A085329 -id:A085329 - cp's OEIS frontend

A062895 Numbers k for which d(k) = d(R(k)), where R(k) is the reversal of k and d(k) is the number of divisors of k.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 22, 24, 26, 31, 33, 37, 39, 42, 44, 51, 55, 58, 62, 66, 71, 73, 77, 79, 85, 88, 93, 97, 99, 101, 107, 111, 113, 115, 117, 121, 122, 123, 129, 131, 141, 143, 149, 151, 155, 157, 158, 159, 161, 165, 167, 169, 171, 177, 178, 179

Offset: 1

Amarnath Murthy, Jun 30 2001

The sequence s of numbers k for which R(d(k)) = d(R(k)) first differs at s(80) = 262 while a(80) = 252. - Mohammed Yaseen, Mar 24 2023

			d(24) = 8 and also d(42) = 8, hence both are members.

Mohammed Yaseen, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

Cf. A000005 (d), A004086 (R), A002113 (palindromes: subsequence).

Cf. A350867 (subsequence of non-palindromic terms), A085329 (similar with sigma).

Select[Range[180],DivisorSigma[0,#]==DivisorSigma[0,FromDigits[Reverse[IntegerDigits[#]]]] &] (* Jayanta Basu, May 17 2013 *)

{ n=0; for (m=1, 10^9, x=m; r=0; while (x>0, d=x-10*(x\10); x\=10; r=r*10 + d); if (numdiv(m) == numdiv(r), write("b062895.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 12 2009

isok(k) = numdiv(fromdigits(Vecrev(digits(k)))) == numdiv(k); \\ Michel Marcus, Jul 06 2021

from sympy import divisor_count as d
def ok(n): return d(n) == d(int(str(n)[::-1]))
print([k for k in range(1, 180) if ok(k)]) # Michael S. Branicky, Mar 24 2023

Corrected and extended by Vladeta Jovovic, Jun 30 2001

A097647 Non-palindromic numbers n such that phi(n) = phi(reversal(n)).

Original entry on oeis.org

190, 427, 429, 724, 924, 4147, 4697, 6276, 6726, 7414, 7964, 9079, 9709, 10040, 10940, 14450, 15860, 19190, 20493, 20553, 28092, 28215, 29082, 35502, 39402, 41847, 42777, 44629, 46899, 49929, 51282, 51845, 53075, 54815, 57035, 57677

Offset: 1

Farideh Firoozbakht, Aug 28 2004

If n is in the sequence and 10 doesn't divide n then reversal(n) is also in the sequence. There exists three terms of this sequence less than 180000000 that reversal of them are primes,i.e. 10040,14450 and 1865170. This sequence has 445 composite terms less than 20000000 and there is no prime term up to 222000000. Has this sequence at least one prime term?
(190/99)*(100^m-1) is in the sequence iff 3 does not divide m (m is a term of A001651). So the sequence is infinite. A229903: 190, 19190, 191919190, 19191919190, ...  are such terms. - Jahangeer Kholdi, Oct 17 2013
There are no prime terms < 10^10. - Donovan Johnson, Oct 18 2013

			10040 is in the sequence because phi(10040)=phi(4001)=4000.

Cf. A097648, A085329.

Cf. A001651, A229903.

Do[If[n!=FromDigits[Reverse[IntegerDigits[n]]]&&EulerPhi[n]==EulerPhi[ FromDigits[Reverse[IntegerDigits[n]]]], Print[n]], {n, 80000}]

A281879 Non-palindromic numbers k such that sigma(k) | sigma(R(k)), where R(k) is the digit reversal of k.

Original entry on oeis.org

15, 16, 17, 59, 129, 165, 176, 187, 205, 273, 276, 299, 429, 446, 478, 528, 599, 825, 1034, 1043, 1135, 1209, 1239, 1515, 1561, 1565, 1616, 1651, 1665, 1717, 1776, 1887, 2086, 2165, 2178, 2255, 2455, 2515, 2618, 2739, 2829, 3489, 4008, 4064, 4475, 4604, 5346, 5795

Offset: 1

Paolo P. Lava, Feb 01 2017

			a(1) = 15 because sigma(51) / sigma(15) = 72 / 24 = 3;
a(2) = 16 because sigma(61) / sigma(16) = 62 / 31 = 2;
a(3) = 17 because sigma(71) / sigma(17) = 72 / 18 = 4.

