A023194 -id:A023194 - cp's OEIS frontend

A229265 Numbers k such that sigma(k) + tau(k) + phi(k) is a prime, where sigma(k) = A000203(k), tau(k) = A000005(k) and phi(k) = A000010(k).

1, 8, 200, 512, 968, 1458, 3200, 4232, 5618, 5832, 6962, 10368, 16928, 26912, 36992, 40328, 53792, 61952, 84050, 101250, 110450, 140450, 147968, 220448, 247808, 249218, 253472, 257762, 279752, 282752, 320000, 336200, 344450, 359552, 361250, 445568, 472392, 512072

Offset: 1

Paolo P. Lava, Sep 18 2013

nonn

			sigma(200) = 465, tau(200) = 12, phi(200) = 80 and 465 + 12 + 80 = 557 is prime.

Cf. A000005, A000010, A000203, A009087, A023194, A038344, A055813, A062700, A064205, A115919, A141242, A229264, A229266-A229268.

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

A249902 Numbers n such that 2n-1 and sigma(n) are both primes.

Original entry on oeis.org

2, 4, 9, 16, 64, 289, 1681, 2401, 3481, 4096, 15625, 65536, 85849, 262144, 491401, 531441, 552049, 683929, 703921, 734449, 1352569, 1885129, 3411409, 3892729, 5470921, 7091569, 7778521, 9247681, 10374841, 12652249, 18139081, 19439281, 22287841, 23902321

Offset: 1

Jaroslav Krizek, Nov 14 2014

nonn

Intersection of A006254 and A023194.
Sequence is a supersequence of the even superperfect numbers m_k (A061652 or even terms from A019279) because sigma(m_k) = 2*(m_k)-1 = k-th Mersenne prime A000668(k) for k>=1.
Conjecture: 2 and 9 are the only numbers n such that 2n - 1, 2n + 1 and sigma(n) are all primes.

			289 is in the sequence because 2*289 - 1 = 577 and sigma(289) = 307 (both primes).

Chai Wah Wu, Table of n, a(n) for n = 1..10622

Cf. A000203, A006254, A023194, A019279, A249903.

[n: n in [2..10000000] | IsPrime(2*n-1) and IsPrime(SumOfDivisors(n))];

Select[Range[10^7], PrimeQ[2 # - 1] && PrimeQ[DivisorSigma[1, #]] &] (* Vincenzo Librandi, Nov 15 2014 *)

for(n=1,10^6,if(isprime(2*n-1)&&isprime(sigma(n)),print1(n,", "))) \\ Derek Orr, Nov 14 2014

from sympy import isprime, divisor_sigma
A249902_list = [2]+[n for n in (d**2 for d in range(1,10**3)) if isprime(2*n-1) and isprime(divisor_sigma(n))] # Chai Wah Wu, Jul 23 2016

A253849 Numbers k such that 2^sigma(k) - 1 is a prime.

Original entry on oeis.org

2, 4, 9, 16, 25, 64

Offset: 1

Jaroslav Krizek, Jan 16 2015

Also numbers n such that sigma(n) is in A000043, i.e., p = 2^sigma(n) - 1 is a Mersenne prime (A000668). The sequence of corresponding primes p reads: 7, 127, 8191, 2147483647, 2147483647, 170141183460469231731687303715884105727, ..., see A253851.
Subsequence of A023194 (numbers n such that sigma(n) is a prime), see there for an explanation why all terms except the first one are squares.
The sequence of values of sigma(a(n)) is 3, 7, 13, 31, 31, 127, ... and each term of this sequence must be a prime from the sequence of Mersenne exponents (A000043). See A253850.
Sequence differs from A023194 because A023194(7) = 289 but if a(7) exists, it must be a number n such that sigma(n) > A000043(43) = 30402457.
a(n) must be an even power of a prime. If it is the square of an odd prime, then this prime must be in A053182. If a(n) is an even power of 2, a(n)=2^(2k), then sigma(a(n))=2^(2k+1)-1. Thus, 2k+1 must be a double Mersenne prime exponent, i.e., such that the corresponding Mersenne prime is again a Mersenne exponent, cf. A103901. Only 4 such primes are known, and a(6)=2^6 (k=3) corresponds to the largest known prime of this type, 2^(2k+1)-1 = 127. - M. F. Hasler, Jan 21 2015

			4 is in the sequence because 2^sigma(4)-1 = 2^7-1 = 127 is prime.

