A023194 -id:A023194 - cp's OEIS frontend

A163162 Numbers k such that sigma(k) is not prime.

1, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Juri-Stepan Gerasimov, Jul 22 2009

			a(1)=1 because A000203(1) is not prime. a(2)=3 because A000203(3)=3 is not prime. a(3)=5 because sigma(5)=6 is not prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Complement of A023194.

Cf. A000203.

[n: n in [1..80]| not IsPrime(SumOfDivisors(n))]; // Vincenzo Librandi, Jan 18 2013

Select[Range[1000], ! PrimeQ[DivisorSigma[1, #]]&] (* Vincenzo Librandi, Jan 18 2013 *)

isok(k) = !isprime(sigma(k)); \\ Michel Marcus, Mar 10 2022

A000027 \ A023194.

a(n) ~ n. - Charles R Greathouse IV, May 27 2011

Edited by R. J. Mathar, Jul 25 2009

A187825 Smallest k such that the partial sums of the divisors of k (in decreasing order) generate n primes.

Original entry on oeis.org

1, 3, 2, 140, 560, 2160, 2772, 2016, 16830, 5148, 20592, 10640, 69300, 31200, 156240, 177840, 288288, 143520, 927360, 1203840, 752400, 1242360, 2702700, 2948400, 3996720, 1884960, 5896800, 2692800, 1244880, 15800400, 4586400, 11060280, 15301440, 14414400

Offset: 0

Michel Lagneau, Dec 27 2012

nonn

It appears that a(n) is even for n > 0 and nonsquarefree for n > 2. The corresponding triangle of k in which row n gives the n primes starts:
k = 1 -> no prime
k = 3  -> 3;
k = 2  -> 2, 3;
k = 140 -> 293, 307, 317;
k = 560 -> 1373, 1451, 1481, 1487.

			a(3) = 140 because the partial sums of the divisors in decreasing order {140, 70, 35, 28, 20, 14, 10, 7, 5, 4, 2, 1} that generate 3 prime numbers are
140 + 70 + 35 + 28 + 20 = 293;
140 + 70 + 35 + 28 + 20 + 14 = 307;
140 + 70 + 35 + 28 + 20 + 14 + 10 = 317.

Amiram Eldar, Table of n, a(n) for n = 0..77

Cf. A023194, A062700, A000203, A187822.

with(numtheory):for n from 0 to 40
do:ii:=0:for k from 1 to 4000000 while(ii=0) do:s:=0:x:=divisors(k):n1:=nops(x):it:=0:for a from n1 by -1 to 1 do:s:=s+x[a]:if type(s,prime)=true then it:=it+1:else fi:od: if it = n then ii:=1: printf ( "%d %d \n",n,k):else fi:od:od:

a[n_] := Catch[ For[k = 1, True, k++, cnt = Count[ Accumulate[ Divisors[k] // Reverse], ?PrimeQ]; If[cnt == n, Print[{n, k}]; Throw[k]]]]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover, Dec 27 2012 *)

a(19)-a(33) by Jean-François Alcover, Dec 28 2012

A193065 Odd numbers N for which numerator(sigma(N)/N) is a prime.

Original entry on oeis.org

9, 25, 289, 729, 1521, 1681, 2401, 3481, 5041, 7921, 10201, 15625, 17161, 27889, 28561, 29929, 83521, 85849, 146689, 257049, 279841, 458329, 491401, 531441, 552049, 579121, 597529, 683929, 703921, 707281, 734449, 829921, 1190281, 1203409, 1352569, 1394761, 1423249, 1481089, 1885129, 2036329, 2211169

Offset: 1

M. F. Hasler, Jul 15 2011

nonn

This sequence includes all odd terms of A023194.
For most of the terms, sigma(N) is prime (i.e., N is in A023194); the first two exceptions are sigma(a(5))=3*13*61 and sigma(a(20))=13*30941. See A193072 for (the square root of) these exceptions.
It is well known that sigma(N) can't be odd unless N is a square (since sigma is multiplicative and sigma(p^e)=1+...+p^e) or twice a square (excluded here).
See A193066 for the square roots of the terms.
The sequence of numbers n for which A002129(n) is prime starts as this sequence here, but excludes a(5), a(20) etc. - R. J. Mathar, Sep 18 2011

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000203.

