A062687 -id:A062687 - cp's OEIS frontend

A355773 Numbers all of whose divisors are members of A333369.

1, 3, 5, 7, 9, 13, 15, 17, 19, 31, 35, 37, 39, 51, 53, 57, 59, 71, 73, 79, 91, 93, 95, 97, 111, 137, 139, 153, 157, 159, 173, 179, 193, 197, 221, 223, 227, 229, 317, 333, 359, 371, 379, 395, 397, 443, 449, 519, 537, 571, 579, 591, 593, 661, 663, 669, 719, 739

Offset: 1

Bernard Schott, Jul 18 2022

All terms are necessarily odd because 2 is not in A333369

			111 is a term since all the divisors of 111, i.e., 1, 3, 37 and 111, are in A333369.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Project Euler, Problem 520: Simbers.

Cf. A333369, A353735, A355770, A355771, A355772.
Similar sequences: A062687, A190217, A329419, A337741
.
Subsequences: A155045, A355853.

simQ[n_] := AllTrue[Tally @ IntegerDigits[n], EvenQ[Plus @@ #] &]; Select[Range[1000], AllTrue[Divisors[#], simQ] &] (* Amiram Eldar, Jul 19 2022 *)

issimber(m) = my(d=digits(m), s=Set(d)); for (i=1, #s, if (#select(x->(x==s[i]), d) % 2 != (s[i] % 2), return (0))); return (1); \\ A333369
isok(k) = fordiv(k, d, if (!issimber(d), return(0))); return(1); \\ Michel Marcus, Jul 19 2022

from sympy import divisors, isprime
def c(n): s = str(n); return all(s.count(d)%2 == int(d)%2 for d in set(s))
def ok(n): return n > 0 and all(c(d) for d in divisors(n, generator=True))
print([k for k in range(740) if ok(k)]) # Michael S. Branicky, Jul 24 2022

A337941 Numbers whose divisors are all Zuckerman numbers (A007602).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 24, 1111111111111111111, 11111111111111111111111

Offset: 1

Bernard Schott, Oct 01 2020

Inspired by A337741.
Zuckerman numbers are numbers that are divisible by the product of their digits (see link).
The next term is the repunit prime R_317 which is too large to include in the data.
Primes in this sequence are 2, 3, 5, 7 and all the repunit primes (see A004023).
This sequence is infinite if and only if there are infinitely many repunit primes.

			6 is a term since all the divisors of 6, i.e., 1, 2, 3 and 6, are Zuckerman numbers.

Giovanni Resta, Zuckerman numbers, Numbers Aplenty

Subsequence of A007602.
Similar sequences: A062687, A190217, A308851, A329419, A337741.
Cf. A004023, A335037, A335038.
Cf. A004022 (subsequence of prime repunits).

zuckQ[n_] := (prod = Times @@ IntegerDigits[n]) > 0 && Divisible[n, prod]; Select[Range[24], AllTrue[Divisors[#], zuckQ] &] (* Amiram Eldar, Oct 01 2020 *)

isok(m) = {fordiv(m, d, my(p=vecprod(digits(d))); if (!p || (d % p), return (0))); return (1);} \\ Michel Marcus, Oct 05 2020

A088001 a(n) is the sum of non-palindromic divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 12, 13, 14, 15, 16, 17, 18, 19, 30, 21, 0, 23, 36, 25, 39, 27, 42, 29, 55, 31, 48, 0, 51, 35, 66, 37, 57, 52, 70, 41, 77, 43, 0, 60, 69, 47, 100, 49, 85, 68, 91, 53, 99, 0, 98, 76, 87, 59, 147, 61, 93, 84, 112, 78, 0, 67, 119, 92, 129, 71, 162, 73

Offset: 1

Labos Elemer, Oct 14 2003

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

Cf. A087999, A088000, A088002.

A088001 := proc(n)
        numtheory[sigma](n)-A088000(n) ;
end proc; # R. J. Mathar, Jul 28 2016

Table[Plus @@ Select[Divisors[k], Reverse[x = IntegerDigits[#]] != x &], {k, 73}] (* Jayanta Basu, Aug 12 2013 *)
Table[Total[Select[Divisors[n],!PalindromeQ[#]&]],{n,80}] (* Harvey P. Dale, May 15 2025 *)

def ispal(n):
    return n==int(str(n)[::-1])
def A088001(n):
    s=0
    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)=0 iff all divisors are palindromic. See A062687.

