A027598 -id:A027598 - cp's OEIS frontend

A286876 Numbers n such that the set of prime divisors of n is equal to the set of prime divisors of sum of proper divisors of n while n is not in A027598.

24, 40, 216, 234, 360, 588, 2016, 3724, 4320, 4680, 6048, 6552, 9720, 11466, 22932, 54432, 58752, 97920, 99200, 108927, 137214, 167580, 185562, 217854, 297600, 309582, 435708, 448335, 524160, 544635, 637000, 804384, 871416, 931840, 1284192, 1384110, 1489752

Offset: 1

Altug Alkan, Aug 02 2017

nonn

108927 is the smallest odd term of this sequence.

			24 is in the sequence because 24 = 2^3*3 and sum of proper divisors of 24 is 1 + 2 + 3 + 4 + 6 + 8 + 12 = 36 = 2^2*3^2 while sigma(24) = 60 is divisible by 5.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A000203, A001065, A007947, A027598.

Select[Range[1500000], And[UnsameQ @@ {#1, #2}, SameQ @@ {#1, #3}] & @@ Map[FactorInteger[#][[All, 1]] &, {#1, #2, #2 - #1} & @@ {#, DivisorSigma[1, #]}] &] (* Michael De Vlieger, Aug 02 2017 *)

rad(n) = factorback(factorint(n)[, 1]);
isok(n) = rad(sigma(n)-n)==rad(n) && rad(sigma(n))!=rad(n);

A055744 Numbers k such that k and phi(k) have the same prime factors.

Original entry on oeis.org

1, 4, 8, 16, 18, 32, 36, 50, 54, 64, 72, 100, 108, 128, 144, 162, 200, 216, 250, 256, 288, 294, 324, 400, 432, 450, 486, 500, 512, 576, 578, 588, 648, 800, 864, 882, 900, 972, 1000, 1014, 1024, 1152, 1156, 1176, 1210, 1250, 1296, 1350, 1458, 1600, 1728, 1764

Offset: 1

Labos Elemer, Jul 11 2000

nonn

Contains products of suitable powers of 2 and Fermat primes. For x = 2^u*3^w, phi(x) = 2^u*3^(w-1) with suitable exponents. Analogous constructions are possible with {2,3,7} prime divisors, etc.
From Ivan Neretin, Mar 19 2015: (Start)
Also, numbers k that meet the following criteria for every prime p dividing k:
  1. All prime divisors of p-1 must also divide k;
  2. If k has no prime divisors of the form m*p+1, and k is divisible by p, then k must be divisible by p^2.
Also, numbers k for which {k, phi(k), phi(phi(k))} is a geometric progression.
(End)
All terms > 1 are even. An even number k is in the sequence iff 2*k is in the sequence. - Robert Israel, Mar 19 2015
For n > 1, the largest prime factor of a(n) has multiplicity >= 2. For all prime factors more than half of the largest prime factor of a(n), the multiplicity differs from 1.
If k = p1^a1 * p2^a2 * ... * pm^am is in the sequence, then so is p1^b1 * p2^b2 * ... * pm^bm for 1 <= i <= m and prime pi and bi >= ai.
If m * p^2 is not in the sequence, for a prime p and some m > 0, then neither is m * p^3. - David A. Corneth, Mar 22 2015
A027748(a(n),j) = A027748(A000010(a(n)),j) for j=1..A001221(a(n)); also numbers k such that k and phi(k) have the same squarefree kernel: A007947(a(n)) = A007947(A000010(a(n))). - Reinhard Zumkeller, Jun 01 2015
Pollack and Pomerance call these numbers "phi-perfect numbers". - Amiram Eldar, Jun 02 2020

			k = 578 = 2*17*17, phi(578) = 272 = 2*2*2*2*17 with 2 and 17 prime factors, so 578 is a term.
k = 588 = 2*2*3*7*7, phi(588) = 168 = 2*2*2*3*7, so 588 is a term.
k = 264196 = 2*2*257*257, phi(264196) = 512*257 = 131584, so 264196 is a term.

