A087990 -id:A087990 - cp's OEIS frontend

A357171 a(n) is the number of divisors of n whose digits are in strictly increasing order (A009993).

1, 2, 2, 3, 2, 4, 2, 4, 3, 3, 1, 6, 2, 4, 4, 5, 2, 6, 2, 4, 3, 2, 2, 8, 3, 4, 4, 6, 2, 6, 1, 5, 2, 4, 4, 9, 2, 4, 4, 5, 1, 6, 1, 3, 6, 4, 2, 10, 3, 4, 3, 5, 1, 7, 2, 8, 4, 4, 2, 8, 1, 2, 4, 5, 3, 4, 2, 6, 4, 6, 1, 11, 1, 3, 5, 5, 2, 8, 2, 6, 4, 2, 1, 9, 3, 2, 3, 4, 2, 9, 3, 5, 2, 3, 3, 10, 1, 5, 3, 5

Offset: 1

Bernard Schott, Sep 16 2022

As A009993 is finite with 512 terms, a(n) is bounded with a(n) <= 511 and not 512, since A009993(1) = 0.

			22 has 4 divisors {1, 2, 11, 22} of which two have decimal digits that are not in strictly increasing order: {11, 22}, hence a(22) = 4-2 = 2.
52 has divisors {1, 2, 4, 13, 26, 52} and a(52) = 5 of them have decimal digits that are in strictly increasing order (all except 52 itself).

Cf. A009993, A357172, A357173, A160218.

Similar: A087990 (palindromic), A355302 (undulating), A355593 (alternating).

f:= proc(n) local d,L,i,t;
  t:= 0;
  for d in numtheory:-divisors(n) do
    L:= convert(d,base,10);
    if `and`(seq(L[i]>L[i+1],i=1..nops(L)-1)) then t:= t+1 fi
  od;
  t
end proc:
map(f, [$1..100]); # Robert Israel, Sep 16 2022

a[n_] := DivisorSum[n, 1 &, Less @@ IntegerDigits[#] &]; Array[a, 100] (* Amiram Eldar, Sep 16 2022 *)

isok(d) = Set(d=digits(d)) == d; \\ A009993
a(n) = sumdiv(n, d, isok(d)); \\ Michel Marcus, Sep 16 2022

from sympy import divisors
def c(n): s = str(n); return s == "".join(sorted(set(s)))
def a(n): return sum(1 for d in divisors(n, generator=True) if c(d))
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Sep 16 2022

G.f.: Sum_{n in A009993} x^n/(1-x^n). - Robert Israel, Sep 16 2022

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n=2..512} 1/A009993(n) = 4.47614714667538759358... (this is a rational number whose numerator and denominator have 1037 and 1036 digits, respectively). - Amiram Eldar, Jan 06 2024

A358099 a(n) is the number of divisors of n whose digits are in strictly decreasing order (A009995).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 1, 5, 1, 3, 3, 4, 1, 5, 1, 6, 4, 2, 1, 6, 2, 2, 3, 4, 1, 7, 2, 5, 2, 2, 3, 6, 1, 2, 2, 8, 2, 7, 2, 3, 4, 2, 1, 6, 2, 5, 3, 4, 2, 6, 2, 5, 2, 2, 1, 10, 2, 4, 6, 6, 3, 4, 1, 3, 2, 6, 2, 8, 2, 3, 4, 4, 2, 4, 1, 9, 4, 4, 2, 9, 3, 4, 3, 4, 1, 9, 3, 4, 4, 3, 3, 8, 2, 4, 3, 7

Offset: 1

Bernard Schott, Oct 29 2022

As A009995 is finite with 1023 terms, a(n) is bounded with a(n) <= 1022 and not 1023, since A009995(1) = 0.

			22 has 4 divisors {1, 2, 11, 22} of which two have decimal digits that are not in strictly decreasing order: {11, 22}, hence a(22) = 4-2 = 2.
52 has 6 divisors {1, 2, 4, 13, 26, 52} of which four have decimal digits that are in strictly decreasing order {1, 2, 4, 52}, hence a(52) = 4.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A009995, A190219, A358100, A358101.

Similar: A086971 (semiprimes), A087990 (palindromic), A355593 (alternating), A357171 (increasing order).

f:= proc(n) local L;
   if n < 10 then return true fi;
   L:= convert(n,base,10);
   andmap(type,L[2..-1]-L[1..-2],positive)
end proc:
g:= n -> nops(select(f,numtheory:-divisors(n))):
map(g, [$1..100]); # Robert Israel, Oct 31 2022

a[n_] := DivisorSum[n, 1 &, Max @ Differences @ IntegerDigits[#] < 0 &]; Array[a, 100] (* Amiram Eldar, Oct 29 2022 *)

a(n) = sumdiv(n, d, my(dd=digits(d)); vecsort(dd, ,12) == dd); \\ Michel Marcus, Oct 30 2022

from sympy import divisors
def c(n): s = str(n); return all(s[i+1] < s[i] for i in range(len(s)-1))
def a(n): return sum(1 for d in divisors(n, generator=True) if c(d))
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Feb 12 2024

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n=2..1023} 1/A009995(n) = 3.89840673699905364734... (this is a rational number whose numerator and denominator have 1292 and 1291 digits, respectively). - Amiram Eldar, Jan 06 2024

A179937 a(n) is the product of the non-palindromic divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 12, 13, 14, 15, 16, 17, 18, 19, 200, 21, 1, 23, 288, 25, 338, 27, 392, 29, 4500, 31, 512, 1, 578, 35, 7776, 37, 722, 507, 8000, 41, 12348, 43, 1, 675, 1058, 47, 221184, 49, 12500, 867, 17576, 53, 26244, 1, 21952, 1083, 1682, 59

Offset: 1

Jaroslav Krizek, Jan 12 2011

			For n = 20, set of non-palindromic divisors is {10, 20}; a(12) = 10*20 = 200.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A087990, A087991, A088000, A088001, A184392.

Table[Times@@Select[Divisors[n],!PalindromeQ[#]&],{n,60}] (* Harvey P. Dale, May 15 2023 *)

def ispal(n):
    return n==int(str(n)[::-1])
def A179937(n):
    s=1
    for i in range(1, n+1):
        if n%i==0 and not ispal(i):
            s*=i
    return s # Indranil Ghosh, Feb 10 2017

a(n) = A007955(n) / A184392(n).

More terms from Indranil Ghosh, Feb 10 2017

A184392 a(n) is the product of palindromic divisors of n.

Original entry on oeis.org

1, 2, 3, 8, 5, 36, 7, 64, 27, 10, 11, 144, 1, 14, 15, 64, 1, 324, 1, 40, 21, 484, 1, 1152, 5, 2, 27, 56, 1, 180, 1, 64, 1089, 2, 35, 1296, 1, 2, 3, 320, 1, 252, 1, 85184, 135, 2, 1, 1152, 7, 10, 3, 8, 1, 324, 3025, 448, 3, 2, 1, 720, 1, 2, 189, 64, 5, 18974736, 1, 8, 3, 70, 1, 10368, 1, 2, 15, 8, 5929, 36, 1, 320, 27, 2, 1, 1008, 5, 2, 3, 59969536, 1, 1620, 7, 8, 3, 2, 5, 1152, 1, 14, 970299, 40

Offset: 1

Jaroslav Krizek, Jan 12 2011

			For n = 20, set of palindromic divisors is {1, 2, 4, 5}; a(12) = 1*2*4*5 = 40.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A007955, A087990, A087991, A088000, A088001, A179937.

palQ[n_]:=Module[{idn=IntegerDigits[n]}, idn==Reverse[idn]]; f[n_]:=Times@@Select[Divisors[n],palQ]; Table[f[n],{n,100}]  (* Harvey P. Dale, Jan 21 2011 *)

def ispal(n):
    return n==int(str(n)[::-1])
def A184392(n):
    s=1
    for i in range(1, n+1):
        if n%i==0 and ispal(i):
             s*=i
    return s # Indranil Ghosh, Feb 10 2017

a(n) = A007955(n) / A179937(n).

