A105402 -id:A105402 - cp's OEIS frontend

A027598 Numbers k such that the set of prime divisors of k is equal to the set of prime divisors of sigma(k).

1, 6, 28, 120, 270, 496, 672, 1080, 1638, 1782, 3780, 8128, 18600, 20580, 24948, 26208, 30240, 32640, 32760, 35640, 41850, 44226, 55860, 66960, 164640, 167400, 185220, 199584, 273000, 293760, 401310, 441936, 446880, 502740, 523776, 614250, 707616, 802620, 819000

Offset: 1

Jud McCranie

nonn

Multiplicities are ignored.
All even perfect numbers are in the sequence. It seems that 1 is the only odd term of the sequence. - Farideh Firoozbakht, Jul 01 2008
sigma() is the multiplicative sum-of-divisors function. - Walter Nissen, Dec 16 2009
Pollack and Pomerance call these "prime-perfect numbers" and show that there are << x^(1/3+e) of these up to x for any e > 0. - Charles R Greathouse IV, May 09 2013
Except for unity for the obvious reason, the primitive terms are the perfect numbers (A000396). - Robert G. Wilson v, Feb 19 2019
If an odd term > 1 exists, it is larger than 5*10^23. - Giovanni Resta, Jun 02 2020

			273000 = 2^3*3*5^3*7*13 and sigma(273000) = 1048320 = 2^8*3^2*5*7*13 so 273000 is in the sequence.

R. K. Guy, Unsolved Problems in Number Theory, B19.

Donovan Johnson, Table of n, a(n) for n = 1..500 (first 100 terms from T. D. Noe)
Paul Pollack and Carl Pomerance, Prime-Perfect Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 12a, Paper A14, 2012.

Intersection of A105402 and A175200. - Amiram Eldar, Jun 02 2020

Cf. A110751, A110819, A055744, A081377, A000203.

Filtered([1..1000000],n->Set(Factors(n))=Set(Factors(Sigma(n)))); # Muniru A Asiru, Feb 21 2019

Select[Range[1000000], Transpose[FactorInteger[#]][[1]] == Transpose[FactorInteger[DivisorSigma[1, #]]][[1]] &] (* T. D. Noe, Dec 08 2012 *)

a(n) = {for (i=1, n, fn = factor(i); fs = factor(sigma(i)); if (fn[,1] == fs[,1], print1(i, ", ")););} \\ Michel Marcus, Nov 18 2012

is(n)=my(f=factor(n),fs=[],t);for(i=1,#f[,1], t=factor((f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1))[,1]; fs=vecsort(concat(fs,t~),,8); if(#setminus(fs,f[,1]~), return(0))); fs==f[,1]~ \\ Charles R Greathouse IV, May 09 2013

Edited by N. J. A. Sloane, Jul 12 2008 at the suggestion of R. J. Mathar

A175200 Numbers k such that rad(k) divides sigma(k).

Original entry on oeis.org

1, 6, 24, 28, 40, 54, 96, 120, 135, 216, 224, 234, 270, 360, 384, 486, 496, 540, 588, 600, 640, 672, 864, 891, 936, 1000, 1080, 1350, 1372, 1521, 1536, 1638, 1782, 1792, 1920, 1944, 2016, 2160, 2176, 3000, 3240, 3375, 3402, 3456, 3564, 3724, 3744, 3780, 4320

Offset: 1

Michel Lagneau, Mar 03 2010

nonn

rad(k) is the product of the distinct primes dividing k (A007947). sigma(k) is the sum of divisors of k (A000203). The odd numbers in this sequence (A336554) are rare: 1, 135, 891, 1521, 3375, 5733, 10935, 11907, 41067, 43875, ...
Also numbers k such that k divides sigma(k)^tau(k). - Arkadiusz Wesolowski, Nov 09 2013
This sequence is infinite. It contains an infinite number of even elements and an infinite number of odd ones. This is due to the fact that for every odd prime p and every prime q dividing p+1, p*q^r is prime-perfect when r = -1 + the multiplicative order of q modulo p. - Emmanuel Vantieghem, Oct 13 2014
For each term, it is possible to find an exponent k such that sigma(n)^k is divisible by n. A007691 (multi-perfect numbers) is a subsequence of terms that have k=1. A263928 is the subsequence of terms that have k=2. - Michel Marcus, Nov 03 2015
Pollack and Pomerance call these numbers "prime-abundant numbers". - Amiram Eldar, Jun 02 2020

