A023194 -id:A023194 - cp's OEIS frontend

A279094 Smallest k such that sigma(k^n) is prime.

2, 2, 4, 2, 25, 2, 59049, 4, 4, 5, 256, 2, 282475249, 243, 4, 2, 729, 2, 1174562876521148458974062689, 8, 64, 16, 25, 1331, 594823321, 16807, 38950081, 151, 361, 2, 470541197898347534873984161, 19902511, 241081, 27, 9, 61, 625, 34271896307633, 73441, 53, 1681

Offset: 1

Jon E. Schoenfield, Mar 11 2017

nonn

For any number k with two or more distinct prime divisors, the sum of divisors of k^n is composite, so each term is of the form p^j where p is prime and j >= 1, i.e., all terms are prime powers (A246655). Additionally, sigma(k^n) = sigma(p^(j*n)) = (p^(j*n + 1) - 1)/(p - 1) is composite when j*n + 1 is composite, so a(n) must be of the form p^j where j*n + 1 is prime.

			a(1) = 2 because sigma(1^1) = sigma(1) = 1 (not prime), but sigma(2^1) = sigma(2) = 1 + 2 = 3 (prime).
a(3) = 4 because sigma(1^3) = 1 (not prime), sigma(2^3) = 1 + 2 + 4 + 8 = 15 (composite), sigma(3^3) = 1 + 3 + 9 + 27 = 40 (composite), but sigma(4^3) = sigma(2^6) = 1 + 2 + 4 + 8 + 16 + 32 + 64 = 127 (prime).
a(19) = 1174562876521148458974062689 = 17^22 because sigma((17^22)^19) is prime and sigma(k^19) is not prime for any smaller value of k.

Jon E. Schoenfield, Table of n, a(n) for n = 1..200

Cf. A000203, A023194, A055638, A246655.

A085379 Greatest prime as sum of distinct divisors of n.

Original entry on oeis.org

3, 3, 7, 5, 11, 7, 13, 13, 17, 11, 23, 13, 23, 23, 31, 17, 37, 19, 41, 31, 23, 23, 59, 31, 41, 37, 53, 29, 71, 31, 61, 47, 53, 47, 89, 37, 59, 53, 89, 41, 89, 43, 83, 73, 71, 47, 113, 7, 83, 71, 97, 53, 113, 71, 113, 79, 89, 59, 167, 61, 31, 103, 127, 83, 139, 67

Offset: 2

Reinhard Zumkeller, Jun 26 2003

nonn

			The divisors of n = 50 are {1,2,5,10,25,50}, the sums of distinct divisors that are prime: 2, 3 = 2+1, 5, 7 = 5+2, 11 = 10+1, 13 = 10+2+1, 17 = 10+5+2, 31 = 25+5+1, 37 = 25+10+2, 41 = 25+10+5+1, 43 = 25+10+5+2+1, 53 = 50+2+1, 61 = 50+10+1, 67 = 50+10+5+2 and 83 = 50+25+5+2+1. Therefore a(50) = 83 < 89 = A070801(50) and A085381(3) = 50.

Amiram Eldar, Table of n, a(n) for n = 2..10001
Eric Weisstein's World of Mathematics, Divisor Function.

Cf. A000203, A023194, A062700, A070801, A085380, A085381.

a[n_] := Module[{d = Divisors[n], c, x}, c = Rest@CoefficientList[Series[Product[1 + x^d[[i]], {i, 1, Length[d]}], {x, 0, Total[d]}], x]; Max[Select[Position[c, ?(# > 0 &)] // Flatten, PrimeQ]]]; Array[a, 100, 2] (* _Amiram Eldar, Mar 01 2024 *)

a(n) <= A070801(n) <= A000203(n).
a(A085380(n)) = A070801(A085380(n)).
a(A085381(n)) < A070801(A085381(n)).
a(A023194(n)) = A000203(A023194(n)) = A062700(n).

A195268 Numbers whose sum of odd divisors is prime.

Original entry on oeis.org

9, 18, 25, 36, 50, 72, 100, 144, 200, 288, 289, 400, 576, 578, 729, 800, 1152, 1156, 1458, 1600, 1681, 2304, 2312, 2401, 2916, 3200, 3362, 3481, 4608, 4624, 4802, 5041, 5832, 6400, 6724, 6962, 7921, 9216, 9248, 9604, 10082, 10201, 11664, 12800

Offset: 1

Michel Lagneau, Sep 14 2011

nonn

Odd numbers k^2 such that sigma(k^2) is prime, times an arbitrary power of two. - Charles R Greathouse IV, Sep 14 2011

			The divisors of 2312 are { 1, 2, 4, 8, 17, 34, 68, 136, 289, 578, 1156, 2312 }, and the sum of the odd divisors 1 + 17 + 289 = 307 is prime. Hence 2312 = 2*34^2 is in the sequence.

