077 a(n) is the number of x such that sigma(x)-1 is 0 or one of the first n-1 primes.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 10, 11, 14, 15, 17, 18, 21, 22, 25, 27, 30, 31, 32, 37, 38, 40, 43, 46, 48, 49, 51, 53, 54, 56, 58, 60, 61, 63, 64, 66, 67, 68, 74, 75, 79, 81, 86, 87, 88, 89, 90, 93, 96, 97, 100, 107, 108, 114, 115, 117, 120, 122, 123, 124, 125, 128, 130, 134, 135

Offset: 1

Labos Elemer, Dec 03 2001

nonn

Former name was: Smallest x such that p(n) = Sigma[x] - 1. That did not match the Data. See A296375 for that sequence.

Iain Fox, Table of n, a(n) for n = 1..10000

Cf. A000040, A000203, A058339, A058340, A066071, A066072, A066073, A066074, A066075, A066076, A066077, A066080.

N:= 100: # To get a(1)..a(N)
P:= ithprime(N-1):
S:= select(t -> isprime(t) and t <= P,map(-1+numtheory:-sigma, [$1..P])):
T:= Statistics:-Tally(sort(S),output=table):
ListTools:-PartialSums([1,seq(T[ithprime(i)],i=1..N-1)]); # Robert Israel, Dec 27 2017

first(n) = my(res = vector(n), a = 1); res[1] = 1; for(k=2, n, for(x=1, prime(k-1), if(prime(k-1) == (sigma(x) - 1), a++)); res[k] = a); res \\ Iain Fox, Dec 28 2017

a(n+1)-a(n) = A066075(n).

Edited by Robert Israel, Dec 27 2017

A066072 Prime numbers arising in A066071.

Original entry on oeis.org

2, 3, 3, 5, 7, 5, 5, 7, 7, 13, 11, 13, 19, 13, 17, 17, 13, 19, 17, 13, 23, 17, 43, 19, 41, 37, 29, 17, 31, 37, 37, 41, 37, 61, 41, 43, 41, 73, 61, 47, 73, 43, 61, 41, 53, 37, 41, 73, 37, 89, 73, 59, 97, 61, 61, 101, 37, 41, 109, 67, 73, 71, 113, 73, 73, 41, 73, 97, 61, 79, 83

Offset: 1

Labos Elemer, Dec 03 2001

nonn

			Solutions to 1+phi(x)=13 are {13, 21, 26, 28, 36, 42}; so 13 occur 5 times in the sequence in positions 10, 13, 15, 18 and 21, obtained as 1+Phi[m] values for 5 composite arguments.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000010, A058340, A066071, A066072, A066073, A066074, A066075, A066076, A066077, A066080.

Select[EulerPhi@ # + 1 & /@ Complement[Range@ #, Prime@ Range@ PrimePi@ #] &@ 168, PrimeQ] (* Michael De Vlieger, Jul 01 2016 *)

{ n=0; for (m=1, 10^9, if (!isprime(m) && isprime(p=(eulerphi(m) + 1)), write("b066072.txt", n++, " ", p); if (n==1000, return)) ) } \\ Harry J. Smith, Nov 10 2009

a(n) = 1 + phi(A066071(n)).

A058339 Number of solutions to 1 + phi(x) = prime(n), where phi is A000010.

Original entry on oeis.org

2, 3, 4, 4, 2, 6, 6, 4, 2, 2, 2, 8, 9, 4, 2, 2, 2, 9, 2, 2, 17, 2, 2, 6, 17, 4, 2, 2, 9, 6, 2, 2, 2, 2, 2, 2, 7, 4, 2, 2, 2, 10, 2, 21, 2, 2, 2, 2, 2, 2, 6, 2, 31, 2, 10, 2, 2, 2, 9, 8, 2, 2, 2, 2, 16, 2, 2, 18, 2, 6, 12, 2, 2, 2, 2, 2, 2, 13, 13, 6, 2, 13, 2, 34

