Manuel Valdivia - cp's OEIS frontend

A187771 Numbers whose sum of divisors is the cube of the sum of its prime divisors.

245180, 612408, 639198, 1698862, 1721182, 5154168, 7824284, 15817596, 20441848, 25969788, 27688078, 28404862, 35860609, 67149432, 77378782, 91397838, 96462862, 179302264, 191550135, 289772221, 306901244, 311657084, 392802179, 441839706, 572673855, 652117774, 988918364

Offset: 1

Manuel Valdivia, Jan 04 2013

This sequence and A187824 and A187761 are winners in the contest held at the 2013 AMS/MAA Joint Mathematics Meetings. - T. D. Noe, Jan 14 2013
The identity sigma(k) = (sopf(k))^m only occurs for m = 3 (this sequence) in the given range, however it is likely that it also occurs for other powers m in higher numbers.
The smallest k such that sigma(k) = sopf(k)^m, for m=4,5,6 are 1056331752 (A221262), 213556659624 (A221263) and 45770980141656, respectively. - Giovanni Resta, Jan 07 2013
Prime divisors are taken without multiplicity. - Harvey P. Dale, Dec 17 2016

			a(13) = 35860609 = 41 * 71 * 97 * 127, then sigma(35860609) = 37933056 = (41 + 71 + 97 + 127)^3.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.

Donovan Johnson and Robert Gerbicz, Table of n, a(n) for n = 1..1105 (first 100 terms from _Donovan Johnson_)
W. Sierpinski, Number Of Divisors And Their Sum, Elementary theory of numbers, Warszawa, 1964.

Cf. A000203, A008472, A020477, A070222.

Cf. A221262 (sigma(k)=sopf(k)^4), A221263 (sigma(k)=sopf(k)^5).

d[n_]:= If[Plus@@Divisors[n]-Power[Plus@@Select[Divisors[n], PrimeQ], 3]==0, n]; Select[Range[2,10^9], #==d[#]&]
Select[Range[2, 10^9],DivisorSigma[1,#]==Total[FactorInteger[#][[All, 1]]]^3&] (* Harvey P. Dale, Dec 17 2016 *)

is(n)=my(f=factor(n));sum(i=1,#f~,f[i,1])^3==sigma(n) \\ Charles R Greathouse IV, Jun 29 2013

a(n) = k if sigma(k) = (sopf(k))^3, where sigma(k) = A000203(k) and sopf(k) = A008472(k).

A182292 Smallest odd number k such that is equal to the sum of its proper divisors greater than k^(1/n), or 0 if none exist.

Original entry on oeis.org

34155, 407715, 8415

Offset: 2

Manuel Valdivia, Apr 24 2012

a(8) = 159030135. There is no n > 4 for which a(n) is smaller unless a(n) = 0. - Charles R Greathouse IV, Apr 25 2012
Other than a(2) to a(4) and a(8), there is no solution < 2*10^10 for a(n) up to a(1000). - Donovan Johnson, Aug 23 2012
From Alexander Violette, Feb 29 2024: (Start)
a(7) <= 7650499534755.
a(14) <= 221753170660847595. (End)

			The sum proper divisors of 407715 greater than 407715^(1/3) is 77 + 105 + 165 + 231 + 353 + 385 + 1059 + 1155 + 1765 + 2471 + 3883 + 5295 + 7413 + 11649 + 12355 + 19415 + 27181 + 37065 + 58245 + 81543 + 135905 = 407715.

See A182147 for more details for 34155.

Cf. A182311, A182286, A005231.

t={}; d[n_]:= Select[Drop[Divisors[n],-1], #1>n^(1/p)&]; Do[s=Select[Range[1,5*10^5,2], #==Plus@@d[#]&];
  AppendTo[t,s], {p,2,4}]; Flatten[t]

a(n)=my(t,k=8413);while(k+=2,if(sigma(k,-1)>2,if(ispower(k,n,&t),,t=k^(1/n)\1);if(sumdiv(k,d,if(d>t,d))==2*k,return(k)))) \\ Charles R Greathouse IV, Apr 25 2012

A182154 Smallest k >= 2 such that k^(2^n)+1 is the lesser member of a twin prime pair.

Original entry on oeis.org

2, 2, 2, 4, 2, 49592, 7132, 532, 333482, 2226686, 3543554, 23379038, 1249625230, 188489906

Offset: 0

Manuel Valdivia, Apr 15 2012

These lesser of twin prime pairs are also generalized Fermat primes, (not possible for greater of twin prime pairs, except for 5).
When extending this sequence, it is useful if the primes b^(2^n)+1 are known in advance (Gallot link). - Jeppe Stig Nielsen, Sep 25 2019
For later terms, the bigger twin is only a probable prime, not a proven prime. - Jeppe Stig Nielsen, Nov 24 2022

			2^(2^4)+1 = 65537 = A001359(861), then a(4) = 2.

Yves Gallot, Generalized Fermat Prime Search.
OEIS Wiki, Generalized Fermat numbers.
PrimeGrid and "Stream", GFN-1x Small Primes search, mentions a(12) and a(13).

Cf. A056993, A001359.

Table[k=2; While[!PrimeQ[k^(2^n)+1]||!PrimeQ[k^(2^n)+3],k++]; k,{n,0,7}]

a(8)-a(10) from Jeppe Stig Nielsen, Sep 25 2019
Name edited by Felix Fröhlich, Sep 25 2019
a(11)-a(13) from Jeppe Stig Nielsen, Nov 24 2022

A182065 Smallest average of twin prime pairs s such that s^(2^n)+1 is prime.

Original entry on oeis.org

4, 4, 4, 198, 30, 102, 3000, 7332, 4482, 187218, 150, 114690, 713310, 1943532, 3467622, 4470420, 23045178, 12529818

Offset: 1

Manuel Valdivia, Apr 09 2012

The averages of this sequence are base values in generalized Fermat primes.

			198^(2^4)+1 = 5580113648647376991977566450378407937 is prime.

Yves Gallot, Generalized Fermat Prime Search.
OEIS Wiki, Generalized Fermat numbers.
PrimeGrid, GFN Prime Search Status and History.

Cf. A056993, A014574.

t=Select[Table[Prime[n]+1,{n,10^5}],PrimeQ[#1+1]&]; s:=t[[m]]; Table[m=1; While[!PrimeQ[s^(2^n)+1],m++]; s,{n,1,9}](* Last five terms obtained by intersection with Yves Gallot records.*)

a(n) = (A014574(k))^(2^n)+1, for k = 1, 1, 1, 15, 5, 9, 82, 166, 117, 2055, 12, 1366, 6162, 14522, ...

a(15)-a(18) from Jeppe Stig Nielsen, Sep 14 2022

A185145 Smallest average of twin prime pairs s such that n*s is also average of twin prime pairs.

Original entry on oeis.org

4, 6, 4, 18, 6, 12, 6, 30, 12, 6, 18, 6, 150, 30, 4, 12, 6, 4, 12, 12, 42, 30, 6, 18, 6, 12, 4, 270, 12, 6, 42, 6, 6, 30, 12, 12, 180, 6, 60, 6, 30, 150, 30, 30, 4, 18, 6, 4, 18, 12, 42, 6, 150, 30, 12, 60, 4, 6, 18, 4, 462, 180, 1230, 18, 30, 108, 60, 180, 12

Offset: 1

Manuel Valdivia, Mar 12 2012

nonn

Probably for all n>1 and also for all average s there are at least an average n*s. Note that this is equivalent to the Twin Prime Conjecture. Verified n to 10^7. First consecutive averages: 4 to 34260.

			A014574(12) = 150, then 13*150 = 1950 = A014574(60).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A066388, A014574.

t=Select[Table[Prime[n] + 1, {n, 10^4}], PrimeQ[#1 + 1] & ]; Table[s:=t[[m]]; m=1; While[!PrimeQ[n*s-1] || !PrimeQ[n*s+1], m++]; s, {n,1,100}]

a(n) = A014574(j) if n*A014574(j) = A014574(k).

A205668 Prime numbers that cannot be expressed as the sum of two lesser primes of twin prime pairs + 1 or two greater primes of twin prime pairs - 1.

Original entry on oeis.org

2, 3, 5, 97, 401, 787, 907, 1117

Offset: 1

Manuel Valdivia, Jan 30 2012

nonn

The occurrence of a pair of twin primes in the sequence would be a counterexample to the conjecture in A134143.

There are probably no more terms. As in Goldbach's conjecture, the number of summands increases rapidly. - Charles R Greathouse IV, Jan 31 2012

			97 is here because neither 96 or 98 is the sum of two primes from the set {2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73}, which are the twin primes less than 100. - _T. D. Noe_, Feb 12 2012

Cf. A000040, A001359, A006512.

k=Insert[Select[Prime[Range[2,10^4]], PrimeQ[#-2]||PrimeQ[#+2]&], 5, 3]; u=Length@k/2; Complement[Prime[Range[4,10^4]], Select[Flatten[Join[Table[k[[2n-1]] + k[[2m-1]] + 1,{n,u}, {m,n}], Table[k[[2n]] + k[[2m]] - 1,{n,u}, {m,n}]]], PrimeQ]]

lower=List();p=2;forprime(q=3,1e8,if(q-p==2,listput(lower,p));p=q)
isk(n)=for(i=1,#lower,if(setsearch(lower,n-lower[i]),return(lower[i]));if(2*lower[i]>n,return(0)));error("ran out")
is(n)=!isk(n-1)&&!isk(n-3) \\ Charles R Greathouse IV, Jan 31 2012

A000040(n) != A001359(j) + A001359(k) + 1 and A000040(n) != A006512(j) + A006512(k) - 1, with n>3 and j<=k.

A200721 Product of two nonconsecutive primes p and q that divides the sum of primes between p and q, exclusively.