More terms from Harvey P. Dale, Jan 21 2011

A355698 a(n) is the number of repdigits divisors of n (A010785).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 3, 2, 5, 1, 3, 3, 4, 1, 5, 1, 4, 3, 4, 1, 6, 2, 2, 3, 4, 1, 5, 1, 4, 4, 2, 3, 6, 1, 2, 2, 5, 1, 5, 1, 6, 4, 2, 1, 6, 2, 3, 2, 3, 1, 5, 4, 5, 2, 2, 1, 6, 1, 2, 4, 4, 2, 8, 1, 3, 2, 4, 1, 7, 1, 2, 3, 3, 4, 4, 1, 5, 3, 2, 1, 6, 2, 2, 2, 8, 1, 6, 2, 3, 2, 2, 2, 6, 1, 3, 6, 4, 1, 4, 1, 4, 4

Offset: 1

Bernard Schott, Jul 14 2022

More than the usual number of terms are displayed in order to show the difference from A087990.
The first 100 terms are the same first 100 terms of A087990, then a(101) = 1 while A087990(101) = 2, because 101 is the smallest palindrome that is not repdigit; the next difference is 121.
Inequalities: 1 <= a(n) <= A087990(n).

			66 has 8 divisors: {1, 2, 3, 6, 11, 22, 33, 66} that are all repdigits, hence a(66) = 8.
121 has 3 divisors: {1, 11, 121} of which 2 are repdigits: {1, 11}, hence a(121) = 2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A010785, A065444, A087990, A190217, A340548, A355699.

Similar sequences: A083230, A087990, A087991, A332268, A355302.

isrepdig:= proc(n) nops(convert(convert(n,base,10),set))=1 end proc:
f:= proc(n) nops(select(isrepdig, numtheory:-divisors(n))) end proc:
map(f, [$1..200]); # Robert Israel, Aug 07 2024

a[n_] := DivisorSum[n, 1 &, Length[Union[IntegerDigits[#]]] == 1 &]; Array[a, 100] (* Amiram Eldar, Jul 14 2022 *)

a(n) = my(ret=0,u=1); while(u<=n, ret+=sum(d=1,9, n%(u*d)==0); u=10*u+1); ret; \\ Kevin Ryde, Jul 14 2022

isrep(n) = {1==#Set(digits(n))}; \\ A010785
a(n) = sumdiv(n, d, isrep(d)); \\ Michel Marcus, Jul 15 2022

from sympy import divisors
def c(n): return len(set(str(n))) == 1
def a(n): return sum(1 for d in divisors(n, generator=True) if c(d))
print([a(n) for n in range(1, 105)]) # Michael S. Branicky, Jul 14 2022

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (7129/2520) * A065444 = 3.11446261209177581335... . - Amiram Eldar, Apr 17 2025

A334392 Numbers m such that the LCM of their palindromic divisors is neither 1 nor m.

Original entry on oeis.org

16, 25, 26, 27, 32, 34, 38, 39, 46, 48, 49, 50, 51, 52, 54, 57, 58, 62, 64, 65, 68, 69, 74, 75, 76, 78, 80, 81, 82, 85, 86, 87, 91, 92, 93, 94, 95, 96, 98, 100, 102, 104, 106, 108, 112, 114, 115, 116, 117, 118, 119, 122, 123, 124, 125, 128, 129, 130, 133, 134

Offset: 1

Bernard Schott, May 04 2020

A334139, A334391 and this sequence form a partition of the set of positive integers N* (A000027).

The integers {2^k, k >= 4, 2^k non-palindrome} form a subsequence whose first few terms are : 16, 32, 64, 128, ...

			50 has 3 palindromic divisors {1, 2, 5} then A087999(50) = 10 and 50 is a term.

Wikipedia, Partition of a set

Cf. A087990, A087999.

Cf. A334391 [LCM(palindromic divisors of m) = 1], A334139 [LCM(palindromic divisors of m) = m], this sequence [LCM(palindromic divisors of m) != 1 and != m].

