A065496 -id:A065496 - cp's OEIS frontend

A065522 Numbers k such that sigma(k) and sigma(k+1) are nontrivial powers (A065496).

21, 93, 381, 1065, 1173, 5065, 5670, 5729, 6603, 8809, 10281, 10680, 15960, 17110, 39286, 40526, 47882, 49951, 61962, 85058, 85261, 99066, 117860, 125985, 126853, 135890, 143241, 159901, 171945, 179556, 185853, 208744, 209585, 210450, 251394, 261767, 288792

Offset: 1

Robert G. Wilson v, Nov 27 2001

nonn

			10680 is in this sequence because sigma(10680)=180^2 and sigma(10681)=108^2. - _Sean A. Irvine_, Sep 04 2023

Cf. A065496.

nn = 300000; SequencePosition[Array[GCD @@ FactorInteger[DivisorSigma[1, #]][[All, -1]] &, nn + 1], {?(# > 1 &), ?(# > 1 &)}][[All, 1]] (* Michael De Vlieger, Sep 04 2023 *)

Missing terms inserted by Sean A. Irvine, Sep 04 2023

A175432 a(n) = the greatest number k such that sigma(n) = m^k for any m >= 1 (sigma = A000203).

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 2, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Jaroslav Krizek, May 10 2010

nonn

a(A175431(n)) = 1 for n >= 1.
a(A065496(n)) > 1 for n >= 1.
It appears that the record values in this sequence, 1, 2, 3, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, ..., is A180221 with a 1 prepended, at least through term #469. Is this a theorem? - Ray Chandler, Aug 20 2010

			For n = 7, a(7) = 3 because sigma(7) = 8 = 2^3.

Antti Karttunen, Table of n, a(n) for n = 1..16384

For locations of records see A169981.

Cf. A000203, A052409, A175433.

Array[Apply[GCD, FactorInteger[DivisorSigma[1, #]][[All, -1]]] &, 105] (* Michael De Vlieger, Nov 05 2017 *)

a(n)=max(ispower(sigma(n)),1) \\ Charles R Greathouse IV, Feb 14 2013

a(n) = A052409(A000203(n)). - N. J. A. Sloane, Aug 19 2010

a(n) = log_A175433(n) [A000203(n)].

A180162 a(n) is the smallest number N such that sigma(N) is an n-th power but not a higher power, with a(n) = 0 if no such number exists.

Original entry on oeis.org

1, 2, 3, 7, 510, 21, 17490, 93, 217, 381, 651, 118879530, 2667, 8191, 11811, 24573, 57337, 82677, 172011, 393213, 761763, 1572861, 2752491, 5332341, 11010027, 21845397, 48758691, 85327221, 199753347, 341310837, 677207307, 1398273429

Offset: 0

Walter Kehowski, Aug 14 2010, Aug 19 2010

nonn

			a(4)=510 since 510=2*3*5*17, sigma(510)=2^4*3^4.
a(11)=2*3*5*7*11*53*971=118879530 since sigma(118879530)=6^11.

Ray Chandler, Table of n, a(n) for n = 0..469

Cf. A065496, A000203, A175432, A169981.

with(numtheory);
egcd:=proc(n::posint) local L; if n>1 then L:=ifactors(n)[2]; L:=map(z-> z[2],L); igcd(op(L)) else 0 fi end:
P:={}: SP:={}:
for w to 1 do
for n from 1 to 12^6 do
sn:=sigma(n);
esn:=egcd(sn);
if not esn in P then
P:=P union {esn};
SP:=SP union {[esn,n]};
printf("n=%d, esn=%d, sn=...\n",n,esn);
print(ifactor(sn));
fi;
od; #n
od; #w
P; SP;

a(n) >= A063869(n). - R. J. Mathar, Aug 20 2010

a(11) found by Walter Kehowski and Artur Jasinski, Aug 16 2010
Edited by N. J. A. Sloane, Aug 19 2010
a(23) onwards from Ray Chandler, Aug 19 2010

A180169 Sigma(A180162(n)), where A180162(n) is the smallest N such that sigma(N) is an n-th power, or 0 if no such N can be found.

Original entry on oeis.org

1, 3, 4, 8, 1296, 32, 46656, 128, 256, 512, 1024, 362797056, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648

Offset: 0

Walter Kehowski, Aug 14 2010

			A180162(4)=510 and so A180169(4)=sigma(510)=2^4*3^4.

Ray Chandler, Table of n, a(n) for n = 0..469

Cf. A065496, A000203, A180162.

Empirical o.g.f.: (-1-1*x+2*x^2-1280*x^4+2560*x^5-46592*x^6+93184*x^7-362795008*x^11+725590016*x^12)/(2*x-1). - Simon Plouffe, Feb 26 2011.

a(23) onwards from Ray Chandler, Aug 20 2010

A281140 Least k such that k is the product of n distinct primes and sigma(k) is an n-th power.