David A. Corneth, Table of n, a(n) for n = 1..117561 (terms <= 10^11; first 10000 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 A073539 and A151999. - Amiram Eldar, Jun 02 2020
Cf. A001221, A000010, A110751, A110819, A027598, A081377.
Cf. A007947, A027748, A055742, A173557, A256248, subsequence of A124240.

a055744 n = a055744_list !! (n-1)
a055744_list = 1 : filter f [2..] where
   f x = all ((== 0) . mod x) (concatMap (a027748_row . subtract 1) ps) &&
         all ((== 0) . mod (a173557 x))
             (map fst $ filter ((== 1) . snd) $ zip ps $ a124010_row x)
         where ps = a027748_row x
-- Reinhard Zumkeller, Jun 01 2015

select(numtheory:-factorset = numtheory:-factorset @ numtheory:-phi,
[1, 2*i $ i=1..2000]); # Robert Israel, Mar 19 2015
isA055744 := proc(n)
    nfs := numtheory[factorset](n) ;
    phinfs := numtheory[factorset](numtheory[phi](n)) ;
    if nfs = phinfs then
        true;
    else
        false;
    end if;
end proc:
A055744 := proc(n)
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA055744(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Sep 23 2016

Select[Range@ 1800,
First /@ FactorInteger@ # == First /@ FactorInteger@ EulerPhi@ # &] (* Michael De Vlieger, Mar 21 2015 *)

is(n)=factor(n)[,1]==factor(eulerphi(n))[,1] \\ Charles R Greathouse IV, Oct 31 2011

is(n)=my(f=factor(n)); f[,1]==factor(eulerphi(f))[,1] \\ Charles R Greathouse IV, May 26 2015

Corrected and extended by James Sellers, Jul 11 2000

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

A081377 Numbers n such that the set of prime divisors of phi(n) is equal to the set of prime divisors of sigma(n).

Original entry on oeis.org

1, 3, 14, 35, 42, 70, 105, 119, 209, 210, 238, 248, 297, 357, 418, 477, 594, 595, 616, 627, 714, 744, 954, 1045, 1178, 1190, 1240, 1254, 1463, 1485, 1672, 1674, 1736, 1785, 1848, 1863, 2079, 2090, 2376, 2385, 2540, 2728, 2926, 2945, 2970, 3080, 3135, 3302

Offset: 1

Labos Elemer, Mar 26 2003

nonn

The multiplicities of the divisors are to be ignored.

Is it true that 1 is the only term in both this sequence and A055744? - Farideh Firoozbakht, Jul 01 2008. Answer from Luke Pebody, Jul 10 2008: No! In fact the numbers 103654150315463023813006470 and 6534150553412193640795377701190 are in both sequences.

			n=418=2*11*19: sigma(418)=720, phi[418]=180, common prime factor set ={2,3,5}
k = 477 = 3*3*53: sigma(477) = 702 = 2*3*3*3*13; phi(477) = 312 = 2*2*2*3*13; common factor set: {2,3,13}.
phi(89999)=66528=2^5*3^3*7*11 and sigma(89999)=118272=2^9*3*7*11 so 89999 is in the sequence.

T. D. Noe, Table of n, a(n) for n = 1..1000
Prime Puzzles, Puzzle 451

Cf. A000010, A000203, A076533, A065642, A081378, A110751, A110819, A027598, A141718.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] Do[s=ba[DivisorSigma[1, n]]; s1=ba[EulerPhi[n]]; If[Equal[s, s1], k=k+1; Print[n]], {n, 1, 10000}]

is(n)=factor(eulerphi(n=factor(n)))[,1]==factor(sigma(n))[,1] \\ Charles R Greathouse IV, Nov 27 2013

Edited by N. J. A. Sloane, Jul 11 2008 at the suggestion of Farideh Firoozbakht

A105402 Positive integers k such that the prime factors of sigma(k) are a subset of the prime factors of k.

Original entry on oeis.org

1, 6, 28, 30, 42, 66, 84, 102, 120, 138, 186, 210, 270, 282, 318, 330, 364, 420, 426, 462, 496, 510, 546, 570, 642, 672, 690, 714, 762, 840, 868, 870, 924, 930, 966, 1080, 1092, 1122, 1146, 1302, 1320, 1410, 1428, 1488, 1518, 1590, 1638, 1722, 1770, 1782, 1890

Offset: 1

Walter Kehowski, May 01 2005

Also numbers k such that k^k/sigma(k) is integral. - Vicente Izquierdo Gomez, Jan 04 2013.

