A110819 -id:A110819 - cp's OEIS frontend

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

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

A110751 Numbers n such that n and its digital reversal have the same prime divisors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 444, 454, 464, 474, 484, 494

Offset: 1

Amarnath Murthy, Aug 11 2005

Contains the palindromes A002113 as a subsequence. 1089 and 2178 are the first two non-palindromic terms. Any number of concatenations of 1089 with itself or 2178 with itself gives a term; e.g. 10891089 etc. Hence there are infinitely many non-palindromic terms. They are given in A110819.

			1089 = 3^2*11^2, 9801 = 3^4*11^2.

Derek Orr, Table of n, a(n) for n = 1..1000

Cf. A002113, A110819.

Select[ Range[ 500], First /@ FactorInteger[ # ] == First /@ FactorInteger[ FromDigits[ Reverse[ IntegerDigits[ # ]]]] &] (* Robert G. Wilson v *)

is_A110751(n)={ local(r=eval(concat(vecextract(Vec(Str(n)),"-1..1")))); r==n || factor(r)[,1]==factor(n)[,1] } /* M. F. Hasler */

from sympy import primefactors
A110751 = [n for n in range(1,10**5) if primefactors(n) == primefactors(int(str(n)[::-1]))] # Chai Wah Wu, Aug 14 2014

Edited and extended by Robert G. Wilson v, Sep 21 2005

Corrected comment, added PARI code. - M. F. Hasler, Nov 16 2008

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

Original entry on oeis.org

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

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

A224252 Nonpalindromic n such that the factorizations of n and its digital reverse differ only for the exponents order.

Original entry on oeis.org

277816, 618772, 14339143, 34193341, 1125355221, 1225535211, 2613391326, 6231933162, 26157457326, 62375475162, 100504263021, 102407325111, 111523704201, 120362405001, 144326261443, 275603902756, 277816277816, 344162623441, 392739273927, 392875758639

Offset: 1

Giovanni Resta, Apr 02 2013

Subset of A110751 and A110819.

			277816 and its reverse 618772 are in the sequence since 277816 = 2^3*7*11^2*41 and 618772 = 2^2*7^3*11*41 have the same prime divisors and the same exponents (1,1,2,3).

Giovanni Resta, Table of n, a(n) for n = 1..34 (terms < 2*10^12)

Cf. A110751, A110819, A224253.

Do[fn = FactorInteger@n; fr = FactorInteger@ FromDigits@ Reverse@ IntegerDigits@n; If[fn != fr && First /@ fn == First /@ fr && Sort[Last /@ fn] == Sort[Last /@ fr], Print[n]], {n, 15*10^6}]

from sympy import primefactors, factorint
A224252 = [n for n in range(1,10**6) if n != int(str(n)[::-1]) and primefactors(n) == primefactors(int(str(n)[::-1])) and sorted(factorint(n).values()) == sorted(factorint(int(str(n)[::-1])).values())] # Chai Wah Wu, Aug 21 2014

A113548 Least non-palindromic number k such that k and its digital reversal both have exactly n prime divisors.

Original entry on oeis.org

13, 12, 132, 1518, 15015, 204204, 10444434, 241879638, 20340535215, 242194868916, 136969856585562, 2400532020354468, 484576809394483806, 200939345091539746692

Offset: 1

Ryan Propper and Robert G. Wilson v, Sep 21 2005

This sequence does not allow ending in 0, else a(8) = 208888680, a(11) = 64635504163230 and a(13) = 477566276048801940. - Michael S. Branicky, Feb 14 2023

			a(1)=13=13 since 31=31,
a(2)=12=2^2*3 since 21=3*7,
a(3)=132=2^2*3*11 since 231=3*7*11,
...
a(7)=10444434=2*3*7*11*13*37*47 since 43444401=3*7*11*13*17*23*37,
a(8)=241879638=2*3*7*11*13*17*23*103 since 836978142=2*3*7*11*13*23*73*83.

Cf. A110843, A110819, A239696.

r[n_] := FromDigits[ Reverse[ IntegerDigits[ n]]]; f[n_] := Block[{k = r[n], len = Length[ FactorInteger[n]]}, If[k != n && len == Length[ FactorInteger[ r[n]]], len, 0]]; t = Table[0, {10}]; Do[ a = f[n]; If[a > 0 && t[[a]] == 0, t[[a]] = n; Print[{a, n}]], {n, 107}]; t

generate(A, B, n) = A=max(A, vecprod(primes(n))); (f(m, p, j) = my(list=List()); forprime(q=p, sqrtnint(B\m, j), if(q==5 && m%2==0, next); my(v=m*q); while(v <= B, if(j==1, my(r=fromdigits(Vecrev(digits(v)))); if(v>=A && r != v && omega(r) == n, listput(list, v)), if(v*(q+1) <= B, list=concat(list, f(v, q+1, j-1)))); v *= q)); list); vecsort(Vec(f(1, 2, n)));
a(n) = my(x=vecprod(primes(n)), y=2*x); while(1, my(v=generate(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Feb 18 2023

a(n) >= A239696(n). - Daniel Suteu, Feb 18 2023

Edited and extended by Giovanni Resta, Jan 16 2006
a(9)-a(10) from Giovanni Resta, Feb 23 2014
a(11)-a(13) from Michael S. Branicky, Feb 14 2023
a(14) from Daniel Suteu, Feb 18 2023