Harvey P. Dale, Table of n, a(n) for n = 1..500
Paolo P. Lava, First 250 terms with the ratio sigma(R(k))/sigma(k)

Cf. A000203, A004086, A085329, A281880.

with(numtheory): T:=proc(w) local x, y, z; x:=w; y:=0;
for z from 1 to ilog10(x)+1 do y:=10*y+(x mod 10); x:=trunc(x/10); od; y; end:
P:=proc(q) local n; for n from 1 to q do
if n<>T(n) then if type(sigma(T(n))/sigma(n),integer) then print(n); fi; fi; od; end: P(10^6);

Select[Range[6000],!PalindromeQ[#]&&Mod[DivisorSigma[1,IntegerReverse[#]],DivisorSigma[ 1,#]] ==0&] (* Harvey P. Dale, Dec 19 2023 *)

A280354 Numbers n such that (i) number of divisors of n equals number of divisors of digit reversal of n, (ii) sum of divisors of n equals sum of divisors of digit reversal of n, and (iii) n is not a palindrome.

Original entry on oeis.org

1561, 1651, 5346, 6435, 157661, 166751, 301134, 321853, 358123, 431103, 507955, 511665, 517055, 537495, 539946, 550715, 559705, 566115, 576908, 594735, 649935, 729287, 765677, 776567, 782927, 809675, 834498, 894438, 896898, 898698, 905289, 982509, 1257912, 1473302

Offset: 1

Ilya Gutkovskiy, Jan 01 2017

Intersection of A062895 and A085329.

Numbers n such that A000005(n) = A000005(A004086(n)), A000203(n) = A000203(A004086(n)) and A136522(n) = 0.

			1561 is in the sequence because 1561 has 4 divisors {1, 7, 223, 1561}, 1 + 7 + 223 + 1561 = 1792 and 1651 has 4 divisors {1, 13, 127, 1651}, 1 + 13 + 127 + 1651 = 1792.

Indranil Ghosh, Table of n, a(n) for n = 1..200
Index entries for sequences related to sums of divisors

Cf. A000005, A000203, A004086, A029742, A062895, A085329, A136522.

Select[Range[1500000], !PalindromeQ[#1] && DivisorSigma[0, #1] == DivisorSigma[0, FromDigits[Reverse[IntegerDigits[#1]]]] && DivisorSigma[1, #1] == DivisorSigma[1,FromDigits[Reverse[IntegerDigits[#1]]]] & ]
fQ[n_]:=With[{irn=IntegerReverse[n]},!PalindromeQ[n]&&DivisorSigma[0,n]==DivisorSigma[0,irn] && DivisorSigma[1,n] == DivisorSigma[ 1,irn]]; Select[Range[1480000],fQ] (* Harvey P. Dale, Dec 17 2024 *)

R(n) = eval(concat(Vecrev(Str(n))));
isok(n) = n != R(n) && numdiv(n) == numdiv(R(n)) && sigma(n) == sigma(R(n));
for(n=1561, 1473302, if(isok(n), print1(n, ", "))) \\ Indranil Ghosh, Mar 06 2017

A259077 Non-palindromic composite numbers such that n' = [Rev(n)]', where n' is the arithmetic derivative of n.

Original entry on oeis.org

366, 663, 3245, 3685, 5423, 5863, 8178, 8718, 14269, 15167, 16237, 18449, 18977, 36679, 73261, 76151, 77981, 94481, 96241, 97663, 140941, 149041, 150251, 152051, 196891, 198691, 302363, 308459, 319853, 335148, 358913, 363203, 841533, 921239, 932129, 954803, 958099, 990859

Offset: 1

Paolo P. Lava, Jun 18 2015

			366' = 311 = 663';
3245' = 999 = 5423'; etc.

Cf. A003415, A004086, A085329, A097647.

with(numtheory): T:=proc(w) local x,y,z; x:=w; y:=0;
for z from 1 to ilog10(x)+1 do y:=10*y+(x mod 10); x:=trunc(x/10);
od; y; end: P:=proc(q) local a,b,p,n;
for n from 1 to q do if not isprime(n) then if n<>T(n) then a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
b:=T(n)*add(op(2,p)/op(1,p),p=ifactors(T(n))[2]);
if a=b then print(n); fi; fi; fi; od; end: P(10^9);

Solutions to A003415(n) = A003415(A004086(n)), with A004086(n) <> n.

cp's OEIS Frontend

A062895 Numbers k for which d(k) = d(R(k)), where R(k) is the reversal of k and d(k) is the number of divisors of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Extensions

A097647 Non-palindromic numbers n such that phi(n) = phi(reversal(n)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A281879 Non-palindromic numbers k such that sigma(k) | sigma(R(k)), where R(k) is the digit reversal of k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A280354 Numbers n such that (i) number of divisors of n equals number of divisors of digit reversal of n, (ii) sum of divisors of n equals sum of divisors of digit reversal of n, and (iii) n is not a palindrome.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A259077 Non-palindromic composite numbers such that n' = [Rev(n)]', where n' is the arithmetic derivative of n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Formula