a(n)+A088000(n) = A000203(n). - R. J. Mathar, Sep 09 2015

A334139 Numbers that are equal to the LCM of their palindromic divisors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 18, 20, 21, 22, 24, 28, 30, 33, 35, 36, 40, 42, 44, 45, 55, 56, 60, 63, 66, 70, 72, 77, 84, 88, 90, 99, 101, 105, 110, 111, 120, 121, 126, 131, 132, 140, 141, 151, 154, 161, 165, 168, 171, 180, 181, 191, 198, 202, 210

Offset: 1

Bernard Schott, Apr 15 2020

These terms are the fixed points of A087999.
All the palindromes are in the sequence.
Now, if m is non-palindromic, then m is a term iff m = q_1^r_1 *...* q_i^r_i *...* q_k^r_k, where q_1 <...=2, r_i >= 1 and every divisor q_i^r_i is a palindrome; these q_i^r_i are in A084092 (see examples).
The first 40 terms, from 1 to 99, are exactly the 40 smallest divisors of 27720, hence the first 40 terms of A178864, but this sequence, which is infinite, is not a duplicate. Also, 27720 is in this sequence.

			2, 5, 131 are terms as palindromic primes.
111 = 3 * 37 is a term because 111 is a palindrome, so LCM(1,3,37,111) = 111.
27720 = 2^3 * 3^2 * 5 * 7 * 11, every 2^3=8, 3^2=9, 5, 7, 11 is a palindrome so 27720 is another term, no palindromic.

Cf. A087999, A178864.

Subsequences: A002113, A002385, A062687, A084092.

Select[Range[200], LCM @@ Select[Divisors[#], PalindromeQ] == # &] (* Amiram Eldar, Apr 15 2020 *)

ispal(x) = my(d=digits(x)); d == Vecrev(d);
isok(n) = lcm(select(ispal,  divisors(n))) == n; \\ Michel Marcus, Apr 16 2020

A355596 Numbers all of whose divisors are alternating numbers (A030141).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 21, 23, 25, 27, 29, 32, 36, 41, 43, 47, 49, 50, 54, 58, 61, 63, 67, 69, 81, 83, 87, 89, 94, 98, 101, 103, 107, 109, 123, 125, 127, 129, 141, 145, 147, 149, 161, 163, 167, 181, 183, 189, 214, 218, 250, 254, 290, 298

Offset: 1

Bernard Schott, Jul 12 2022

The smallest alternating number that is not a term is 30, because of 15.

			32 is a term since all the divisors of 32, i.e., 1, 2, 4, 8, 16 and 32, are alternating numbers

Subsequence of A030141.
Cf. A355593, A355594, A355595.
Similar sequences: A062687, A190217, A329419, A337941.

q[n_] := AllTrue[Divisors[n], !MemberQ[Differences[Mod[IntegerDigits[#], 2]], 0] &]; Select[Range[300], q] (* Amiram Eldar, Jul 12 2022 *)

isokd(n, d=digits(n))=for(i=2, #d, if((d[i]-d[i-1])%2==0, return(0))); 1; \\ A030141
isok(m) = sumdiv(m, d, isokd(d)) == numdiv(m); \\ Michel Marcus, Jul 12 2022

from sympy import divisors
def p(d): return 0 if d in "02468" else 1
def c(n):
    if n < 10: return True
    s = str(n)
    return all(p(s[i]) != p(s[i+1]) for i in range(len(s)-1))
def ok(n):
    return c(n) and all(c(d) for d in divisors(n, generator=True))
print([k for k in range(1, 200) if ok(k)]) # Michael S. Branicky, Jul 12 2022

a(51) and beyond from Michael S. Branicky, Jul 12 2022

A196677 Numbers n such that sum of the divisors of n equals the sum of the reversals of the divisors of n. Numbers with all palindrome divisors are not in the sequence.

Original entry on oeis.org

30, 42, 330, 462, 681, 772, 824, 890, 989, 2180, 3030, 4242, 4542, 4722, 8074, 9775, 17331, 23980, 33330, 35823, 36213, 43263, 46662, 47324, 55805, 62121, 62421, 65301, 65451, 66441, 66741, 68181, 68331, 68631, 68781, 69171, 71215, 71452, 73565, 74391, 74417, 74572, 74972