Original entry on oeis.org

2, 22, 102, 510, 90510, 995610, 11616990, 130258590, 1483974030, 18404105922510, 428454465915630, 10195374973815570, 240871269907008510, 94467020965716904490370, 4580445736068712946096430, 7027383957579235221501981990, 419420669769073022876839238610, 24967450935148397377034326845390

Offset: 1

Altug Alkan, Jan 15 2017

nonn

Freiberg (Theorem 1.2) shows that there are >> (n*x^(1/n))/(log x)^(n+2) such values of k up to x. He calls the set of such numbers B*(x;+1;n). In particular, a(n) exists for each n.
Corresponding values of sigma(k) are 3 = 3^1, 36 = 6^2, 216 = 6^3, 1296 = 6^4, 248832 = 12^5, 2985984 = 12^6, 12^7, 12^8, 12^9, 24^10, 24^11, 24^12, 24^13, 48^14, 48^15, 60^16, 60^17, 60^18, 60^19, 84^20, 84^21, 84^22, 84^23, ...
a(14) <= 94467020965716904490370. - Daniel Suteu, Mar 28 2021

			a(3) = 102 because 102 = 2 * 3 * 17 and (2 + 1)*(3 + 1)*(17 + 1) = 6^3.

David A. Corneth, Table of n, a(n) for n = 1..23
Tristan Freiberg, Products of shifted primes simultaneously taking perfect power values, arXiv:1008.1978 [math.NT], 2010.

Cf. A000203, A001597, A005117, A065496, A281069.

a(n) = my(k=2); while(!issquarefree(k) || !ispower(sigma(k), n) || omega(k)!=n, k++); k \\ Felix Fröhlich, Jan 17 2017

a(10)-a(13) from Jinyuan Wang, Nov 08 2020
a(14) from Daniel Suteu and David A. Corneth, Mar 28 2021
a(15)-a(18) from David A. Corneth, Mar 29 2021

A065523 Numbers n such that sigma(n) is a prime power (A025475).

Original entry on oeis.org

3, 7, 21, 31, 81, 93, 127, 217, 381, 400, 651, 889, 2667, 3937, 8191, 11811, 24573, 27559, 57337, 82677, 131071, 172011, 253921, 393213, 524287, 761763, 917497, 1040257, 1572861, 1777447, 2752491, 3120771, 3670009, 4063201, 5332341

Offset: 1

Robert G. Wilson v, Nov 27 2001

nonn

Donovan Johnson, Table of n, a(n) for n = 1..80

A proper subset of A065496.

s[n_] := DivisorSigma[1, n ]; Select[ Range[2, 10^6], !PrimeQ[ s[ # ]] && Mod[ s[ # ], s[ # ] - EulerPhi[ s[ # ]]] == 0 & ]

a(31)-a(35) from Donovan Johnson, Sep 05 2008

A124142 Abundant numbers k such that sigma(k) is a perfect power.

Original entry on oeis.org

66, 70, 102, 210, 282, 364, 400, 510, 642, 690, 714, 770, 820, 930, 966, 1080, 1092, 1146, 1164, 1200, 1416, 1566, 1624, 1672, 1782, 2130, 2226, 2250, 2346, 2460, 2530, 2586, 2652, 2860, 2910, 2912, 3012, 3198, 3210, 3340, 3498, 3522, 3560, 3710, 3810

Offset: 1

Walter Kehowski, Dec 01 2006

nonn

Positive integers k such that sigma(k) > 2*k and sigma(k) = a^b where both a and b are greater than 1.

If k is a term with sigma(k) a square, and p and q are members of A066436 that do not divide k, then k*p*q is in the sequence. Thus if A066436 is infinite, so is this sequence. - Robert Israel, Oct 29 2018

			a(1) = 66 since sigma(66) = 144 = 12^2.

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

Cf. A001597, A005101, A065496, A066436.

with(numtheory); egcd := proc(n::posint) local L; if n>1 then L:=ifactors(n)[2]; L:=map(z->z[2],L); return igcd(op(L)) else return 1 fi; end; L:=[]: for w to 1 do for n from 1 to 10000 do s:=sigma(n); if s>2*n and egcd(s)>1 then print(n,s,ifactor(s)); L:=[op(L),n]; fi od od;

filterQ[n_] := With[{s = DivisorSigma[1, n]}, s > 2n && GCD @@ FactorInteger[s][[All, 2]] > 1];
Select[Range[4000], filterQ] (* Jean-François Alcover, Sep 16 2020 *)

is(k) = {my(s = sigma(k)); s > 2*k && ispower(s);} \\ Amiram Eldar, Aug 02 2024

A124143 Perfect powers pp such that sigma(k) = pp for some positive integer k.