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

Subsequence of A028982.

Cf. A023194, A055638.

with(numtheory):for n from 1 to 20000 do:x:=divisors(n):n1:=nops(x):s:=0:for m from 1 to n1 do:if irem(x[m],2)=1 then s:=s+x[m]:fi:od:if type(s,prime)=true  then printf(`%d, `,n): else fi:od:

Select[Range[13000], PrimeQ[DivisorSigma[1, #/2^IntegerExponent[#, 2]]] &] (* Amiram Eldar, Jul 31 2022 *)

list(lim)=my(v=List(),t);forstep(k=3,sqrt(lim),2,if(isprime(sigma(t=k^2)),listput(v,t);while((t<<=1)<=lim,listput(v,t)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Sep 14 2011

A200981 Numbers n such that the sum of non-divisors of n is prime.

Original entry on oeis.org

3, 4, 10, 21, 34, 46, 58, 70, 85, 93, 118, 129, 130, 144, 178, 201, 226, 237, 262, 298, 310, 322, 324, 325, 333, 334, 346, 382, 406, 418, 430, 466, 478, 502, 513, 514, 517, 549, 598, 622, 633, 634, 657, 658, 669, 706, 730, 742, 813, 826, 837, 838

Offset: 1

Paolo P. Lava, Dec 13 2011

			Non-divisors of 10 are 3, 4, 6, 7, 8, 9 and their sum is 37 that is prime.

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Cf. A023194, A024816.

with(numtheory);
P:=proc(i)
local a,n;
for n from 1 to i do
  a:=n*(n+1)/2-sigma(n); if isprime(a) then print(n); fi;
od;
end:
P(1000000);

okQ[n_] := (n > 0) && PrimeQ[n]; Select[Range[1000], okQ[# (#+1)/2 - DivisorSigma[1, #]] &] (* T. D. Noe, Dec 15 2011 *)

A229266 Primes of the form sigma(k) + tau(k) + phi(k), where sigma(k) = A000203(k), tau(k) = A000005(k) and phi(k) = A000010(k).

Original entry on oeis.org

3, 23, 557, 1289, 2447, 3779, 9209, 10331, 11351, 18367, 14051, 34351, 42953, 67883, 95717, 96587, 134807, 164249, 193057, 310553, 253159, 321397, 383723, 548213, 657311, 499151, 630023, 516251, 732181, 713927, 927013, 932431, 784627, 906473, 855331, 1121987

Offset: 1

Paolo P. Lava, Sep 18 2013

nonn

			The third term of A229265 is 200 and sigma(200) +  tau(200) + phi(200) = 465 + 12 + 80 = 557 is prime.

Cf. A000005, A000010, A000203, A009087, A023194, A038344, A055813, A062700, A064205, A115919, A141242, A229264, A229265, A065061, A229268.

with(numtheory); P:=proc(q) local a, n; for n from 1 to q do a:=sigma(n)+tau(n)+phi(n);
if isprime(a) then print(a); fi; od; end: P(10^6);

Select[Table[DivisorSigma[0,n]+DivisorSigma[1,n]+EulerPhi[n],{n,10^6}],PrimeQ] (* Harvey P. Dale, Oct 03 2023 *)

A232444 Numbers n such that sigma(n) and sigma(n^2) are primes.

Original entry on oeis.org

2, 4, 64, 289, 729, 15625, 7091569, 7778521, 11607649, 15912121, 43546801, 56957209, 138980521, 143688169, 171845881, 210801361, 211673401, 253541929, 256224049, 275792449, 308810329, 329386201, 357172201, 408807961, 499477801, 531625249, 769341169, 1073741824, 1260747049

Offset: 1

Alex Ratushnyak, Nov 24 2013

nonn

Intersection of A023194 and A055638.
Sigma(n) = A000203(n) = sum of divisors of n.
Terms a(2)...a(29) are squares of 2, 8, 17, 27, 125, 2663, 2789, 3407, 3989, 6599, 7547, 11789, 11987, 13109, 14519, 14549, 15923, 16007, 16607, 17573, 18149, 18899, 20219, 22349, 23057, 27737, 32768, 35507.