Offset: 1

Paolo P. Lava, Oct 05 2011

Subset of A080716.
The numbers that are not considered here belong to A062687, numbers all of whose divisors are palindromic. - Michel Marcus, Oct 10 2014
The sequence contains the terms palindromic numbers: 989, 97079, 98789, 99299, 1226221, 1794971, 13488431,…. Divisors(97079) = {1, 193, 503, 97079} and 193 + 503 = 696 = 391 + 305. Divisors(1794971) = {1, 1031, 1741, 1794971} and 1031 + 1741 = 2772 = 1301 + 1471. - Marius A. Burtea, Nov 20 2019

			Divisors of 989 are 1, 23, 43, 989 and 1+23+43+989=1+32+34+989=1056.
Divisors of 8074 are 1, 2, 11, 22, 367, 734, 4037, 8074 and 1+2+11+22+367+734+4037+8074=1+2+11+22+763+437+7304+4708=13248.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (terms 1..57 from R. J. Mathar, 58..1000 from Amiram Eldar)

Cf. A000203, A069192, A069942, A080716.

f:=func; g:=func; [k:k in [1..80000]| g(k) and not forall{d:d in Divisors(k)|f(d)}]; // Marius A. Burtea, Nov 20 2019

Rev:=proc(n)
local a,i,k;
i:=convert(n,base,10); a:=0;
for k from 1 to nops(i) do a:=a*10+i[k]; od;
a;
end:
P:=proc(j)
local h,m,n,ok,p,r,t;
for m from 1 to j  do
  p:=divisors(m); t:=0; ok:=0;
  for r from 1 to nops(p) do t:=t+Rev(p[r]); if p[r]<>Rev(p[r]) then ok:=1; fi;     od;
  if ok=1 and sigma(m)=t then print(m); fi;
od;
end:
P(100000);
# alternative
isA196677 := proc(n)
    isA080716(n) and not isA062687(n) ;
end proc:
n := 1;
for i from 1 do
    if isA196677(i) then
        printf("%d %d\n",n,i) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Sep 09 2015

A331896 Positive numbers all of whose divisors are negabinary palindromes (A331891).

Original entry on oeis.org

1, 3, 5, 7, 11, 17, 21, 23, 31, 43, 51, 77, 85, 103, 127, 155, 211, 217, 233, 257, 301, 341, 479, 635, 683, 739, 771, 857, 889, 937, 1117, 1229, 1285, 1333, 1367, 1799, 1951, 2111, 2159, 2383, 2395, 2459, 2731, 2827, 3187, 3251, 3347, 3937, 4001, 4273, 4369

Offset: 1

Amiram Eldar, Jan 30 2020

			21 is a term since all the divisors of 21, {1, 3, 7, 21}, are palindromes in negabinary representation: {1, 111, 11011, 10101}.

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

Cf. A039724, A062687, A329419, A331891.

negabin[n_] := negabin[n] = If[n==0, 0, negabin[Quotient[n-1, -2]]*10 + Mod[n, 2]]; nbPalinQ[n_] := PalindromeQ @ negabin[n]; negaBinAllDivPalQ[n_] := nbPalinQ[n] && AllTrue[Most @ Divisors[n], nbPalinQ]; Select[Range[5000], negaBinAllDivPalQ]

A246545 Numbers k with at least one nonpalindromic divisor such that the sum of phi(d) = the sum of phi(reverse(d)), where d runs over the divisors of k and phi is the Euler totient function.

Original entry on oeis.org

80, 880, 1920, 3140, 3880, 7305, 8080, 57755, 63405, 88880, 193920, 1188031, 1226221, 1794971, 7966197, 8339125, 13488431, 63007844, 123848321, 165387961, 312256913, 698621186

Offset: 1

Paolo P. Lava, Oct 01 2014

In general Sum_{d|k} phi(d) = k.

The numbers that are not considered here belong to A062687, numbers all of whose divisors are palindromic. - Michel Marcus, Oct 10 2014

			Divisors of 3140 are 1, 2, 4, 5, 10, 20, 157, 314, 628, 785, 1570, 3140.
Adding the Euler totient function of the reverse of the divisors: phi(1) + phi(2) + phi(4) + phi(5) + phi(01) + phi(02) + phi(751) + phi(413) + phi(826) + phi(587) + phi(0751) + phi(0413) = 3140.

