A023194 -id:A023194 - cp's OEIS frontend

A330832 Numbers of the form p*q, where p is prime and q=(p^k-1)/(p-1) is also prime for some integer k>1.

6, 14, 39, 62, 155, 254, 3279, 5219, 16382, 19607, 70643, 97655, 208919, 262142, 363023, 402233, 712979, 1040603, 1048574, 1508597, 2265383, 2391483, 4685519, 5207819, 6728903, 21243689, 25239899, 56328959, 61035155, 67977559, 150508643, 310747739, 344964203

Offset: 1

Walter Kehowski, Jan 08 2020

Also numbers with power-spectral basis {q,p^k}. The equation q=(p^k-1)/(p-1) is equivalent to the decomposition of the identity q + p^k = pq + 1 in Z/pqZ, and it is now easily verified that {q,p^k} is the spectral basis of p*q, consisting of primes and powers.

The numbers p^(r^e)*q, where p, q, r are primes, and q=(p^(r^e)-1)/(p^(r^(e-1))-1), e>0, have power-spectral basis {q,p^(r^e)}. However, the primes q for e>1 are usually quite large, while e=1 is accessible. For example, the table in A003424 has 4738 entries with all primes q<10^12, but only 8 have y>1.

			a(5) = 5*(5^3-1)/(5-1) = 5*31 = 155. The number 155 has spectral basis {31,125}.

Walter Kehowski, Table of n, a(n) for n = 1..4731

Cf. A003424, A023194, A023195, A085104, A330833, A330834, A330835.

a(n) = A330833(n) * A330835(n).

A330833 a(n) = first prime factor p of the term A330832(n) = p*q.

Original entry on oeis.org

2, 2, 3, 2, 5, 2, 3, 17, 2, 7, 41, 5, 59, 2, 71, 13, 89, 101, 2, 17, 131, 3, 167, 173, 23, 29, 293, 383, 5, 13, 43, 677, 701, 743, 17, 761, 773, 827, 839, 857, 911, 1091, 1097, 5, 1163, 1181, 1193, 1217, 73, 1373, 1427, 79, 1487, 1559, 1583, 83, 2, 1709, 1811, 1847, 1931, 1973, 2129, 2273, 2309, 2339, 2411, 2663, 2729, 2789, 2957

Offset: 1

Walter Kehowski, Jan 08 2020

			a(5) = 5 and, since A330834(5) = 3, then A330835(5) = (5^3-1)/(5-1) = 31 is prime.

Walter Kehowski, Table of n, a(n) for n = 1..4731

Cf. A003424, A023194, A023195, A085104, A330832, A330834, A330835.

A330834 The exponents k of A330832, that is, if A330832(n)=p*q, where p is prime and q=(p^k-1)/(p-1) is prime, then a(n)=k.

Original entry on oeis.org

2, 3, 3, 5, 3, 7, 7, 3, 13, 5, 3, 7, 3, 17, 3, 5, 3, 3, 19, 5, 3, 13, 3, 3, 5, 5, 3, 3, 11, 7, 5, 3, 3, 3, 7, 3, 3, 3, 3, 3, 3, 3, 3, 13, 3, 3, 3, 3, 5, 3, 3, 5, 3, 3, 3, 5, 31, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 7, 3, 5, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Walter Kehowski, Jan 08 2020

nonn

			a(5) = 3, and, since A330833(5)=5, then A330835(5)=(5^3-1)/(5-1) = 31 is prime.

Walter Kehowski, Table of n, a(n) for n = 1..4731
G. Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly 108(4), April 2001.

Cf. A003424, A023194, A023195, A085104, A330832, A330833, A330835.

A330835 Primes q appearing in A330832: that is, if A330832(n)=p*q, where p is prime and q=(p^k-1)/(p-1) is prime, then a(n)=q.

Original entry on oeis.org

3, 7, 13, 31, 31, 127, 1093, 307, 8191, 2801, 1723, 19531, 3541, 131071, 5113, 30941, 8011, 10303, 524287, 88741, 17293, 797161, 28057, 30103, 292561, 732541, 86143, 147073, 12207031, 5229043, 3500201, 459007, 492103, 552793, 25646167, 579883, 598303, 684757

Offset: 1

Walter Kehowski, Jan 08 2020

nonn

The terms in the b-file are the same as those of A003424 with y=1, but with an ordering based on that of A330832. The ordering allows the inclusion of the only duplicate 2^5-1=31 and (5^3-1)/(5-1)=31.

			a(5)=31 since A330833(5)=5, A330834(5)=3, and (5^3-1)/(5-1) = 31 is prime.

Walter Kehowski, Table of n, a(n) for n = 1..4731
G. Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly 108(4), April 2001.

Cf. A003424, A023194, A023195, A085104, A330832, A330833, A330834.

a(n) = (A330833(n) ^ A330834(n) - 1) / (A330833(n) - 1).

