A062895 -id:A062895 - cp's OEIS frontend

A085869 Numbers n such that n and its digit reversal have the same prime signature.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 22, 26, 31, 33, 37, 39, 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, 181

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jul 06 2003

			15 is a member 15 = 3*5 and 51 = 3*17 both have the prime signature p*q, p and q are primes.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A062895.

s:= n-> sort(map(i-> i[2], ifactors(n)[2])):
a:= proc(n) option remember; local k; for k from 1+a(n-1) while
      s(k)<>s((p-> parse(cat(seq(p[-i], i=1..length(p)))))(""||k)) do od; k
    end: a(0):=0:
seq(a(n), n=1..80);  # Alois P. Heinz, Mar 09 2018

Corrected and extended by Ray Chandler, Aug 08 2003

A350867 Non-palindromic 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.

Original entry on oeis.org

13, 15, 17, 24, 26, 31, 37, 39, 42, 51, 58, 62, 71, 73, 79, 85, 93, 97, 107, 113, 115, 117, 122, 123, 129, 143, 149, 155, 157, 158, 159, 165, 167, 169, 177, 178, 179, 183, 185, 187, 199, 203, 205, 221, 226, 246, 264, 265, 285, 286, 288, 294, 302, 311, 314, 319

Offset: 1

Daniel Tsai, Feb 18 2022

			264 and 462 are non-palindromic and also d(264) = 16 = d(462), and so both are members.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A000005 (d), A004086 (R).

Intersection of A029742 (non-palindromes) and A062895 (d(R(k)) = d(k)).

isok(k) = my(R = fromdigits(Vecrev(digits(k)))); R != k && numdiv(R) == numdiv(k);

from sympy import divisor_count as d
def ok(k): Rk = int(str(k)[::-1]); return Rk != k and d(k) == d(Rk)
print([k for k in range(320) if ok(k)]) # Michael S. Branicky, Feb 20 2022

A346113 Base-10 numbers k whose number of divisors equals the number of divisors in R(k), where k is written in all bases from base-2 to base-10 and R(k), the digit reversal of k, is read as a number in the same base.

Original entry on oeis.org

1, 9077, 10523, 10838, 30182, 58529, 73273, 77879, 83893, 244022, 303253, 303449, 304853, 329893, 332249, 334001, 334417, 335939, 336083, 346741, 374617, 391187, 504199, 512695, 516982, 595274, 680354, 687142, 758077, 780391, 792214, 854669, 946217, 948539, 995761, 1008487, 1377067, 1389341

Offset: 1

Scott R. Shannon, Jul 05 2021

There are 633 terms below 50 million and 1253 terms below 100 million. All of those have tau(k), the number of divisors of k, equal to 1, 2, 4, 8 or 16. The first term where tau(k) = 2 is n = 93836531, a prime, which is also the first term of A136634. All terms in A136634 will appear in this sequence, as will all terms in A228768(n) for n>=10. The first term with tau(k) = 4 is 9077, the first with tau(k) = 8 is 595274, and the first with tau(k) = 16 is 5170182. It is possible tau(k) must equal 2^i, with i>=0, although this is unknown.

All known terms are squarefree. - Michel Marcus, Jul 07 2021

			9077 is a term as the number of divisors of 9077 = tau(9077) = 4, and this equals the number of divisors of R(9077) when written and then read as a base-j number, with 2 <= j <= 10. See the table below for k = 9077.
.
  base | k_base         | R(k_base)      | R(k_base)_10  | tau(R(k_base)_10)
----------------------------------------------------------------------------------
   2   | 10001101110101 | 10101110110001 | 11185         | 4
   3   | 110110012      | 210011011      | 15421         | 4
   4   | 2031311        | 1131302        | 6002          | 4
   5   | 242302         | 203242         | 6697          | 4
   6   | 110005         | 500011         | 38887         | 4
   7   | 35315          | 51353          | 12533         | 4
   8   | 21565          | 56512          | 23882         | 4
   9   | 13405          | 50431          | 33157         | 4
  10   | 9077           | 7709           | 7709          | 4

Scott R. Shannon, Table of n, a(n) for n = 1..1253

Cf. A000005, A004086.
Cf. A030102, A030103, A030104, A030105, A030106, A030107, A030108.
Cf. A136634 (prime terms), A228768.
Subsequence of A062895.

Select[Range@100000,Length@Union@DivisorSigma[0,Join[{s=#},FromDigits[Reverse@IntegerDigits[s,#],#]&/@Range[2,10]]]==1&] (* Giorgos Kalogeropoulos, Jul 06 2021 *)

isok(k) = {my(t= numdiv(k)); for (b=2, 10, my(d=digits(k, b)); if (numdiv(fromdigits(Vecrev(d), b)) != t, return (0));); return(1);} \\ Michel Marcus, Jul 06 2021

A087093 Numbers that have the same number of divisors as their digit reversal, but with different prime signatures.