Pollack and Pomerance call these numbers "prime-deficient numbers". - Amiram Eldar, Jun 02 2020

			102 is a term since 102 = 2*3*17 and sigma(102) = 2^3*3^3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Pollack and Carl Pomerance, Prime-Perfect Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 12a, Paper A14, 2012.

Cf. A000203, A027598, A058063, A054027, A014567, A023194, A175200.

A:=select(proc(z) numtheory[factorset](sigma(z)) subset numtheory[factorset](z) end,[$1..100000]); has 716 members.

Select[Range[2000],IntegerQ[#^#/DivisorSigma[1,#]] &] (* Vicente Izquierdo Gomez, Jan 04 2013 *)

Extended by R. J. Mathar, Dec 08 2008

A081381 Numbers n such that n and tau(n) = A000005(n) have the same prime factors (ignoring multiplicity).

Original entry on oeis.org

1, 2, 8, 9, 12, 18, 72, 80, 96, 108, 128, 288, 448, 486, 625, 720, 768, 864, 972, 1152, 1200, 1250, 1620, 1944, 2000, 2025, 2560, 4032, 4050, 5000, 5625, 6144, 6561, 6912, 7500, 7776, 8748, 9408, 10800, 11250, 11264, 12960, 13122, 16200, 18000, 18432, 19440

Offset: 1

Labos Elemer, Mar 26 2003

nonn

			n = 5000 = 2*2*2*5*5*5*5, tau(5000) = 20 = 2*2*5, common prime factors: {2,5}

Donovan Johnson and Giovanni Resta, Table of n, a(n) for n = 1..1096 (terms < 10^11, first 500 terms from Donovan Johnson)

Cf. A000005, A027598, A055744, A065642, A081377-A081380.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] Do[s=ba[DivisorSigma[0, n]]; If[Equal[s, ba[n]], Print[n]], {n, 1, 10000}]

is(n)=my(f=factor(n)); factor(numdiv(f))[,1]==f[,1] \\ Charles R Greathouse IV, Oct 19 2017

A295078 Numbers n > 1 such that n and sigma(n) have the same smallest and simultaneously largest prime factors.

Original entry on oeis.org

6, 28, 40, 84, 120, 140, 224, 234, 270, 420, 468, 496, 672, 756, 936, 1080, 1120, 1170, 1372, 1488, 1550, 1638, 1782, 1862, 2176, 2340, 2480, 2574, 3100, 3250, 3276, 3360, 3472, 3564, 3724, 3744, 3780, 4116, 4464, 4598, 4650, 4680, 5148, 5456, 5586, 6048, 6200

Offset: 1

Jaroslav Krizek, Nov 13 2017

nonn

All even perfect numbers are terms.
Conjecture: A007691 (multiply-perfect numbers) is a subsequence.
Note that an odd perfect number (if it exists) would be a counterexample to the conjecture. - Robert Israel, Jan 08 2018
Intersection of A071834 and A295076.
Numbers n such that A020639(n) = A020639(sigma(n)) and simultaneously A006530(n) = A006530(sigma(n)).
Numbers n such that A020639(n) = A071189(n) and simultaneously A006530(n) = A071190(n).
Supersequence of A027598.

			40 = 2^3*5 and sigma(40) = 90 = 2*3^2*5 hence 40 is in the sequence.
The first odd term is 29713401 = 3^2 * 23^2 * 79^2; sigma(29713401) = 45441669 = 3*7^3*13*43*79.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A000203, A006530, A007691, A020639, A027598, A071189, A071190, A295076.

[n: n in [2..10000] | Minimum(PrimeDivisors(n)) eq Minimum(PrimeDivisors(SumOfDivisors(n))) and Maximum(PrimeDivisors(n)) eq Maximum(PrimeDivisors(SumOfDivisors(n)))]

filter:= proc(n) local f, s; uses numtheory;
  f:= factorset(n);
  s:= factorset(sigma(n));
  min(f) = min(s) and max(f)=max(s)
end proc:
select(filter, [$2..10^4]); # Robert Israel, Jan 08 2018

Rest@ Select[Range@ 6200, SameQ @@ Map[{First@ #, Last@ #} &@ FactorInteger[#][[All, 1]] &, {#, DivisorSigma[1, #]}] &] (* Michael De Vlieger, Nov 13 2017 *)

isok(n) = if (n > 1, my(fn = factor(n)[,1], fs = factor(sigma(n))[,1]); (vecmin(fn) == vecmin(fs)) && (vecmax(fn) == vecmax(fs))); \\ Michel Marcus, Jan 08 2018

Added condition n>1 to definition. Corrected b-file. - N. J. A. Sloane, Feb 03 2018

A329858 Numbers k such that k and usigma(k) have the same set of prime divisors, where usigma(k) is the sum of unitary divisors of k (A034448).