A348659 Numbers whose numerator and denominator of the harmonic mean of their divisors are both prime numbers.

Original entry on oeis.org

3, 5, 13, 14, 15, 37, 42, 61, 66, 73, 92, 114, 157, 182, 193, 258, 277, 308, 313, 397, 402, 421, 457, 476, 477, 541, 546, 570, 613, 661, 673, 733, 744, 757, 812, 877, 978, 997, 1093, 1148, 1153, 1201, 1213, 1237, 1266, 1278, 1321, 1381, 1428, 1453, 1621, 1657

Offset: 1

Amiram Eldar, Oct 28 2021

nonn

The prime terms of this sequence are the primes p such that (p+1)/2 is also a prime (A005383).

If p is in A109835, then p*(2*p-1) is a semiprime term.

			3 is a term since the harmonic mean of its divisors is 3/2 and both 2 and 3 are primes.

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

Cf. A005383, A099377, A099378, A109835.

Similar sequences: A023194, A048968, A074266, A348659.

q[n_] := Module[{h = DivisorSigma[0, n]/DivisorSigma[-1, n]}, And @@ PrimeQ[{Numerator[h], Denominator[h]}]]; Select[Range[2000], q]

A368651 Numbers k such that 2^sigma(k) - k is a prime.

Original entry on oeis.org

3, 5, 17, 49, 53, 185, 503, 1301, 1689, 1797, 5929, 14747, 20433, 29903, 42137, 64763

Offset: 1

J.W.L. (Jan) Eerland, Jan 02 2024

If it exists, a(17) > 120000. - Michael S. Branicky, Aug 19 2024

			5 is in the sequence because 2^sigma(5)-5 = 2^6-5 = 59 is prime.

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

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

a[n_] := Select[Range@ n, PrimeQ[2^DivisorSigma[1, #] - #] &]; a[20000]
DeleteCases[ParallelTable[If[PrimeQ[2^DivisorSigma[1,k]-k],k,n],{k,1,10^4}],n]

a(16) from J.W.L. (Jan) Eerland, Jan 25 2024

A371421 Numbers whose aliquot-like sequence based on the largest aliquot divisor of the sum of divisors of n (A371418) terminates in a fixed point.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 63, 64, 67, 68, 71, 73, 74, 79, 80, 81, 82, 89, 93, 96, 97, 98, 100, 101

Offset: 1

Amiram Eldar, Mar 23 2024

nonn

It is unknown whether 222 is a term of this sequence or not (see A371423).

			3 is a term because when we start with 3 and repeatedly apply the mapping x -> A371418(x), we get the sequence 3, 2, 1, 1, 1, ...
40 is a term because when we start with 40 and repeatedly apply the mapping x -> A371418(x), we get the sequence 40, 45, 39, 28, 28, 28, ...

Amiram Eldar, Table of n, a(n) for n = 1..113
Robert D. Carmichael, Empirical Results in the Theory of Numbers, The Mathematics Teacher, Vol. 14, No. 6 (1921), pp. 305-310; alternative link. See p. 309.

Cf. A371418, A371422, A371423.
A023194 is a subsequence.
Cf. A063769, A080907.

r[n_] := n/FactorInteger[n][[1, 1]]; f[n_] := r[DivisorSigma[1, n]]; q[n_] := Module[{m = NestWhileList[f, n, UnsameQ, All][[-1]]}, f[m] == m]; Select[Range[221], q]

A063784 Primes that are the sum of cubes of divisors of some integer.

Original entry on oeis.org

73, 757, 1772893, 48551233240513, 378890487846991, 3156404483062657, 17390284913300671, 280343912759041771, 319913861581383373, 487014306953858713, 7824668707707203971, 8443914727229480773, 32564717507686012813, 48095468363380957093, 54811417636756749151

Offset: 1

Labos Elemer, Aug 17 2001

nonn

Primes of the form p^6 + p^3 + 1 where p is a prime. - Amiram Eldar, Aug 16 2024

			sigma_3(9) = 1 + 27 + 729 = 757, a prime.

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

Cf. A001158, A023194, A060883, A063783, A066100.

Select[Table[p^6 + p^3 + 1, {p, Prime[Range[500]]}], PrimeQ] (* Amiram Eldar, Aug 16 2024 *)

{ n=0; p=0; for (m=1, 10^9, p=nextprime(p+1); if(isprime(q=p^6 + p^3 + 1), write("b063784.txt", n++, " ", q); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 31 2009

Primes of form p = sigma_3(k).
From Amiram Eldar, Aug 16 2024: (Start)
a(n) = A001158(A063783(n)).
a(n) = A060883(A066100(n)). (End)

A074386 Numbers k such that sigma(k) is the square of a prime.