			4 is in the sequence because both sigma(4)=7 and sigma(4^2)=31 are primes.

Donovan Johnson and Chai Wah Wu, Table of n, a(n) for n = 1..10385 [Terms from 1 to 500 from Donovan Johnson]

Cf. A000203, A023194, A055638.

isok(n) = isprime(sigma(n)) && isprime(sigma(n^2)); \\ Michel Marcus, Nov 26 2013

from sympy import isprime, divisor_sigma
A232444_list = [2]+[n for n in (d**2 for d in range(1,10**4)) if isprime(divisor_sigma(n)) and isprime(divisor_sigma(n**2))] # Chai Wah Wu, Jul 23 2016

a(6)-a(12) from Michel Marcus, Nov 26 2013

a(13)-a(29) from Alex Ratushnyak, Nov 26 2013

A247838 Numbers k such that sigma(sigma(k)) is prime.

Original entry on oeis.org

3, 2667, 3937, 57337, 172011, 253921, 677207307, 1073602561, 732959441001382539, 750688035198863979, 1000923107604038521, 1108158528150703969

Offset: 1

Jaroslav Krizek, Sep 28 2014

Numbers k such that A051027(k) is a prime p.
Prime 3 is the only prime p such that sigma(sigma(p)) is a prime q.
Conjecture: Subsequence of A046528 (numbers that are a product of distinct Mersenne primes).
Corresponding values of primes p: 7, 8191, 8191, 131071, 524287, 524287, ... (A247822). Conjecture: values of primes p is equal to Mersenne primes (A000668).
732959441001382539, 750688035198863979, 1000923107604038521, 1108158528150703969 and 196751176038481899983340171 are terms. - Jaroslav Krizek, Mar 25 2015
a(9) > 10^10. - Michel Marcus, Feb 13 2020
a(13) > 10^19. - Giovanni Resta, Feb 14 2020

			2667 is a term because sigma(sigma(2667)) = sigma(4096) = 8191 (i.e., prime).

Cf. A000203, A023194, A063103, A000668, A046528, A051027, A247821, A247822, A247954.

[n: n in [1..10000000] | IsPrime(SumOfDivisors(SumOfDivisors(n)))];

with(numtheory): A247838:=n->`if`(isprime(sigma(sigma(n))),n,NULL): seq(A247838(n), n=1..10^5); # Wesley Ivan Hurt, Oct 02 2014

Select[Range[260000],PrimeQ[DivisorSigma[1,DivisorSigma[1,#]]]&] (* The program generates the first six terms of the sequence. *) (* Harvey P. Dale, Jan 18 2024 *)

isok(n) = isprime(sigma(sigma(n))); \\ Michel Marcus, Oct 01 2014

a(n) = 2*A247821(n)-1.

a(7)-a(8) from Michel Marcus, Oct 02 2014

a(9)-a(12) from Giovanni Resta, Feb 14 2020

A259417 Even powers of the odd primes listed in increasing order.

Original entry on oeis.org

1, 9, 25, 49, 81, 121, 169, 289, 361, 529, 625, 729, 841, 961, 1369, 1681, 1849, 2209, 2401, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6561, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 14641, 15625, 16129, 17161, 18769, 19321, 22201, 22801, 24649

Offset: 1

Hartmut F. W. Hoft, Jun 26 2015

nonn

Each of the following sequences, p^(q-1) with p >= 2 and q > 2 primes, except their respective first elements, powers of 2, is a subsequence:
A001248(p) = p^2,  A030514(p) = p^4,  A030516(p) = p^6,
A030629(p) = p^10, A030631(p) = p^12, A030635(p) = p^16,
A030637(p) = p^18, A137486(p) = p^22, A137492(p) = p^28,
A139571(p) = p^30, A139572(p) = p^36, A139573(p) = p^40,
A139574(p) = p^42, A139575(p) = p^46, A173533(p) = p^52,
A183062(p) = p^58, A183085(p) = p^60.
See also the link to the OEIS Wiki.
The sequences A053182(n)^2, A065509(n)^4, A163268(n)^6 and A240693(n)^10 are subsequences of this sequence.
The odd numbers in A023194 are a subsequence of this sequence.

			a(11) = 5^4 = 625 is followed by a(12) = 3^6 = 729 since no even power of an odd prime falls between them.