Select[Range[1,23*10^5,2],PrimeQ[Numerator[DivisorSigma[1,#]/#]]&] (* Harvey P. Dale, Sep 17 2017 *)

forstep(N=1,1e7,2,isprime(numerator(sigma(N)/N)) && print1(N","))

a(n) = A193066(n)^2.

A193071 Odd numbers N for which sigma(N^2) is not prime.

Original entry on oeis.org

1, 7, 9, 11, 13, 15, 19, 21, 23, 25, 29, 31, 33, 35, 37, 39, 43, 45, 47, 51, 53, 55, 57, 61, 63, 65, 67, 69, 73, 75, 77, 79, 81, 83, 85, 87, 91, 93, 95, 97, 99, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 127, 129, 133, 135, 137, 139, 141, 143, 145, 147

Offset: 1

M. F. Hasler, Jul 15 2011

nonn

The function sigma(n) (=A000203(n)) takes odd values when n is a square or twice a square. Thus, odd numbers n for which sigma(n) is prime (cf. A023194) must be odd squares. This sequence gives the odd numbers whose square yields a composite sum of divisors (or 1).

This is the complement of A193070 in the odd numbers A005408.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Select[Range[1,151,2],!PrimeQ[DivisorSigma[1,#^2]]&] (* Harvey P. Dale, Dec 13 2012 *)

forstep(N=1, 1e7, 2, !isprime(sigma(N^2)) && print1(N", "))

A193071 = A005408 \ A193070.

A248963 Prime powers p^m for which sigma(p^2m) is not prime.

Original entry on oeis.org

1, 7, 9, 11, 13, 16, 19, 23, 25, 29, 31, 32, 37, 43, 47, 53, 61, 67, 73, 79, 81, 83, 97, 103, 107, 109, 113, 121, 127, 128, 137, 139, 149, 151, 157, 163, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, 257, 263, 269, 271, 277, 281, 283, 307, 311, 313, 317, 331

Offset: 1

M. F. Hasler, Oct 18 2014

nonn

sigma(x) cannot be prime unless x is a square of a prime power, x = p^2m, cf. A055638 and A023194. This sequence lists the complement: prime powers whose square does not have a prime sum of divisors.

Although generally 1 is not considered a prime power, it seemed logical for various good reasons to include the initial term a(1)=1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000961, A055638, A023194, A023195, A062700.

for(n=1,999,isprimepower(n)||next;isprime(sigma(n^2))||print1((n)","))

A248963 = A000961 \ A055638, i.e., the complement of A055638 in A000961.

A249760 Numbers k such that k+1 and sigma(k) are both primes.

Original entry on oeis.org

2, 4, 16, 65536

Offset: 1

Jaroslav Krizek, Nov 13 2014

4 is the only number k such that k-1 and sigma(k) are both primes.
Corresponding values of k+1 and sigma(k) are in A249759 and A249761.
Conjectures: (1) sequence is finite; (2) a(n) + 1 is a Fermat prime (A019434); (3) sigma(a(n)) is a Mersenne prime (A000668).
Subsequence of A023194, and by a comment in that entry it follows that each term is a prime power. From that conjectures (2) and (3) above easily follow. - Jeppe Stig Nielsen, Jan 13 2015

			16 is a term because 16+1=17 and sigma(16)=31 are both primes.

Cf. A000203, A000668, A019434, A249759, A249761.

[n: n in [1..10^8] | IsPrime(n+1) and IsPrime(SumOfDivisors(n))];

Select[Range[10^5], PrimeQ[# + 1]&& PrimeQ[DivisorSigma[1, #]] &] (* Vincenzo Librandi, Nov 14 2014 *)

a(n) = A249759(n) - 1.

A252040 Least number k such that the sum of the n-th powers of divisors of k is prime, or 0 if there is no such k.