Cf. A000043, A000203, A000668, A023194, A023195, A253850, A253851.

[n: n in[1..10000] | IsPrime((2^SumOfDivisors(n)) - 1)];

a253849[n_] := Select[Range@ n, PrimeQ[2^DivisorSigma[1, #] - 1] &]; a253849[20000] (* Michael De Vlieger, Jan 19 2015 *)

A274962 Numbers n such that sigma(n) and sigma(n) + 2 are both primes.

Original entry on oeis.org

2, 2401, 19356878641, 46904541018721, 119601542190001, 360371335935601, 16472757578830081, 26835157974988801, 59879777952495601, 147669280778756881, 170589096345900241, 219660193449998401, 1103765757989399761, 1515946818108402241, 2044393722679974961, 2608728003079029841, 2805689752523610241, 3071293995460971361, 4537323492222149281, 9583348094642219041, 9982134924573725761

Offset: 1

Jaroslav Krizek, Jul 12 2016

nonn

Intersection of A249763 and A023194.
The next term, if it exists, must be greater than 5*10^12.
Each term > 2 is a square.
From Chai Wah Wu, Jul 13 2016: (Start)
Every term > 2 is of the form p^(2m) with p prime and m > 1. Proof: from the discussion in A023194, a term is of the form p^(2m). An odd term cannot be of the form n = p^2. If p = 6k+1, then sigma(n) = 36k^2 + 18k + 3 is composite. If p = 6k-1, then sigma(n) + 2 = 36k^2 - 6k + 3 is composite. Finally, 4 is not a term.
This could be the reason this sequence is so much sparser than A274963.
(End)
Terms cannot be of the form 2^(2m) since sigma(2^(2m)) + 2 = 2^(2m+1) + 1 is divisible by 3. - Altug Alkan, Jul 14 2016
Terms cannot be of the form 3^(2m) since sigma(3^(3m)) + 2 = 3(3^(2m) + 1)/2 is divisible by 3, i.e., all terms are of the form (6*k+1)^(2m) or (6*k-1)^(2m) - Chai Wah Wu, Aug 06 2016
Terms cannot be of the form p^6 since if p = 6*k+1, then sigma((6*k+1)^6) + 2 = 9*(5184*k^6 + 6048*k^5 + 3024*k^4 + 840*k^3 + 140*k^2 + 14*k + 1) and if p = 6*k-1 then sigma((6*k-1)^6) + 2 = 3*(15552*k^6 - 12960*k^5 + 4752*k^4 - 936*k^3 + 108*k^2 - 6*k + 1). Also note that terms cannot be of the form p^8 since if p = 6*k-1 then sigma((6*k-1)^8) = (1 - 6*k + 36*k^2)*(1 - 18*k + 432*k^2 - 4104*k^3 + 19440*k^4 - 46656*k^5 + 46656*k^6) and if p = 6*k+1 then sigma((6*k+1)^8) = 9*(186624*k^8 + 279936*k^7 + 186624*k^6 + 72576*k^5 + 18144*k^4 + 3024*k^3 + 336*k^2 + 24*k + 1). The least term that is of the form p^10 is 2089^10. So this partially explains why numbers of the form p^4 appear in this sequence most of the time in limited range. - Altug Alkan, Jul 15 2016
From Chai Wah Wu, Jul 20 2016: (Start)
If p^m > 2 is a term, then m == 4 mod 6 and p == 1 mod 6. Proof: Let q(k) be sigma(p^m) expressed as a polynomial in k. If p = 6k-1, then q(k) = 1 + (6k-1) + (6k-1)^2 + ... + (6k-1)^m.
The constant term of q(k) is 1-1+1-1+...-1+1 = 1 whereas the other coefficients are multiples of 6, i.e., q(k) = 1 + 6k*(...), thus sigma(p^m) + 2 is a multiple of 3.
Suppose p = 6k+1, then q(k) = 1 + (6k+1) + (6k+1)^2 + ... + (6k+1)^m. The constant term is m+1 and the other coefficients are multiples of 6, i.e., q(k) = (m+1) + 6k*(...).
This means that if m = 6r+2, then sigma(p^m) is a multiple of 3 and if m = 6r, then sigma(p^m) + 2 is a multiple of 3. End of Proof.
The following table lists the minimal k for r <= 4.
r   | smallest k such that (6k+1)^(6r+4) is a term (A275237)
------------------------------------------------------------
0   |      1
1   |    348
2   |    436
3   |   6018
4   |   5880
For every prime p = 6k+1, does there exist r >= 0 such that(6k+1)^(6r+4) is a term?
(End)
Altug Alkan found that sigma((6k+1)^34) (i.e., the r = 5 case) is always composite (see comment in A275237). - Chai Wah Wu, Jul 21 2016