Original entry on oeis.org

3, 81, 400

Offset: 1

Labos Elemer, Aug 22 2002

The next term, if it exists, is > 10^11. - Donovan Johnson, Aug 24 2012
a(4), if it exists, satisfies sigma(a(4)) > 10^36.  - Hiroaki Yamanouchi, Sep 10 2014
If n belongs to this sequence, it may have at most two distinct prime divisors. If n=p^k, then sigma(p^k) = (p^(k+1)-1)/(p-1) = r^2 for some prime r. For k=1, it trivially has the only solution n=3; while for k>1 it is a partial case of the Nagell-Ljunggren equation and has the only prime solution r=11 (see Bennett-Levin 2015) corresponding to n=3^4=81. If n=p^k*q^m, then sigma(n) = (p^(k+1)-1)/(p-1) * (q^(m+1)-1)/(q-1) = r^2 for some prime r, implying that (p^(k+1)-1)/(p-1) = (q^(m+1)-1)/(q-1) = r. Here k+1 and m+1 must be odd distinct primes. The Goormaghtigh conjecture would imply that its only solution is n=400 with (p,k,q,m)=(5,2,2,4). - Max Alekseyev, Apr 24 2015

			sigma[{3,81,400}]={4,121,961}.

M. A. Bennett and A. Levin, The Nagell-Ljunggren equation via Runge’s method, Monatshefte für Mathematik 177:1 (2015), 15-31.
Wikipedia, Goormaghtigh conjecture

Cf. A000203, A001248, A008848, A023194, A028982.
Cf. A084738, A065854, A128164.
Subsequence of A006532.

Do[s=DivisorSigma[1, n]; If[PrimeQ[Sqrt[s]], Print[n]], {n, 1, 1000000}] (* Corrected by N. J. A. Sloane, May 26 2008 *)

Definition corrected by Juan Lopez, May 26 2008

Edited by N. J. A. Sloane, May 26 2008

A187822 Smallest k such that the partial sums of the divisors of k (taken in increasing order) contain exactly n primes.

Original entry on oeis.org

1, 2, 4, 16, 64, 140, 440, 700, 1650, 2304, 5180, 3960, 3900, 14400, 15600, 43560, 39600, 57600, 56700, 81900, 25200, 112896, 100100, 177840, 198000, 411840, 222768, 226800, 637560, 752400, 556920, 907200, 409500, 565488, 1306800, 1984500, 1884960

Offset: 0

Michel Lagneau, Dec 27 2012

nonn

It appears that a(n) is even for n > 0 and nonsquarefree for n > 1. We also conjecture that there is an infinite subsequence of squares 1, 4, 16, 64, 2304, 14400, 57600, 112896, ....
The corresponding triangle in which row n gives the n primes starts with:
k =   1 -> no prime
k =   2 -> 3;
k =   4 -> 3, 7;
k =  16 -> 3, 7, 31;
k =  64 -> 3, 7, 31, 127;
k = 140 -> 3, 7, 19, 29, 43;
k = 440 -> 3, 7, 41, 61, 83, 167; ...

			a(4) = 64 because the partial sums of the divisors {1, 2, 4, 8, 16, 32, 64} that generate 4 prime numbers are:
1 + 2 = 3;
1 + 2 + 4 = 7;
1 + 2 + 4 + 8 + 16  = 31;
1 + 2 + 4 + 8 + 16 + 32 + 64 = 127.

Amiram Eldar, Table of n, a(n) for n = 0..126

Cf. A023194, A062700, A000203.

read("transforms") :
A187822 := proc(n)
    local k,ps,pct ;
    if n = 0 then
        return 1;
    end if;
    for k from 1 do
        ps := sort(convert(numtheory[divisors](k),list)) ;
        ps := PSUM(ps) ;
        pct := 0 ;
        for p in ps do
            if isprime(p) then
                pct := pct+1 ;
            end if;
        end do:
        if pct = n then
            return k ;
        end if;
    end do:
end proc: # R. J. Mathar, Jan 18 2013

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

A187822(n)={n<1||for(k=1,9e9,numdiv(k)M. F. Hasler, Dec 29 2012

cp's OEIS Frontend

A330832 Numbers of the form p*q, where p is prime and q=(p^k-1)/(p-1) is also prime for some integer k>1.

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A330833 a(n) = first prime factor p of the term A330832(n) = p*q.

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A330834 The exponents k of A330832, that is, if A330832(n)=p*q, where p is prime and q=(p^k-1)/(p-1) is prime, then a(n)=k.

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A330835 Primes q appearing in A330832: that is, if A330832(n)=p*q, where p is prime and q=(p^k-1)/(p-1) is prime, then a(n)=q.

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A348659 Numbers whose numerator and denominator of the harmonic mean of their divisors are both prime numbers.

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A368651 Numbers k such that 2^sigma(k) - k is a prime.

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Extensions

A371421 Numbers whose aliquot-like sequence based on the largest aliquot divisor of the sum of divisors of n (A371418) terminates in a fixed point.

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Mathematica

A063784 Primes that are the sum of cubes of divisors of some integer.

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A074386 Numbers k such that sigma(k) is the square of a prime.

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