Original entry on oeis.org

2, 2, 4, 2, 3418801, 0, 64, 2, 4, 0, 21611482313284249, 0, 38580382095460899155325996786287338133521, 0, 0, 2, 2908327322588596409721563830760333292497745890881, 0, 366742604046618684582110328334808145525757868881, 0, 0, 0, 264638735999480827156185738343112840094453729311824181089

Offset: 1

Derek Orr, Dec 12 2014

nonn

a(n) = 2 if and only if 2^n + 1 is in A019434.
From Jinyuan Wang, Jan 30 2021: (Start)
a(n) = 0 if n > 1 is not a prime power. Proof: note that sigma_n(k) = Product_{i=1..m} (1 + p_i^n + ... + p_i^(n*e_i)), where k = Product_{i=1..m} p_i^e_i. We only need to prove when n > 1 is not a prime power and e > 1, s = Sum_{i=0..e-1} p^(n*i) = (p^(n*e) - 1)/(p^n - 1) is composite. If e is prime, then s is divisible by (p^(e^(t+1)) - 1)/(p^(e^t) - 1), where t is the e-adic valuation of n. If e is composite, then s is divisible by (p^(n*q) - 1)/(p^n - 1), where q is a prime factor of e.
Corollary: k must be of the form p^(e - 1) when n = e^t, where p and e are primes. Therefore, a(2^t) = 0 if 2^2^t + 1 is composite. (End)

			2 has two divisors, 2 and 1. 2^3 + 1^3 = 9 is not prime.
3 has two divisors, 3 and 1. 3^3 + 1^3 = 28 is not prime.
4 has three divisors, 4, 2, and 1. 4^3 + 2^3 + 1^3 = 73 is prime. So, a(3) = 4.

Jinyuan Wang, Table of n, a(n) for n = 1..50

Cf. A000203, A001157, A001158, A001159, A001160, A019434.

Cf. A023194 (sigma(n) is prime), A063783 (sigma_3(n) is prime).

a252040[n_Integer] := If[PrimePowerQ[n] && (p=First@ First@ FactorInteger[n])>2, q=2; While[!PrimeQ[DivisorSigma[n, q^(p-1)]], q=NextPrime[q]]; q^(p-1), 2*Boole[PrimeQ[2^n+1]]]; a252040 /@ Range[10] (* Michael De Vlieger, Dec 13 2014 *) (* modified by Jinyuan Wang, Jan 30 2021 *)

a(n) = if(isprimepower(n, &p) && p>2, my(q=2); while(!ispseudoprime(sigma(q^(p-1), n)), q=nextprime(q+1)); q^(p-1), 2*isprime(2^n+1)); \\ Modified by Jinyuan Wang, Jan 25 2021

Name edited by and more terms from Jinyuan Wang, Jan 30 2021

A275938 Numbers m such that d(m) is prime while sigma(m) is not prime (where d(m) = A000005(m) and sigma(m) = A000203(m)).

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277

Offset: 1

Altug Alkan, Aug 12 2016

From Robert Israel, Aug 12 2016: (Start)
d(m) is prime iff m = p^k where p is prime and k+1 is prime.
For such m, sigma(m) = 1 + p + ... + p^k = (p*m-1)/(p-1).
The sequence contains 2^(q-1) for q in A054723,
3^(q-1) for q prime but not in A028491,
5^(q-1) for q prime but not in A004061,
7^(q-1) for q prime but not in A004063, etc.
In particular, it contains all odd primes. (End)

			49 is a term because A000005(49) = 3 is prime while sigma(49) = 57 is not.

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

Cf. A000005, A000203, A004061, A004063, A009087, A023194, A028491.

Cf. A000040, A286095.

N:= 1000: # to get all terms <= N
P:= select(isprime, {2,seq(p,p=3..N,2)}):
fp:= proc(p) local q,res;
  q:= 2;
  res:= NULL;
  while p^(q-1) <= N do
     if not isprime((p^q-1)/(p-1)) then res:= res, p^(q-1) fi;
     q:= nextprime(q);
  od;
  res;
end proc:
sort(convert(map(fp, P),list)); # Robert Israel, Aug 12 2016

lista(nn) = for(n=1, nn, if(isprime(numdiv(n)) && !isprime(sigma(n)), print1(n, ", ")));

UNION of A000040 and A286095 (except for the term 2). - Bill McEachen, Jul 16 2024

A278913 a(n) is the smallest number k with prime sum of divisors such that tau(k) = n-th prime.