Original entry on oeis.org

4, 8, 32, 36, 121, 128, 144, 216, 256, 324, 400, 512, 576, 784, 900, 961, 1024, 1296, 1600, 1728, 1764, 1936, 2304, 2704, 2744, 2916, 3136, 3600, 3844, 4096, 4356, 4624, 4900, 5184, 5776, 5832, 6084, 6400, 7056, 7744, 7776, 8000, 8100, 8192, 9216, 9604

Offset: 1

Walter Kehowski, Dec 01 2006

nonn

			a(1) = 4 since sigma(3) = 4 = 2^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: Inversion of Multiplicative Functions (invphi.gp).

Intersection of A001597 and A002191 \ {1}.

Cf. A065496.

Set(Sort([SumOfDivisors(k): k in[1..10000], b in [2..15], a in [2..100] | SumOfDivisors(k) eq a^b])); // Jaroslav Krizek, Mar 10 2015

Set(Sort([SumOfDivisors(k): k in[A065496(n)]])); // Jaroslav Krizek, Mar 10 2015

with(numtheory); egcd := proc(n) local L; if n>1 then L:=ifactors(n)[2]; L:=map(z->z[2],L); return igcd(op(L)) else return 1 fi; end; L:=[]: P:={}: for w to 1 do for n from 1 to 10000 do s:=sigma(n); if egcd(s)>1 then print(n,s,ifactor(s)); L:=[op(L),n]; P:=P union {s}; fi od od; L; P;

powerQ[n_] := Block[{pf = FactorInteger@ n, min}, min = Min @@ Last /@ pf; min > 1 && AllTrue[Last /@ pf/min, IntegerQ]]; lim = 10000; Intersection[Select[Range@ lim, powerQ], DeleteDuplicates@ Sort[DivisorSigma[1, #] & /@ Range@ lim]] (* Michael De Vlieger, Mar 10 2015 *)

is(n) = ispower(n) && invsigmaNum(n) > 0; \\ Amiram Eldar, Aug 02 2024, using Max Alekseyev's invphi.gp

A124144 Perfect powers pp such that sigma(k) = pp for some abundant number k.

Original entry on oeis.org

144, 216, 576, 784, 961, 1296, 1728, 1764, 2304, 2744, 3136, 3600, 3844, 4356, 5184, 6084, 7056, 7776, 8100, 9216, 11664, 12544, 13824, 14400, 15376, 15876, 17424, 19600, 20736, 21952, 24336, 27000, 28224, 32400, 34596, 36864, 38416, 39204, 41616, 44100, 46656, 50176

Offset: 1

Walter Kehowski, Dec 01 2006

nonn

			a(1) = 144 since sigma(66) = 144 > 2*66 = 132.

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

Cf. A001597, A005101, A065496.

with(numtheory); egcd := proc(n::posint) local L; if n>1 then L:=ifactors(n)[2]; L:=map(z->z[2],L); return igcd(op(L)) else return 1 fi; end; L:=[]: P:={}: for w to 1 do for n from 1 to 10000 do s:=sigma(n); if s>2*n and egcd(s)>1 then print(n,s,ifactor(s)); L:=[op(L),n]; P:=P union {s}; fi od od; L; P;

ppQ[n_] := GCD @@ FactorInteger[n][[;; , 2]] > 1;
f[n_] := Module[{s = DivisorSigma[1, n]}, If[s > 2*n, s, Nothing]];
seq[max_] := Union[Select[Array[f, max], # < max && ppQ[#] &]]; seq[60000] (* Amiram Eldar, Mar 11 2024 *)

a(32) inserted and more terms added by Amiram Eldar, Mar 11 2024

A175431 Numbers m such that sigma(m) is not a nontrivial power, i.e., sigma(m) = A000203(n) is not equal a^b where a and b are greater than 1.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Jaroslav Krizek, May 10 2010

nonn

Complement of A065496.

A175432(a(n)) = 1 for n >= 1.

			Number 3 is not in the sequence because sigma(3) = 4 = 2^2.

Cf. A000203, A065496, A175432.

Extended by Ray Chandler, Aug 20 2010

cp's OEIS Frontend

A065522 Numbers k such that sigma(k) and sigma(k+1) are nontrivial powers (A065496).

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A175432 a(n) = the greatest number k such that sigma(n) = m^k for any m >= 1 (sigma = A000203).

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A180162 a(n) is the smallest number N such that sigma(N) is an n-th power but not a higher power, with a(n) = 0 if no such number exists.

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A180169 Sigma(A180162(n)), where A180162(n) is the smallest N such that sigma(N) is an n-th power, or 0 if no such N can be found.

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A281140 Least k such that k is the product of n distinct primes and sigma(k) is an n-th power.

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A065523 Numbers n such that sigma(n) is a prime power (A025475).

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A124142 Abundant numbers k such that sigma(k) is a perfect power.

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A124143 Perfect powers pp such that sigma(k) = pp for some positive integer k.

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