			2401 is in the sequence because sigma(2401) = 2801 and sigma(2401) + 2 = 2803 are both primes.

Chai Wah Wu, Table of n, a(n) for n = 1..10337

Cf. A000203, A023194, A249763, A274963, A275237.

[n: n in[1..10^7] | IsPrime(SumOfDivisors(n)) and IsPrime(SumOfDivisors(n)+2)]

isok(n) = isprime(s=sigma(n)) && isprime(s+2); \\ Michel Marcus, Jul 14 2016

from sympy import isprime, divisor_sigma
A274962_list = [2]+[n for n, s in ((d**2, divisor_sigma(d**2)) for d in range(1,10**3)) if isprime(s) and isprime(s+2)] # Chai Wah Wu, Jul 13 2016

a(4)-a(21) from Chai Wah Wu, Jul 13 2016

A274963 Numbers n such that both sigma(n) and sigma(n) - 2 are primes.

Original entry on oeis.org

4, 9, 16, 25, 729, 1681, 3481, 7921, 10201, 17161, 552049, 579121, 1423249, 5812921, 7091569, 7447441, 9066121, 9765625, 10374841, 10569001, 11895601, 22572001, 38105929, 43546801, 46689889, 52258441, 75151561, 82065481, 86918329, 90649441, 94458961, 94926049

Offset: 1

Jaroslav Krizek, Jul 12 2016

nonn

Intersection of A249485 and A023194.
The next term, if it exists, must be greater than 10^8.
Each term is a square.
Most of the terms seem to be of the form p^2 for some prime p. Out of the first 10539 terms, 6 of them are not of the form p^2. - Chai Wah Wu, Jul 13 2016

			729 is in the sequence because sigma(729) = 1093 and 1091 are both primes.

Chai Wah Wu, Table of n, a(n) for n = 1..10539

Cf. A000203, A023194, A249485, A274962.

[n: n in[1..10^7] | IsPrime(SumOfDivisors(n)) and IsPrime(SumOfDivisors(n)-2)]

from sympy import isprime, divisor_sigma
A274963_list = [n for n, s in ((d**2, divisor_sigma(d**2)) for d in range(1,10**3)) if isprime(s) and isprime(s-2)] # Chai Wah Wu, Jul 13 2016

A278914 a(n) is the smallest odd number k with prime sum of divisors such that tau(k) = n-th prime.

Original entry on oeis.org

9, 2401, 729, 9765625, 531441, 45949729863572161, 5559917313492231481, 1471383076677527699142172838322885948765175969, 10264895304762966931257013446474591264089923314972889033759201, 230466617897195215045509519405933293401

Offset: 2

Jaroslav Krizek, Nov 30 2016

nonn

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

For n >= 7; a(n) > A023194(10000) = 5896704025969.

			a(2) = 9 because 9 is the smallest odd number with prime values of sum of divisors (sigma(9) = 13) such that tau(9) = 3 = 2nd prime.