Original entry on oeis.org

2, 4, 16, 64, 9765625, 4096, 65536, 262144, 1471383076677527699142172838322885948765175969, 10264895304762966931257013446474591264089923314972889033759201, 1073741824, 18701397461209715023927088008788055619800417991632621566284510161

Offset: 1

Jaroslav Krizek, Nov 30 2016

nonn

tau(n) = A000005(n) = the number of divisors of n.

a(11) = 1073741824; a(n) > A023194(10000) = 5896704025969 for n = 9, 10 and n >= 12.

			a(3) = 16 because 16 is the smallest number with prime values of sum of divisors (sigma(16) = 31) such that tau(16) = 5 = 3rd prime.

Davin Park, Table of n, a(n) for n = 1..50

Cf. A000005, A000203, A023194, A278914.

A278913:=func; [A278913(n): n in[1..8]];

A278913[n_] := NestWhile[NextPrime, 2, ! PrimeQ[Cyclotomic[Prime[n], #]] &]^(Prime[n] - 1) (* Davin Park, Dec 28 2016 *)

a(n) = {my(k=1); while(! (isprime(sigma(k)) && isprime(p=numdiv(k)) && (primepi(p) == n)), k++); k;} \\ Michel Marcus, Dec 03 2016

a(n) = A123487(n)^(prime(n)-1). - Davin Park, Dec 10 2016

More terms from Davin Park, Dec 08 2016

A286095 Composite numbers n such that tau(n) (number of divisors of n) is prime and sigma(n) (sum of divisors of n) is not prime.

Original entry on oeis.org

49, 81, 121, 169, 361, 529, 625, 841, 961, 1024, 1369, 1849, 2209, 2809, 3721, 4489, 5329, 6241, 6889, 9409, 10609, 11449, 11881, 12769, 14641, 16129, 18769, 19321, 22201, 22801, 24649, 26569, 32041, 32761, 36481, 37249, 38809, 39601, 44521, 49729, 51529, 52441, 54289

Offset: 1

Bernard Schott, May 22 2017

nonn

If sigma(n) is prime (A023194) then tau(n) is prime too. (See Crux Mathematicorum link.)
But the reverse is false; the numbers which verify tau(n) prime and sigma(n) not prime are in the sequence A275938.
All odd primes belong to the sequence A275938, but there are also in this sequence composite numbers which are all prime powers, these prime powers are here.

			tau(49) = 3 and sigma(49) = 57 = 3 * 19.

Robert Israel, Table of n, a(n) for n = 1..10000
Peter A. Lindstrom and Andy Liu, Problem 465, Crux Mathematicorum, page 188, Vol.6 , Jun. 80.

Cf. A000005, A000203, A023194, A275938.

for n from 2 to 550000 do p(n):=tau(n);
if not isprime(n) and is prime(p(n)) and not isprime(sigma(n)) then print (n,p(n),sigma(n)) else fi; od:
# alternative
N:= 10^5: # to get all terms <= N
P:= select(isprime, [2,seq(i,i=3..isqrt(N),2)]):
S:= {}:
for p in P do
  k:= 1:
  do
    k:= nextprime(k+1)-1;
    if p^k > N then break fi;
    if not isprime((p^(k+1)-1)/(p-1)) then S:= S union {p^k} fi
  od
od:
sort(convert(S,list)); # Robert Israel, Jun 05 2017

Select[Range[10^5], Function[n, And[CompositeQ@ n, Map[PrimeQ@ DivisorSigma[#, n] &, {0, 1}] == {True, False}]]] (* Michael De Vlieger, May 24 2017 *)

lista(nn) = {forcomposite(n=1, nn, if (isprime(numdiv(n)) && !isprime(sigma(n)), print1(n, ", ")););} \\ Michel Marcus, May 24 2017

cp's OEIS Frontend

A163162 Numbers k such that sigma(k) is not prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A187825 Smallest k such that the partial sums of the divisors of k (in decreasing order) generate n primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A193065 Odd numbers N for which numerator(sigma(N)/N) is a prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A193071 Odd numbers N for which sigma(N^2) is not prime.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

PARI

Formula

A248963 Prime powers p^m for which sigma(p^2m) is not prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A249760 Numbers k such that k+1 and sigma(k) are both primes.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

Formula

A252040 Least number k such that the sum of the n-th powers of divisors of k is prime, or 0 if there is no such k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions