A128607 -id:A128607 - cp's OEIS frontend

A128608 a(n)=sigma(A128607(n)), where A128607(n) is the sequence of perfect (or pure) powers such that a(n) is a perfect power.

1, 121, 400, 961, 116281, 2989441, 7958041, 361722361, 962922961, 1902442689, 1891467081, 5168743489, 4755619521, 5215583961, 6835486329, 7496615889, 13884144561, 13884144561, 35018011161, 120776405841, 120776405841, 230195565369, 253358202409, 171651947481

Offset: 1

Walter Kehowski, Mar 20 2007

nonn

Denote by egcd(n) the gcd of all the powers in the prime factorization of n. In our context, a square has egcd=2, a cube has egcd=3 and so on. The only elements n in the sequence for which egcd(n)>2 are 81 and 343. Are there any others? Conjecture I: egcd(A128607(n))=2 for all n>2. Let a(n)=sigma(A128607(n)). Note that A128607(11)=1857437604=(2^2)*(3^2)*(11^2)*(653^2) has a(11)=5168743489=(7^3)*(13^3)*(19^3). Any other cubes or higher egcd's in this sequence? Conjecture II: egcd(a(n))=2 for all n ne 11.

			a(2) = sigma(A128607(2)) = sigma(343) = 1+7+7^2+7^3 = 400 = 2^4*5^2.

Robert Israel, Table of n, a(n) for n = 1..58

Cf. A000203, A001597, A128607.

N:= 10^13: # to get all terms where A128607(n) <= N
pows:= {1, seq(seq(n^k, n = 2 .. floor(N^(1/k))), k = 2 .. floor(log[2](N)))}:
filter:= proc(n) local s, F;
   s:= numtheory:-sigma(n);
   F:= map(t -> t[2], ifactors(s)[2]);
   igcd(op(F)) >= 2
end proc:
filter(1):= true: A128608:= sort(convert((filter, pows), list)):
map(numtheory:-sigma,A128608); # Robert Israel, Feb 14 2016

M = 10^13; (* to get all terms where A128607(n) <= M *)
pows = {1, Table[Table[n^k, {n, 2, Floor[M^(1/k)]}], {k, 2, BitLength[M]-1}]} // Flatten // Union;
okQ[n_] := Module[{s, F}, s = DivisorSigma[1, n]; F = FactorInteger[s][[All, 2]]; GCD @@ F >= 2];
okQ[1] = True;
DivisorSigma[1, #]& /@ Select[pows, okQ] (* Jean-François Alcover, Feb 09 2023, after Robert Israel *)

1, 13884144561, 35018011161, 120776405841, added by Zak Seidov, Feb 14 2016

Edited by Robert Israel, Feb 14 2016

A180090 Sigma-powerful numbers: powerful numbers n such that sigma(n) is also powerful. An incomplete version of A337045.

Original entry on oeis.org

81, 343, 400, 1705636, 3648100, 13645088, 25992000, 26680500, 29184800, 80802000, 110215125, 178054848, 180093375, 213444000, 310144500, 408632609, 575664500, 1055340196, 1120504500, 1476326929, 1667329664, 2066544500

Offset: 1

Walter Kehowski, Aug 09 2010

dead

The list is not exhaustive since the search was restricted to sigma(p^k) such that p<1728 and all powers k such that all primes in the factorization of sigma(p^k) are also less than 1728.

			sigma(3^4)=11^2, sigma(7^3)=2^4*5^2, sigma(2^4*5^2)=31^2, sigma(2^2*653^2)=7^2*13^2*19^2.

Walter A. Kehowski, Table of n, a(n) for n = 1..1536
Walter A. Kehowski, Sigma-powerful numbers

Cf. A001694, A000203, A128607.

A337045 Indecomposable sigma-powerful numbers: powerful numbers k such that sigma(k) is also powerful, but restricted to terms that are not the product of 2 terms > 1 of A337044.

Original entry on oeis.org

81, 343, 400, 9261, 189728, 224939, 972000, 1705636, 2205472, 3087000, 3591200, 3648100, 7968032, 13645088, 15350724, 21161304, 24240600, 25992000, 26680500, 29184800, 32832900, 48586824, 51595489, 80802000, 103617387, 109215352, 110215125, 119604096, 122805792

Offset: 1

Hugo Pfoertner, Aug 15 2020

nonn

This is an implementation of the suggestion that Walter A. Kehowski made on his website (see link) with regard to so-called indecomposable sigma-powerful numbers. However, the results deviate from the table linked there. The table is considered to be deficient.

			From _David A. Corneth_, Aug 29 2020: (Start)
No two proper divisors of 400 are sigma-powerful and have the product of those divisors 400 so 400 is in the sequence.
27783 = 81 * 343 is sigma-powerful but 81 and 343 are sigma-powerful as well so 27783 can be decomposed into two sigma-powerful factors. So 27783 is not in the sequence. (End)

David A. Corneth, Table of n, a(n) for n = 1..6421 (terms < 10^18)
Walter A. Kehowski, Sigma-powerful numbers, Aug 09 2010.

Cf. A000203, A001694, A128607, A180090, A337044.

v=vector(50); n=0;
for(m=2, 150000000, my(is); if(ispowerful(m) && ispowerful(sigma(m)), v[n++]=m; is=1; for(j=1, n-1, if(v[n]%v[j], , if(vecsearch(v[1..n-1], v[n]/v[j]), is=0; break))); if(is, print1(v[n], ", "))))

A065302 Squarefree nonprime numbers whose sum of divisors is also squarefree.

Original entry on oeis.org

1, 26, 74, 122, 146, 218, 314, 362, 386, 458, 554, 626, 746, 794, 818, 842, 866, 914, 1082, 1202, 1226, 1322, 1346, 1418, 1466, 1514, 1538, 1658, 1706, 1754, 1874, 1994, 2018, 2042, 2066, 2138, 2186, 2234, 2258, 2306, 2402, 2426, 2474, 2594, 2642, 2762

Offset: 1

Labos Elemer, Oct 29 2001

nonn

All elements except the first, a(1)=1, are of the form 2*p, where p is a prime and p == 1 (mod 12). Also, sigma(2*p) = (1+2)*(1+p) = 6m where m = (1+p)/2 and m == 1 (mod 6). A squarefree composite number not of the form 2*p cannot be in the sequence since sigma is multiplicative. For example, sigma(p*q) = (1+p)*(1+q) is divisible by 4 for p,q > 2. - Walter Kehowski, Mar 21 2007

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000203, A008683, A065299, A065300, A065301, A065303.

Cf. A128607, A128608.

Select[Range[3000], !PrimeQ[#] && SquareFreeQ[#] && SquareFreeQ[DivisorSigma[1, #]] &] (* Amiram Eldar, Jun 05 2025 *)

isok(m) = !isprime(m) && moebius(m) && moebius(sigma(m)); \\ Harry J. Smith, Oct 16 2009

Name corrected by Amiram Eldar, Jun 05 2025

cp's OEIS Frontend

A128608 a(n)=sigma(A128607(n)), where A128607(n) is the sequence of perfect (or pure) powers such that a(n) is a perfect power.

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A180090 Sigma-powerful numbers: powerful numbers n such that sigma(n) is also powerful. An incomplete version of A337045.

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A337045 Indecomposable sigma-powerful numbers: powerful numbers k such that sigma(k) is also powerful, but restricted to terms that are not the product of 2 terms > 1 of A337044.

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A065302 Squarefree nonprime numbers whose sum of divisors is also squarefree.

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Mathematica

PARI

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