Original entry on oeis.org

26, 1133, 20309, 51159, 3246905, 28673661, 5201685791

Offset: 1

Manuel Valdivia, Nov 21 2011

nonn

Prime p is approximately q/((2*log(q)-1)*k), for k = 1, 1, 3, 307, 5041, 102378,..(quotients).

a(8) > 2*10^10. 3235398421447 is also a term. - Donovan Johnson, Nov 23 2011

			51159 = 3*17053, (5+ ... +17047)/51159 = 307.

Cf. A055514, A055233, A086447, A086448, A001358.

ss[n_] := Module[{f = Transpose[FactorInteger[n]], p, q, s}, If[f[[2]] == {1, 1}, {p, q} = PrimePi[f[[1]]]; s = Total[Table[Prime[i], {i, p + 1, q - 1}]]; s != 0 && Mod[s, n] == 0, False]]; Select[Range[2, 21000], ss] (* T. D. Noe, Nov 21 2011 *)

a(7) from Donovan Johnson, Nov 23 2011

A186129 Numbers that can be partitioned into four parts s, t, u, v such that s+k = t-k = u*k = v/k for some k > 1.

Original entry on oeis.org

18, 27, 32, 36, 45, 48, 50, 54, 63, 64, 72, 75, 80, 81, 90, 96, 98, 99, 100, 108, 112, 117, 125, 126, 128, 135, 144, 147, 150, 153, 160, 162, 171, 175, 176, 180, 189, 192, 196, 198, 200, 207, 208, 216, 224, 225, 234, 240, 242, 243, 245, 250, 252, 256, 261

Offset: 1

Manuel Valdivia, Feb 13 2011

nonn

Equivalently, solutions n to a*(b+1)^2 = b*n with a > b >= 2.
The general rule to obtain such a partition is to start with any number b > 1 and one of its multiples a = k*b (k > 1 and a < n) and let s = a-b, t = a+b, u = a/b and v = a*b.
Sequence appears to be a subsequence of A013929, of A046790, and of A072903.

			18 = 2+6+2+8; for k=2 we have 2+2 = 6-2 = 2*2 = 8/2 = 4, hence 18 is a term.
45 = 8+12+5+20; for k=2 we have 8+2 = 12-2 = 5*2 = 20/2 = 10, hence 45 is a term.

José Estalella, Ciencia Recreativa. Gustavo Gili - Editor. Barcelona, 1918, pp. 5-6.

Klaus Brockhaus, Table of n, a(n) for n = 1..1178 (terms <= 5000)

Cf. A000041, A013929, A046790, A072903.

[ n: n in [1..300] | exists{ b: b in [2..n] | exists{ a: a in [b+1..n div 4] | n*b eq a*(b+1)^2 } } ]; // Klaus Brockhaus, Feb 15 2011

A175906 Numbers n of the form 2^(A000043-1)*A046528 such that sigma(n) is a perfect number.

Original entry on oeis.org

12, 10924032, 16125952, 3757637632, 45091651584, 66563866624, 727145809044307968, 1152771972099211264, 845044701535107443245558061611352064

Offset: 1

Manuel Valdivia, Oct 12 2010

nonn

sigma(sigma(A046528(2,12,13,18,21,22,56,57,175,176,177,..))) is Mersenne prime.

			45091651584=2^18*3*7*8191, sigma(45091651584)=137438691328 is perfect number.

Cf. A000396, A000203, A000668, A000043, A046528, A063103.

a(n)= 2^( A000043(2,5,5,6,7,7,8,8,9,9,9,..)-1)*A046528(2,12,13,18,21,22,56,57,175,176,177,..).

A175410 a(n) = (b(m)-1)*b(m) = Sum_{n=b(m)+1,...,c(m)}n, b=A046174, c=A046175, m=n+1.

Original entry on oeis.org

132, 27060, 5269320, 1022496552, 198362899020, 38481433358940, 7465200453136272, 1448210416843244880, 280945355811546307860, 54501950819034511436292, 10573097513564898455783640

Offset: 1

Manuel Valdivia, May 05 2010

nonn

Solution to (b-1)*b = (c^2+2bc+c)/2.

			A046174(2) = 12, then 11*12 = 13+14+15+16+17+18+19+20 = 132, is a term. A046174(3) = 165, then 164*165 = 166+167+,....,+285 = 27060, is a term.

A046174, A046175.

lst={};k=1;j=0;Do[b=14*k-j-2;AppendTo[lst,(b-1)*b];j=k;k=b,{n,1,16}];lst

cp's OEIS Frontend

User: Manuel Valdivia

A187771 Numbers whose sum of divisors is the cube of the sum of its prime divisors.

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A182292 Smallest odd number k such that is equal to the sum of its proper divisors greater than k^(1/n), or 0 if none exist.

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A182154 Smallest k >= 2 such that k^(2^n)+1 is the lesser member of a twin prime pair.

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A182065 Smallest average of twin prime pairs s such that s^(2^n)+1 is prime.

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A185145 Smallest average of twin prime pairs s such that n*s is also average of twin prime pairs.

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A205668 Prime numbers that cannot be expressed as the sum of two lesser primes of twin prime pairs + 1 or two greater primes of twin prime pairs - 1.

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A200721 Product of two nonconsecutive primes p and q that divides the sum of primes between p and q, exclusively.

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