Robert G. Wilson v, Table of n, a(n) for n = 2..66 (first 49 terms from Davin Park)

Cf. A000005, A000203, A278911, A278913.

A278914:=func; [A278914(n): n in[2..6]];

A278914[n_] := NestWhile[NextPrime, 3, ! PrimeQ[Cyclotomic[Prime[n], #]] &]^(Prime[n] - 1); Array[A278914, 10, 2] (* Davin Park, Dec 28 2016 *)

a(n) = {my(k=1); while(! (isprime(sigma(k)) && isprime(p=numdiv(k)) && (primepi(p) == n)), k+=2); k;} \\ Michel Marcus, Dec 03 2016

a(n) = A101636(n)^(prime(n)-1). - Davin Park, Dec 10 2016

More terms from Davin Park, Dec 11 2016

A279096 Numbers k such that sigma(k^3) is prime.

Original entry on oeis.org

4, 9, 16, 25, 64, 81, 169, 289, 625, 961, 1024, 2401, 3721, 5329, 7921, 22201, 26569, 63001, 121801, 124609, 212521, 273529, 358801, 418609, 744769, 885481, 896809, 1048576, 1181569, 1247689, 1510441, 1630729, 1666681, 1682209, 1771561, 1874161, 1985281

Offset: 1

Jon E. Schoenfield, Mar 12 2017

nonn

All terms are square. Moreover, each term is of the form p^j where both p and j*3 + 1 are prime (see A279094).

			4 is in the sequence because sigma(4^3) = sigma(2^6) = 1 + 2 + 4 + 8 + 16 + 32 + 64 = 127, which is prime.
16 is in the sequence because sigma(16^3) = sigma(2^12) = Sum_{m=0..12} 2^m = (2^13 - 1)/(2 - 1) = 8191, which is prime.
36 is not in the sequence because sigma(36^3) = sigma(2^6*3^6) = ((2^7 - 1)/(2 - 1))*((3^7 - 1)/(3 - 1)) = 127*1093, which is not prime. (36 is not of the form p^j where p is prime.)
361 is not in the sequence (even though 361 = 19^2 is of the form p^j where both p and 3*j + 1 are prime) because sigma(361^3) = sigma(19^6) = (19^7 - 1)/(19 - 1) = 49659541 = 701 * 70841.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000203 (sigma(k)), A023194 (sigma(k) is prime), A055638 (sigma(k^2) is prime), A279094 (smallest k such that sigma(k^n) is prime).

mx = 10^7; ee = Select[Range@ Log2@ mx, PrimeQ[3 # + 1] &]; Union@ Reap[ Do[ Do[ If[(v = p^e) <= mx, If[ PrimeQ[(p v^3 - 1)/ (p-1)], Sow@ v], Break[]], {e, ee}], {p, Prime@ Range@ PrimePi@ Sqrt@ mx}]][[2, 1]] (* Giovanni Resta, Mar 12 2017 *)
Select[Range[2*10^6],PrimeQ[DivisorSigma[1,#^3]]&] (* Harvey P. Dale, Jan 10 2024 *)

isok(n) = isprime(sigma(n^3)); \\ Michel Marcus, Mar 12 2017

A367460 Numbers k such that 3^sigma(k) - k is a prime.

Original entry on oeis.org

1, 10, 52, 400, 2480, 7202, 28222

Offset: 1

J.W.L. (Jan) Eerland, Jan 26 2024

a(8) > 67569.

			10 is in the sequence because 3^sigma(10) - 10 = 3^18 - 10 = 387420479 is prime.

Cf. A000043, A000203, A000668, A023194, A023195, A253850, A253851, A368651.

[n: n in[1..10000] | IsPrime((3^SumOfDivisors(n)) - n)]

a[n_] := Select[Range@ n, PrimeQ[3^DivisorSigma[1, #] - #] &]; a[20000]
DeleteCases[ParallelTable[If[PrimeQ[3^DivisorSigma[1,k]-k],k,n],{k,1,10^4}],n]

a(7) from Michael S. Branicky, Jan 27 2024

A387406 Numbers k such that sigma(A253560(k)) / A253560(k) is equal to (sigma(k)+1) / k, where A253560(k) = k multiplied by its largest prime factor.