Original entry on oeis.org

1, 6, 24, 60, 90, 180, 360, 378, 816, 1056, 1512, 3780, 9100, 10500, 12240, 13230, 15750, 15840, 26460, 31500, 36750, 40950, 46494, 51408, 52920, 63000, 63700, 66528, 73500, 87360, 94500, 95550, 110250, 145600, 145920, 147000, 163800, 181632, 185976, 220500

Offset: 1

Amiram Eldar, Nov 22 2019

nonn

			6 is in the sequence since 6 = 2 * 3 and usigma(6) = 12 = 2^2 * 3 both have the same set of prime divisors, {2, 3}.

Amiram Eldar, Table of n, a(n) for n = 1..435 (terms below 5*10^10)

The unitary version of A027598.

Cf. A007947, A034448.

rad[n_] := Times @@ (First@# & /@ FactorInteger@ n); usigma[1]=1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[2*10^5], rad[#] == rad[usigma[#]] &]

A329879 Numbers k such that k and nusigma(k) have the same set of prime divisors, where nusigma(k) is the sum of nonunitary divisors of k (A048146).

Original entry on oeis.org

4, 9, 24, 25, 49, 54, 112, 121, 150, 169, 289, 294, 361, 480, 529, 726, 750, 841, 961, 1014, 1369, 1681, 1734, 1849, 1984, 2058, 2166, 2209, 2430, 2520, 2688, 2809, 3174, 3481, 3721, 3780, 4489, 5041, 5046, 5329, 5760, 5766, 6241, 6889, 7921, 7986, 8214, 8700

Offset: 1

Amiram Eldar, Nov 23 2019

nonn

Numbers k such that rad(nusigma(k)) = rad(k), where rad(k) is the squarefree kernel of k (A007947).

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

Cf. A007947, A027598, A048146, A329858.

rad[n_] := Times @@ (First@# & /@ FactorInteger@ n); usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); nusigma[n_] := DivisorSigma[1, n] - usigma[n]; Select[Range[10^4], rad[nusigma[#]] == rad[#] &]

A214266 Numbers n such that n, phi(n) and sigma(n) have same set of prime factors.

Original entry on oeis.org

1, 103654150315463023813006470, 207308300630926047626012940, 414616601261852095252025880, 518270751577315119065032350, 829233202523704190504051760, 982794906694760522078876160, 1036541503154630238130064700

Offset: 1

Michel Marcus, Jul 09 2012

			For n = 207308300630926047626012940 = 2^2*3^4*5*7*11*13^3*17^3*29^2*31^3*37^2*67^2
phi(n) = 2^18*3^9*5^2*7*11*13^2*17^2*29*31^2*37*67 and
sigma(n) = 2^16*3^6*5^2*7^5*11^2*13^2*17*29*31*37*67^2
have the same prime factors.

Michel Marcus, Table of n, a(n) for n = 1..100
Prime Puzzles, Puzzle 451

Cf. A027598, A055744, A081377.

cp's OEIS Frontend

A286876 Numbers n such that the set of prime divisors of n is equal to the set of prime divisors of sum of proper divisors of n while n is not in A027598.

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A055744 Numbers k such that k and phi(k) have the same prime factors.

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

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A081377 Numbers n such that the set of prime divisors of phi(n) is equal to the set of prime divisors of sigma(n).

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A105402 Positive integers k such that the prime factors of sigma(k) are a subset of the prime factors of k.

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Maple

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A081381 Numbers n such that n and tau(n) = A000005(n) have the same prime factors (ignoring multiplicity).

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A295078 Numbers n > 1 such that n and sigma(n) have the same smallest and simultaneously largest prime factors.

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