Hartmut F. W. Hoft, Table of n, a(n) for n = 1..473
OEIS Wiki, Index entries for number of divisors.

a259417[bound_] := Module[{q, h, column = {}}, For[q = Prime[2], q^2 <= bound, q = NextPrime[q], For[h = 1, q^(2*h) <= bound, h++, AppendTo[column, q^(2*h)]]]; Prepend[Sort[column], 1]]
a259417[25000] (* data *)
With[{upto=25000},Select[Union[Flatten[Table[Prime[Range[2,Floor[ Sqrt[ upto]]]]^n,{n,0,Log[2,upto],2}]]],#<=upto&]] (* Harvey P. Dale, Nov 25 2017 *)

Sum_{n>=1} 1/a(n) = 1 + Sum_{k>=1} (P(2*k) - 1/2^(2*k)) = 1.21835996432366585110..., where P is the prime zeta function. - Amiram Eldar, Jul 10 2022

A270413 Numbers m such that sigma(m-1) is a prime.

Original entry on oeis.org

3, 5, 10, 17, 26, 65, 290, 730, 1682, 2402, 3482, 4097, 5042, 7922, 10202, 15626, 17162, 27890, 28562, 29930, 65537, 83522, 85850, 146690, 262145, 279842, 458330, 491402, 531442, 552050, 579122, 597530, 683930, 703922, 707282, 734450, 829922, 1190282, 1203410

Offset: 1

Jaroslav Krizek, Mar 16 2016

nonn

Prime terms are in A249759.
Corresponding values of sigma(n-1): 3, 7, 13, 31, 31, 127, 307, 1093, ...
Conjecture: supersequence of A256438.
Conjecture: 31 is the only prime p such that p = sigma(x-1) = sigma(y-1) for distinct numbers x and y; 31 = sigma(17-1) = sigma(26-1).
Supersequence of A270414 and A270415.

			17 is in the sequence because sigma(17-1) = sigma(16) = 31 (prime).

Cf. A000203, A023194, A249759, A256438, A270414, A270415.

[n: n in [2..2000000] |  IsPrime(SumOfDivisors(n-1))];

Select[Range[10^6], PrimeQ@ DivisorSigma[1, # - 1] &] (* Michael De Vlieger, Mar 17 2016 *)

isok(n) = isprime(sigma(n-1)); \\ Michel Marcus, Mar 17 2016

a(n) = A023194(n) + 1.

A278911 Odd numbers with prime sum of divisors.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Nov 30 2016

nonn

Also odd numbers with prime number and sum of divisors; if the sum of divisors is prime, then the number of divisors is prime.
Values of prime sums are sorted in A247837.
Subsequence of A050150 (odd numbers with prime number of divisors).
Odd terms of A023194.
All terms are squares of the form p^e such that p is odd prime and e+1 is a prime.

			sigma(9) = 13 (prime).

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

Cf. A000203, A005408, A023194, A050150, A193070, A247837, A278914.

[n: n in[2..10^7] | IsOdd(n) and IsPrime(SumOfDivisors(n)) and IsPrime(NumberOfDivisors(n))];

N:= 10^7: # to get all terms <= N
Ps:= select(isprime, [seq(i,i=3..floor(N^(1/2)),2)]):
es:= map(`-`,select(isprime, [seq(i,i=3..floor(log[3](N))+1,2)]),1):
Pes:= [seq(seq([p,e],p=Ps),e=es)]:
filter:= proc(pe) local v; v:= (pe[1]^(pe[2]+1)-1)/(pe[1]-1); pe[1]^pe[2] <= N and isprime(v) end proc:
sort(map(pe -> pe[1]^pe[2], select(filter, Pes))); # Robert Israel, Jan 22 2019

Select[Range[1, 2*10^6, 2], PrimeQ@DivisorSigma[1, #] &] (* Michael De Vlieger, Dec 01 2016 *)

isok(n) = (n % 2) && isprime(sigma(n)); \\ Michel Marcus, Dec 01 2016

a(n) = A193070(n)^2. - Michel Marcus, Dec 01 2016

cp's OEIS Frontend

A279094 Smallest k such that sigma(k^n) is prime.

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A085379 Greatest prime as sum of distinct divisors of n.

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A195268 Numbers whose sum of odd divisors is prime.

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A200981 Numbers n such that the sum of non-divisors of n is prime.

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A229266 Primes of the form sigma(k) + tau(k) + phi(k), where sigma(k) = A000203(k), tau(k) = A000005(k) and phi(k) = A000010(k).

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A232444 Numbers n such that sigma(n) and sigma(n^2) are primes.

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A247838 Numbers k such that sigma(sigma(k)) is prime.

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A259417 Even powers of the odd primes listed in increasing order.

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