Original entry on oeis.org

6, 18, 28, 54, 117, 162, 196, 486, 496, 775, 1372, 1458, 1521, 4374, 8128, 9604, 13122, 15376, 19773, 24025, 39366, 67228, 88723, 118098, 257049, 354294, 470596, 476656, 744775, 796797, 1032256, 1062882, 2896363, 3188646, 3294172, 3341637, 6725201, 9565938, 12326221, 14776336, 23059204, 23088025, 25774633, 27237961

Offset: 1

Antti Karttunen, Aug 30 2025

Terms k for which sigma(k/A053585(k)) = A006530(k). This further entails that A001221(k) = 2 [See A023194].

Antti Karttunen, Table of n, a(n) for n = 1..58 (larger b-file needed)
C. A. Holdener and J. A. Holdener, Characterizing Quasi-Friendly Divisors, Journal of Integer Sequences, Vol. 23 (2020), Article 20.8.4.
Index entries for sequences related to sigma(n)

Cf. A000203, A001221, A006530, A023194, A053585, A253560, A387405.

Subsequences: A000396 (even terms only), A240991 (conjectured, if true, then A000396 has only even terms).

fk[k_]:=k*FactorInteger[k][[-1,1]];Select[Range[10^6],DivisorSigma[1,fk[#]]/fk[#]==(DivisorSigma[1,#]+1)/#&] (* James C. McMahon, Aug 31 2025 *)

A253560(n) = if (n==1, 1, n*vecmax(factor(n)[, 1]));
isA387406(n) = { my(x=A253560(n)); ((sigma(x)/x) == ((sigma(n)+1)/n)); };

A071166 a(n) = n - A006530(A000203(n)), difference between n and largest prime factor of the sum of its divisors.

Original entry on oeis.org

-1, 1, -3, 2, 3, 5, 3, -4, 7, 8, 5, 6, 11, 12, -15, 14, 5, 14, 13, 19, 19, 20, 19, -6, 19, 22, 21, 24, 27, 29, 25, 30, 31, 32, 23, 18, 33, 32, 35, 34, 39, 32, 37, 32, 43, 44, 17, 30, 19, 48, 45, 50, 49, 52, 51, 52, 53, 54, 53, 30, 59, 50, -63, 58, 63, 50, 61, 66, 67, 68, 59, 36, 55, 44, 69, 74, 71, 74, 49, 70, 75, 76, 77, 82, 75, 82, 83

Offset: 2

Labos Elemer, May 15 2002

sign

Terms are mostly positive. At cases when sigma(n) is prime the differences are negative. See A071167.

			n=12, divisors={1,2,3,4,6,12}, sigma(12)=28, its largest prime factor is 7, so a(12)=12-7=5.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A000203, A006530, A062700, A023194, A071167.

gpf:= n -> max(numtheory:-factorset(n)):
seq(n - gpf(numtheory:-sigma(n)), n=2..100); # Robert Israel, Feb 12 2017

ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] t=Table[w-ma[DivisorSigma[1, w]], {w, 2, 128}]

a(n)=n-factor(sigma(n))[1,1] \\ Charles R Greathouse IV, Feb 19 2013

cp's OEIS Frontend

A229265 Numbers k such that sigma(k) + tau(k) + phi(k) is a prime, where sigma(k) = A000203(k), tau(k) = A000005(k) and phi(k) = A000010(k).

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A249902 Numbers n such that 2n-1 and sigma(n) are both primes.

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A253849 Numbers k such that 2^sigma(k) - 1 is a prime.

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A274962 Numbers n such that sigma(n) and sigma(n) + 2 are both primes.

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A274963 Numbers n such that both sigma(n) and sigma(n) - 2 are primes.

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A278914 a(n) is the smallest odd number k with prime sum of divisors such that tau(k) = n-th prime.

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A279096 Numbers k such that sigma(k^3) is prime.

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