Original entry on oeis.org

24, 42, 264, 288, 462, 658, 856, 882, 1071, 1128, 1314, 1464, 1701, 2058, 2130, 2132, 2312, 2324, 2424, 2510, 2590, 2616, 2664, 2744, 2765, 2782, 2786, 2872, 2904, 2938, 2975, 3159, 3194, 4010, 4090, 4092, 4094, 4125, 4131, 4136, 4168, 4184, 4220, 4228

Offset: 1

Ray Chandler, Aug 09 2003

Terms in A062895 but not in A085869.

			24 and 42 are members as 24 and 42 each have 8 divisors but with different prime signatures.

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

Cf. A000005, A062895, A085869, A085870.

psQ[n_]:=Sort[Transpose[FactorInteger[n]][[2]]]!=Sort[Transpose[ FactorInteger[FromDigits[Reverse[IntegerDigits[n]]]]][[2]]];ndQ[n_]:= DivisorSigma[0,n]==DivisorSigma[0,FromDigits[Reverse[ IntegerDigits[ n]]]]; Select[Range[5000],psQ[#]&&ndQ[#]&] (* Harvey P. Dale, Nov 11 2011 *)

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

A354745 Non-repdigit numbers k such that every permutation of the digits of k has the same number of divisors.

Original entry on oeis.org

13, 15, 17, 24, 26, 31, 37, 39, 42, 51, 58, 62, 71, 73, 79, 85, 93, 97, 113, 117, 131, 155, 171, 177, 178, 187, 199, 226, 262, 288, 311, 337, 339, 355, 373, 393, 515, 535, 551, 553, 558, 585, 622, 711, 717, 718, 733, 771, 781, 817, 828, 855, 871, 882, 899, 919, 933, 989, 991, 998

Offset: 1

Metin Sariyar, Jun 05 2022

After a(93) = 84444, no further terms < 10^18. - Michael S. Branicky, Jun 08 2022

			871 is a term because d(871) = d(817) = d(178) = d(187) = d(718) = d(781) = 4, where d(n) is the number of divisors of n.

Cf. A000005, A003459, A067012, A062895, A350867, A354746.

Select[Range[10000],CountDistinct[DivisorSigma[0,FromDigits /@ Permutations[IntegerDigits[#]]]]==1&&CountDistinct[IntegerDigits[#]]>1&]

from sympy import divisor_count
from itertools import permutations
def ok(n):
    s, d = str(n), divisor_count(n)
    if len(set(s)) == 1: return False
    return all(d==divisor_count(int("".join(p))) for p in permutations(s))
print([k for k in range(5500) if ok(k)]) # Michael S. Branicky, Jun 05 2022

A346141 Numbers k whose number of divisors equals the number of divisors in each of R(k), k+R(k), R(k+R(k)), abs(k-R(k)), and R(abs(k-R(k))), where R(m) is the digit reversal of m and where the reversals of m do not equal m.

Original entry on oeis.org

117858, 129138, 137976, 138222, 194838, 201569, 222831, 281256, 302844, 439415, 448203, 454016, 500638, 514934, 516378, 526486, 533938, 552926, 560766, 562936, 595016, 607499, 607597, 610454, 610595, 629255, 639265, 652182, 654018, 659358, 667065, 679731, 684625, 795706, 810456, 813179

Offset: 1

Scott R. Shannon, Jul 05 2021

There are 324 terms below 50 million. In that range the number of divisors of the terms is 8,12,16,24,32 or 48. The first term with 8 divisors is 129138, the first with 12 is 302844, the first with 16 is 117858, the first with 24 is 138222, the first with 32 is 26739192, and the first with 48 is 19245366.

			117858 is a term as the number of divisors of 117858 = tau(117858) = 16, and this equals tau(R(117858)) = tau(858711) = 16, tau(117858+R(117858)) = tau(976569) = 16, tau(R(117858+R(117858))) = tau(965679) = 16, tau(abs(117858-R(117858))) = tau(740853) = 16, and tau(R(abs(117858-R(117858)))) = tau(358047) = 16.

Cf. A000005, A004086, A346113, A056964, A056965.

Subsequence of A062895.

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A085869 Numbers n such that n and its digit reversal have the same prime signature.

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A350867 Non-palindromic 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.

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A346113 Base-10 numbers k whose number of divisors equals the number of divisors in R(k), where k is written in all bases from base-2 to base-10 and R(k), the digit reversal of k, is read as a number in the same base.

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A087093 Numbers that have the same number of divisors as their digit reversal, but with different prime signatures.

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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.

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A354745 Non-repdigit numbers k such that every permutation of the digits of k has the same number of divisors.

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A346141 Numbers k whose number of divisors equals the number of divisors in each of R(k), k+R(k), R(k+R(k)), abs(k-R(k)), and R(abs(k-R(k))), where R(m) is the digit reversal of m and where the reversals of m do not equal m.

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