A110751 -id:A110751 - cp's OEIS frontend

A110819 Non-palindromes in A110751; that is, non-palindromic numbers n such that n and R(n) have the same prime divisors, where R(n) = digit reversal of n.

1089, 2178, 4356, 6534, 8712, 9801, 10989, 21978, 24024, 26208, 42042, 43956, 48048, 61248, 65934, 80262, 84084, 84216, 87912, 98901, 109989, 219978, 231504, 234234, 242424, 253344, 255528, 264264, 272646, 275184, 277816, 288288, 405132, 424242, 432432, 439956

Offset: 1

Ryan Propper, Sep 15 2005

Trivially, if integer k is a term of this sequence, then R(k) is a term as well.

If n is in the sequence, then so is (10^m+1)*n where 10^m > n. In particular, the sequence is infinite. - Robert Israel, Aug 14 2014

			The prime divisors of 87912 and R(87912) = 21978 are both {2, 3, 11, 37}, so 87912 and 21978 are both in the sequence.

Donovan Johnson, Table of n, a(n) for n = 1..500

Cf. A110751.

revdigs:= proc(n)
local L,nL,i;
L:= convert(n,base,10);
nL:= nops(L);
add(L[i]*10^(nL-i),i=1..nL);
end:
filter:= proc(n) local r;
  r:= revdigs(n);
  r <> n and numtheory:-factorset(r) = numtheory:-factorset(n)
end proc:
select(filter, [$10 .. 10^6]); # Robert Israel, Aug 14 2014

r[n_] := FromDigits[Reverse[IntegerDigits[n]]]; Do[If[r[n] != n && Select[Divisors[n], PrimeQ] == Select[Divisors[r[n]], PrimeQ], Print[n]], {n, 1, 10^6}]

from sympy import primefactors
A110819 = [n for n in range(1,10**6) if str(n) != str(n)[::-1] and primefactors(n) == primefactors(int(str(n)[::-1]))] # Chai Wah Wu, Aug 14 2014

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

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

A227858 Numbers which start and end with the same digit in decimal.

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, 505, 515, 525, 535, 545, 555, 565, 575, 585, 595, 606, 616, 626

Offset: 1

Brian E. Foose and Todd R. Haldeman, Nov 01 2013

Supersequence of A002113 and A208259.
The first term that is not a palindromic number is a(109) = 1011. - Alonso del Arte, Nov 01 2013
Primes in this sequence are A077652. - Jonathan Vos Post, Nov 02 2013
Differs from A110751 which contains for example 1089. - R. J. Mathar, Jun 29 2025

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A002113, A077652, A208259.

Select[Range[700],IntegerDigits[#][[1]]==IntegerDigits[#][[-1]]&] (* Harvey P. Dale, Nov 28 2019 *)

is(n)=n=digits(n);n[1]==n[#n] \\ Charles R Greathouse IV, Nov 01 2013

a(n) = 10n + O(1). - Charles R Greathouse IV, Nov 01 2013

A110752 a(n) is the number of divisors of the concatenation of 1089 with itself n times.

Original entry on oeis.org

9, 36, 192, 144, 288, 1536, 576, 4608, 3840, 4608, 3072, 12288, 1152, 9216, 1572864, 73728, 2304, 245760, 144, 73728, 4718592, 49152, 2304, 6291456, 294912, 18432, 589824, 294912, 18432, 100663296

Offset: 1

Amarnath Murthy, Aug 11 2005

1089 has the property that any number of concatenations of it with itself and its digit reversal have the same set of distinct prime factors.

			a(2) = tau(10891089) = 36.

Cf. A110751, A110753.

k = 0; Do[k = 10^4 * k + 1089; Print[DivisorSigma[0, k]], {n, 1, 30}] (* Ryan Propper, Oct 15 2005 *)

a(19)-a(30) from Ryan Propper, Oct 15 2005

Name edited by Jon E. Schoenfield, Nov 19 2018

A110753 a(n) is the number of divisors of the concatenation of 2178 with itself n times.

Original entry on oeis.org

18, 72, 384, 288, 576, 3072, 1152, 9216, 7680, 9216, 6144, 24576, 2304, 18432, 3145728, 147456, 4608, 491520, 288, 147456, 9437184, 98304, 4608, 12582912, 589824, 36864, 1179648, 589824, 36864, 201326592

Offset: 1

Amarnath Murthy, Aug 11 2005

2178 has the property that any number of concatenations of it with itself and its digit reversal have the same set of distinct prime factors.

			a(2) = tau(21782178) = 72.

Cf. A110751, A110752.

k = 0; Do[k = (10^4 * k) + 2178; Print[DivisorSigma[0, k]], {n, 1, 30}] (* Ryan Propper, Aug 28 2005 *)
Table[DivisorSigma[0,FromDigits[PadRight[{},4n,{2,1,7,8}]]],{n,30}] (* Harvey P. Dale, Jun 23 2014 *)

More terms from Ryan Propper, Aug 28 2005

A266140 Palindromes such that removing at most one digit will result in a term in A110784.