Offset: 1

Labos Elemer, Dec 14 2000

nonn

			The equation phi(x) = p-1 always has at least 2 solutions: p and 2p a prime and a composite. Many times more than 2 x gives phi(x) = p-1. For p-1 = 96 there are 17 (that is, an odd number of) solutions: {97, 119, 153, 194, 195, 208, 224, 238, 260, 280, 288, 306, 312, 336, 360, 390, 420}, 4 odd and 13 even numbers while for p-1 = 100 there are 4 (an even number of) solutions: {101, 125, 202, 250}. For all odd solutions x, 2x is also a solution.
1+phi(x) = 11 has 2 solutions: 11 and 22; 1+phi(x) = 241 has 31 solutions: x = {241, 287, 305, 325, 369, 385, 429, 465, 482, 488, 495, 496, 525, 572, 574, 610, 616, 620, 650, 700, 732, 738, 744, 770, 792, 858, 900, 924, 930, 990, 1050}.

T. D. Noe, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
David M. Bressoud, CNT.m, Computational Number Theory Mathematica package.

Cf. A000010, A000040, A006093, A058340, A066071, A066072, A066073, A066074, A066075, A066076, A066077, A066080.

with(numtheory): >[seq(nops(invphi(-1+ithprime(i))),i=1..256)];

Needs["CNT`"]; Table[Length[PhiInverse[Prime[n] - 1]], {n, 100}] (* T. D. Noe, Dec 11 2013 *)
Take[Length /@ Values@ KeySelect[KeyMap[# + 1 &, PositionIndex@ Array[EulerPhi, 10^4]], PrimeQ], 84] (* Michael De Vlieger, Dec 29 2017 *)

a(n) = invphiNum(prime(n) - 1); \\ Amiram Eldar, Aug 18 2024, using Max Alekseyev's invphi.gp

a(n) = A210500(n) + A210501(n). - Arkadiusz Wesolowski, Jan 19 2013

Offset corrected by Arkadiusz Wesolowski, Jan 19 2013

A138537 Primes p_n for which A140141(n) = 2p_n, where p_n = n-th prime (A000040).

Original entry on oeis.org

11, 23, 29, 31, 47, 53, 59, 67, 71, 79, 83, 103, 107, 127, 131, 137, 139, 149, 151, 167, 173, 179, 191, 197, 199, 211, 223, 227, 229, 239, 251, 263, 269, 271, 283, 293, 307, 311, 317, 331, 347, 359, 367, 373, 379, 383, 389, 419, 431, 439, 443, 463, 467, 479

Offset: 1

Vladimir Shevelev, May 10 2008

nonn

Perhaps the same as A058340, but need proof. - Ray Chandler, May 20 2008

The first member of this sequence not in A058340 is 295937. - Robert Israel, Aug 12 2016

Robert Israel, Table of n, a(n) for n = 1..20000

Cf. A000010, A000040, A039649, A058340, A140141.

filter:= n -> isprime(n) and numtheory:-invphi(numtheory:-phi(n))[2] = 2*n:
select(filter, [seq(i,i=2..1000)]); # Robert Israel, Aug 12 2016

Corrected and extended by Ray Chandler, May 20 2008

A138539 Primes p_n for which A140141(n) < 2p_n, where p_n = n-th prime (A000040).

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 37, 41, 43, 61, 73, 89, 97, 101, 109, 113, 157, 163, 181, 193, 233, 241, 257, 277, 281, 313, 337, 349, 353, 397, 401, 409, 421, 433, 449, 457, 461, 487, 521, 541, 577, 593, 601, 613, 617, 641, 661, 673, 701, 733, 757, 761, 769, 821, 829

Offset: 1

Vladimir Shevelev, May 10 2008

nonn

Apparently this is the union of {2} and A058341. - R. J. Mathar, Jul 03 2009

This sequence is the complement of A058340. - Torlach Rush, Jun 29 2018

Cf. A140141, A039649, A000010, A000040, A058340.

Corrected and extended by Ray Chandler, May 20 2008

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A066073 Composite numbers k such that sigma(k) - 1 is prime.

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A066071 Nonprime numbers k such that phi(k) + 1 is prime.

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A066076 Primes p such that there is a unique solution to p = sigma(x) - 1.

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A066074 Primes arising in A066073.

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A066075 Number of solutions x to prime(n) = sigma(x) - 1, where prime(n) is the n-th prime.

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A066077 a(n) is the number of x such that sigma(x)-1 is 0 or one of the first n-1 primes.

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A066072 Prime numbers arising in A066071.

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