Select[Range[125], !MemberQ[{1, #}, LCM @@ Select[Divisors[#], PalindromeQ]] &] (* Amiram Eldar, May 05 2020 *)

ispal(x) = my(d=digits(x)); d == Vecrev(d);
isok(m) = my(d=divisors(m), lcmpd = lcm(select(x->ispal(x), d))); (lcmpd != 1) && (lcmpd != m); \\ Michel Marcus, May 05 2020

A339624 Perfect powers p^k, k >= 2 of palindromic primes p when p^k is not a palindrome.

Original entry on oeis.org

16, 25, 27, 32, 49, 64, 81, 125, 128, 243, 256, 512, 625, 729, 1024, 2048, 2187, 2401, 3125, 4096, 6561, 8192, 15625, 16384, 16807, 17161, 19683, 22801, 32761, 32768, 36481, 59049, 65536, 78125, 97969, 117649, 124609, 131072, 139129, 146689, 161051, 177147, 262144

Offset: 1

Bernard Schott, Dec 10 2020

Equivalently: numbers m with only one prime factor such that the LCM of their palindromic divisors is neither 1 nor m: subsequence of A334392.
G. J. Simmons conjectured there are no palindromes of form n^k for k >= 5 (and n > 1) (see Simmons p. 98). According to this conjecture, these perfect powers are terms: {2^k, k>=4}, {3^k, k>=3}, {5^k, k>=2}, {7^k, k=2 and k>=4}, {11^k, k>=5}, {101^k, k>= 5}, {131^k, k>=2}, ...
From a(1) = 16 to a(17) = 2187, the data is the same as A056781(10) until A056781(26), then a(18) = 2401 and A056781(27) = 4096.

			5^2 = 25, 2^6 = 64, 3^4 = 81 are terms.
7^2 = 49 is a term, 7^3 = 343 is not a term, and 7^4 = 2401 is a term.
101^2 = 10201 and 11^4 = 14641 are not terms.

Murray S. Klamkin, Problems in applied mathematics: selections from SIAM review, (1990), p. 520.

Gustavus J. Simmons, Palindromic Powers, J. Rec. Math., 3 (No. 2, 1970), 93-98 [Annotated scanned copy].
Wikipedia, Palindromic number, Perfect Powers.

Intersection of A025475 and A334392.
Cf. A087990, A087999, A334139, A334391.
Subsequences: A000079 \ {1,2,4,8}, A000244 \ {1,3,9}, A000351 \ {1,5}, A000420 \ {1,7,343}, A001020 \ {1,11,121,1331,14641}, A096884 \ {1,101, 10201, 1030301, 104060401}.

q[n_] := Module[{f = FactorInteger[n]}, Length[f] == 1 && f[[1, 2]] > 1 && PalindromeQ[f[[1, 1]]]]; Select[Range[10^5], !PalindromeQ[#] && q[#] &] (* Amiram Eldar, Dec 10 2020 *)

ispal(n) = my(d=digits(n)); Vecrev(d)==d;
isok(k) = my(p); isprimepower(k, &p) && isprime(p) && ispal(p) &&!ispal(k); \\ Michel Marcus, Dec 10 2020

More terms from Amiram Eldar, Dec 10 2020

A348152 Hyperpalindions (palindromes in A093036).

Original entry on oeis.org

1, 2, 4, 6, 66, 2772

Offset: 1

Ivan N. Ianakiev, Oct 03 2021

Clifford A. Pickover, A passion for mathematics: numbers, puzzles, madness, religion, and the quest for reality, John Wiley & Sons, 2005, pages 107-108.

Intersection of A002113 and A093036.

Cf. A087990.

Select[Cases[Import["https://oeis.org/A093036/b093036.txt","Table"],{,}][[All,2]],PalindromeQ]

cp's OEIS Frontend

A357171 a(n) is the number of divisors of n whose digits are in strictly increasing order (A009993).

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A358099 a(n) is the number of divisors of n whose digits are in strictly decreasing order (A009995).

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A179937 a(n) is the product of the non-palindromic divisors of n.

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A184392 a(n) is the product of palindromic divisors of n.

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A355698 a(n) is the number of repdigits divisors of n (A010785).

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A334392 Numbers m such that the LCM of their palindromic divisors is neither 1 nor m.

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A339624 Perfect powers p^k, k >= 2 of palindromic primes p when p^k is not a palindrome.

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