			rad(6) = 6, sigma(6) = 12 = 6*2.
rad(24) = 6, sigma(24) = 60 = 6*10.
rad(43875) = 195, sigma(43875) = 87360 = 195*448.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 827.

Donovan Johnson, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Paul Pollack and Carl Pomerance, Prime-Perfect Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 12A, Paper A14, 2012.

Cf. A000203, A007947, A027598, A069235, A105402, A173615, A336554 (odd terms).

[n: n in [1..5000] | IsZero(DivisorSigma(1, n)^n mod n)];// Vincenzo Librandi, Aug 07 2018

for n from 1 to 5000 do : p1:= ifactors(n)[2] :p2 :=mul(p1[i][1], i=1..nops(p1)): if irem(sigma(n),p2) =0 then print (n): else fi: od :

Select[Range@5000, Divisible[DivisorSigma[1, #]^#, # ]&] (* Vincenzo Librandi, Aug 07 2018 *)

isok(n) = {fs = Set(factor(sigma(n))[,1]); fn = Set(factor(n)[,1]); fn == setintersect(fn, fs);} \\ Michel Marcus, Nov 03 2015

A336353 Numbers k such that sigma(k) does not have any prime factor larger than the largest prime factor of k.

Original entry on oeis.org

1, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95

Offset: 1

Antti Karttunen, Jul 19 2020

nonn

Note that the terms after 1 do not form a subsequence of A102750: the first counterexample is 343 = 7^3. See A336354.

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Positions of zeros in A336352.
Cf. A102750, A336317.
Subsequences: A000396, A001599, A065091, A105402, A333646 (see comment there), A336354.

PARI
```
isA336353(n) = !A336352(n);
```

A333646 Numbers k divisible by the largest prime factor of the sum of divisors of k; a(1) = 1.

Original entry on oeis.org

1, 6, 15, 28, 30, 33, 40, 42, 51, 66, 69, 84, 91, 95, 102, 105, 117, 120, 135, 138, 140, 141, 145, 159, 165, 182, 186, 190, 210, 213, 224, 231, 234, 255, 270, 273, 280, 282, 285, 287, 290, 295, 308, 318, 321, 330, 345, 357, 364, 395, 420, 426, 435, 440, 445, 455

Offset: 1

Amiram Eldar, Jun 05 2020

nonn

Pomerance (1973) proved that all the harmonic numbers (A001599) are in this sequence.
If m is a product of distinct Mersenne primes (A046528), m > 1 and 3 | m, then 2*m is a term.
If p is a term of A005105 then, 6*p is a term for p > 3, and 3*p is a term if p is not a Mersenne prime (A000668).

			15 is a term since sigma(15) = 24, 3 is the largest prime factor of 24, and 15 is divisible by 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Carl Pomerance, On a Problem of Ore: Harmonic Numbers, unpublished manuscript, 1973; abstract *709-A5, Notices of the American Mathematical Society, Vol. 20, 1973, page A-648, entire volume.

A001599 and A105402 are subsequences.

Cf. A000203, A000668, A005105, A006530, A046528, A071190.

Select[Range[500], Divisible[#, FactorInteger[DivisorSigma[1, #]][[-1, 1]]] &]

Numbers k such that A071190(k) | k.

cp's OEIS Frontend

A027598 Numbers k such that the set of prime divisors of k is equal to the set of prime divisors of sigma(k).

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A175200 Numbers k such that rad(k) divides sigma(k).

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A336353 Numbers k such that sigma(k) does not have any prime factor larger than the largest prime factor of k.

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A333646 Numbers k divisible by the largest prime factor of the sum of divisors of k; a(1) = 1.

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