A201009 Numbers m such that the set of distinct prime divisors of m is equal to the set of distinct prime divisors of the arithmetic derivative m'.

Original entry on oeis.org

1, 4, 16, 27, 108, 144, 256, 432, 500, 784, 972, 1323, 1728, 2700, 2916, 3125, 3456, 5292, 8788, 11664, 12500, 13068, 15376, 16875, 19683, 20736, 23328, 25000, 27648, 28125, 31212, 34300, 47916, 54000, 57132, 65536, 72000, 78732, 97556, 102400, 103788, 104544

Offset: 1

Paolo P. Lava, Jan 09 2013

nonn

A027748(n,k) = A027748(A003415(n),k) for k=1..A001221(n). - Reinhard Zumkeller, Jan 16 2013

A051674 is a subsequence of this sequence.

			n = 1728 = 2^6*3^3, n' = 6912 = 2^8*3^3 have the same prime factors 2 and 3.

Paolo P. Lava and Donovan Johnson, Table of n, a(n) for n = 1..500 (first 100 terms from Paolo P. Lava)

Cf. A001221, A003415, A027748, A051674, A055744, A081377, A110751, A110819.

a201009 = a201009_list
a201009_list = 1 : filter
   (\x -> a027748_row x == a027748_row (a003415 x)) [2..]
-- Reinhard Zumkeller, Jan 16 2013

with(numtheory);
A201009:=proc(q)
local a,b,k,n;
for n from 1 to q do
  a:=ifactors(n)[2]; b:=ifactors(n*add(op(2,p)/op(1,p),p=ifactors(n)[2]))[2];
  if nops(a)=nops(b) then
    if product(a[k][1],k=1..nops(a))=product(b[k][1],k=1..nops(a)) then print(n);
fi; fi; od; end:
A201009(100000); # Paolo P. Lava, Jan 09 2013

from sympy import primefactors, factorint
A201009 = [n for n in range(1,10**5) if primefactors(n) == primefactors(sum([int(n*e/p) for p,e in factorint(n).items()]) if n > 1 else 0)] # Chai Wah Wu, Aug 21 2014

A371307 The smallest nonpalindromic number in base n that shares the same prime divisors as its digit reversal in base n.

Original entry on oeis.org

135, 32, 8, 8, 245, 12, 16, 16, 1089, 15, 72, 24, 468, 28, 32, 32, 108, 24, 48, 40, 98, 44, 144, 39, 1800, 52, 392, 35, 27869, 60, 64, 64, 45, 68, 216, 72, 162, 76, 400, 48, 75809, 48, 968, 88, 4590, 92, 288, 60, 238, 100, 1352, 104, 242, 63, 120, 112, 143370, 72, 1800, 120, 640, 124, 105

Offset: 2

Scott R. Shannon, May 03 2024

The largest value in the first 500 terms is a(478) = 109443357.

			a(2) = 135 as 135 = 3^3 * 5 = 10000111_2 whose digit reversal is 11100001_2 = 225 = 3^2 * 5^2, both of which have 3 and 5 as prime divisors.
a(10) = 1089 as 1089 = 3^2 * 11^2 whose digit reversal is 9801 = 3^4 * 11^2, both of which have 3 and 11 as prime divisors. See also A110819.
a(14) = 468 as 468 = 2^2 * 3^2 * 13 = 256_14 whose digit reversal is 652_14 = 1248 = 2^5 * 3 * 13, both of which have 2, 3, and 13 as prime divisors.

Scott R. Shannon, Table of n, a(n) for n = 2..500

Cf. A110819, A372488, A372384, A027746.

cp's OEIS Frontend

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

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A110751 Numbers n such that n and its digital reversal have the same prime divisors.

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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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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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A224252 Nonpalindromic n such that the factorizations of n and its digital reverse differ only for the exponents order.

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A113548 Least non-palindromic number k such that k and its digital reversal both have exactly n prime divisors.

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A201009 Numbers m such that the set of distinct prime divisors of m is equal to the set of distinct prime divisors of the arithmetic derivative m'.