Original entry on oeis.org

0, 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

Offset: 1

Chai Wah Wu, Dec 25 2015

Union of A266139 and A110784. Every palindrome p can have its digits permuted to produce a term m <= p in this sequence, and this sequence is the minimal such sequence (i.e., no term in the sequence can have its digits permuted to form another term in the sequence). Palindromes modulo permutation of digits.
a(n) = A110751(n-1) for 2<=n<=109, which means A110751 contains numbers like 1001 and 1089 which are absent here. - R. J. Mathar, Aug 07 2025
The first entry of A002113 not in this sequence here is 1001. - R. J. Mathar, Aug 07 2025

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A110784, A266139.

A342826 Numbers k such that d(1)^0 + d(2)^1 + ... + d(p)^(p-1) = d(1)^(p-1) + d(2)^(p-2) + ... + d(p)^0, where d(i), i=1..p, are the digits of k.

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

Offset: 1

Carole Dubois, Mar 23 2021

This sequence starts off like other palindromic sequences such as A178354, A002113, A110751, and A227858 but it differs starting at the 110th term: 109th: 1001, 110th: 1011, 111th: 1101, ..., 119th: 1863, etc.
Differs from A297271 (which e.g. has 1021, 1031, 1041,.. 1091 which are absent here). - R. J. Mathar, Sep 27 2021
Contains the palindromes, and palindromes where pairs of digits have been substituted by d(i)=0, d(p-i)=1 or d(i)=1, d(p-1)=0, and "genuine" numbers like 1863, 2450, 2804, 2814, 3681, 4081, 4182, 103221, 113221, 122301, 122311, 142023,.. - R. J. Mathar, Sep 27 2021

			1863 is in this sequence because 1^0 + 8^1 + 6^2 + 3^3 = 1^3 + 8^2 + 6^1 + 3^0 = 72.

R. J. Mathar, Table of n, a(n) for n = 1..1137 (replacing older b-file which did not contain a(101))
Carole Dubois, Scatterplot of terms of A178354, A342826 vs Sum of powers in definition.

Cf. A002113 (subset), A179309, A110751, A227858.

isA342826 := proc(n)
    local dgs ;
    dgs := convert(n,base,10) ;
    if  add(op(i,dgs)^(i-1),i=1..nops(dgs)) = add(op(-i,dgs)^(i-1),i=1..nops(dgs)) then
        true;
    else
        false;
    end if;
end proc:
A342826 := proc(n)
    option remember ;
    if n =1 then
        1;
    else
        for a from procname(n-1)+ 1 do
            if isA342826(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Sep 27 2021

Select[Range[500],Mod[#,10]!=0&&Total[IntegerDigits[#]^Range[0,IntegerLength[ #]-1]]==Total[IntegerDigits[#]^Range[IntegerLength[#]-1,0,-1]]&] (* Harvey P. Dale, Jan 18 2023 *)

def digpow(s): return sum(int(d)**i for i, d in enumerate(s))
def aupto(limit):
  alst = []
  for k in range(1, limit+1):
    s = str(k)
    if digpow(s) == digpow(s[::-1]): alst.append(k)
  return alst
print(aupto(464)) # Michael S. Branicky, Mar 23 2021

A110754 a(n) = tau(N), where N = the number obtained as a concatenation of 8712 with itself n times and tau(n) = number of divisors of n.

Original entry on oeis.org

36, 144, 768, 576, 1152, 6144, 2304, 18432, 15360, 18432, 12288, 49152, 4608, 36864, 6291456, 294912, 9216, 983040, 576, 294912, 18874368, 196608, 9216, 25165824, 1179648, 73728, 2359296, 1179648, 73728, 402653184, 2304, 2359296, 33554432, 147456, 75497472, 31457280, 147456, 36864

Offset: 1

Amarnath Murthy, Aug 11 2005

8712 has the property that any number of concatenation of it with self and the digit reversal have same prime divisors.

			a(2) = tau(87128712) = 144.

Cf. A110751, A110752, A110753.

A110754 := proc(n) local pow8712,i ; pow8712 := 8712*add(10^(4*i),i=0..n-1) ; numtheory[tau](pow8712) ; end: seq(A110754(n),n=1..22) ; # R. J. Mathar, Aug 17 2007

Table[DivisorSigma[0,FromDigits[PadRight[{},4n,{8,7,1,2}]]],{n,25}] (* Harvey P. Dale, Dec 29 2016 *)

a(n)={numdiv(8712*(10^(4*n)-1)/9999)} \\ Andrew Howroyd, Nov 09 2019

a(n) = A000005(8712*Sum_{i=0..n-1} 10^(4i)). - R. J. Mathar, Aug 17 2007

More terms from R. J. Mathar, Aug 17 2007

a(23)-a(38) from Andrew Howroyd, Nov 09 2019

cp's OEIS Frontend

A110819 Non-palindromes in A110751; that is, non-palindromic numbers n such that n and R(n) have the same prime divisors, where R(n) = digit reversal of n.

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

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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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A227858 Numbers which start and end with the same digit in decimal.

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A110752 a(n) is the number of divisors of the concatenation of 1089 with itself n times.

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A110753 a(n) is the number of divisors of the concatenation of 2178 with itself n times.

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