Cf. A000010, A004086, A062687, A247826.

with(numtheory); T:=proc(h) local x,y,w; x:=h; y:=0;
for w from 1 to ilog10(h)+1 do y:=10*y+(x mod 10); x:=trunc(x/10); od; y; end:
P:=proc(q) local a,b,k,n,ok;
for n from 1 to q do a:=divisors(n); b:=0; ok:=0;
for k from 1 to nops(a) do b:=b+phi(T(a[k]));
if a[k]<>T(a[k]) then ok:=1; fi; od;
if ok=1 and n=b then print(n); fi; od; end: P(10^9);

isok(n) = {d = divisors(n); rd = vector(#d, i, subst(Polrev(digits(d[i])), x, 10)); (d != rd) && (n == sum(i=1, #rd, eulerphi(rd[i])));} \\ Michel Marcus, Oct 10 2014

a(11)-a(16) from Michel Marcus, Oct 10 2014

Name clarified and a(17)-a(22) from Jinyuan Wang, Apr 08 2025

A329100 Composite palindromes whose divisors > 1 are all nontrivial palindromes (i.e., palindromes with at least two digits).

Original entry on oeis.org

121, 1111, 1331, 1441, 1661, 1991, 3443, 3883, 7997, 10201, 12221, 13231, 14641, 15251, 15851, 18281, 19291, 31613, 35653, 37673, 37873, 38683, 112211, 113311, 115511, 116611, 124421, 125521, 134431, 136631, 139931, 145541, 146641, 157751, 167761, 169961, 176671

Offset: 1

Maxim Veselov, Nov 04 2019

This is the intersection of A062687 and A038511.
From Chai Wah Wu, Nov 08 2019 : (Start)
All terms start and end with the digits 1,3,7 or 9.
First term with 3 prime factors: 1331 = 11^3.
First term with 3 distinct prime factors: 145541 = 11*101*131.
First term with 4 prime factors: 14641 = 11^4.
First term with 5 prime factors: 1478741 = 11^4*101.
No term with more than 3 distinct prime factors or more than 5 prime factors among first 10000 terms.
(End)

			For k = 1331, its divisors > 1 are 11, 121 and 1331, all of which are palindromes with at least two digits, so 1331 is a term.
For k = 167761, its divisors > 1 are 11, 101, 151, 1111, 1661, 15251 and 167761, all of which are palindromes with at least two digits, so 167761 is a term.

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

Cf. A008364, A038511, A062687.

aQ[n_] := CompositeQ[n] && AllTrue[Rest @ Divisors[n], # > 10 && PalindromeQ @ IntegerDigits[#] &]; Select[Range[200000], aQ] (* Amiram Eldar, Nov 06 2019 *)

isA329100(n) = if((n>1) && !isprime(n) && gcd(n,210)==1, {d = divisors(n); rd = vector(#d, i, subst(Polrev(digits(d[i])), x, 10)); (d == rd); }, 0) \\ Jianing Song, Nov 06 2019, based on the program of A062687

More terms from Jianing Song, Nov 06 2019

A330616 Palindromes that are the product of 2 non-palindromic numbers.

Original entry on oeis.org

252, 272, 323, 414, 434, 444, 464, 494, 525, 555, 575, 585, 595, 636, 646, 656, 666, 676, 686, 696, 767, 777, 828, 848, 868, 888, 949, 969, 989, 999, 2002, 2112, 2772, 2992, 3003, 3553, 4004, 4224, 4554, 4774, 4884, 5005, 5115, 5225, 5775, 6006, 6336, 6996, 7007

Offset: 1

Bernard Schott, Mar 01 2020

			272=16*17 is a term but 282=6*47 is not a term.

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

Subsequence of A084325 (palindromes such that at least one divisor is non-palindromic).

Cf. A062687 (numbers all of whose divisors are palindromic).

seqQ[n_] := PalindromeQ[n] && AnyTrue[Divisors[n], ! PalindromeQ[#] && ! PalindromeQ[n/#] &]; Select[Range[7007], seqQ] (* Amiram Eldar, Mar 01 2020 *)

More terms from Amiram Eldar, Mar 01 2020

cp's OEIS Frontend

A355773 Numbers all of whose divisors are members of A333369.

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A337941 Numbers whose divisors are all Zuckerman numbers (A007602).

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A088001 a(n) is the sum of non-palindromic divisors of n.

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A334139 Numbers that are equal to the LCM of their palindromic divisors.

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A355596 Numbers all of whose divisors are alternating numbers (A030141).

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A196677 Numbers n such that sum of the divisors of n equals the sum of the reversals of the divisors of n. Numbers with all palindrome divisors are not in the sequence.

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A331896 Positive numbers all of whose divisors are negabinary palindromes (A331891).

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A246545 Numbers k with at least one nonpalindromic divisor such that the sum of phi(d) = the sum of phi(reverse(d)), where d runs over the divisors of k and phi is the Euler totient function.

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