A084092 -id:A084092 - cp's OEIS frontend

A282845 Number of ways to write n as an ordered sum of 6 prime power palindromes (A084092).

0, 0, 0, 0, 0, 0, 1, 6, 21, 56, 126, 246, 432, 702, 1077, 1576, 2232, 3072, 4112, 5352, 6801, 8422, 10197, 12102, 14117, 16146, 18177, 20112, 21882, 23382, 24661, 25566, 26136, 26316, 26181, 25560, 24677, 23436, 21981, 20226, 18486, 16536, 14642, 12702, 10962, 9166, 7662, 6222, 5042, 3912, 3096, 2306, 1746, 1236, 921, 600

Offset: 0

Ilya Gutkovskiy, Feb 22 2017

Is there k which satisfies a(n) > 0 for all n > k?

			a(7) = 6 because we have:
[2, 1, 1, 1, 1, 1]
[1, 2, 1, 1, 1, 1]
[1, 1, 2, 1, 1, 1]
[1, 1, 1, 2, 1, 1]
[1, 1, 1, 1, 2, 1]
[1, 1, 1, 1, 1, 2]

Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Prime Power
Eric Weisstein's World of Mathematics, Palindromic Number
Index entries for sequences related to palindromes

Cf. A000961, A002113, A002385, A084092, A260254, A261131, A261132, A261422, A282585.

nmax = 55; CoefficientList[Series[(x + Sum[Boole[PrimePowerQ[k] && PalindromeQ[k]] x^k, {k, 1, nmax}])^6, {x, 0, nmax}], x]

G.f.: (Sum_{k>=1} x^A084092(k))^6.

A192137 Numbers m such that their concatenation of prime divisors are palindromic numbers.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 16, 25, 27, 32, 39, 49, 64, 69, 81, 101, 117, 119, 121, 125, 128, 129, 131, 151, 159, 181, 191, 207, 219, 243, 249, 256, 259, 313, 329, 339, 343, 351, 353

Offset: 1

Jaroslav Krizek, Jun 24 2011

The corresponding values of palindromic concatenation in A192138. Superset of A002385 (palindromic primes), A192139 and A192140.

			Concatenation of prime divisors of 39 = 3 * 13 is 313 (palindromic number).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A002113, A002385, A084317, A084092, A192138, A192139, A192140, A192141.

Select[Range[2,500],PalindromeQ[FromDigits[Flatten[IntegerDigits/@ FactorInteger[ #][[All,1]]]]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 02 2017 *)

A192139 Powers p^m, m >= 0, of palindromic primes p (A002385).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 25, 27, 32, 49, 64, 81, 101, 121, 125, 128, 131, 151, 181, 191, 243, 256, 313, 343, 353, 373, 383, 512, 625, 727, 729, 757, 787, 797, 919, 929, 1024, 1331, 2048, 2187, 2401, 3125, 4096, 6561, 8192, 10201, 10301, 10501, 10601, 11311, 11411, 12421, 12721

Offset: 1

Jaroslav Krizek, Jun 24 2011

Superset of A002385 and A084092. Subset of A192137.

M. F. Hasler, Table of n, a(n) for n = 1..858 (all terms up to 10^7).

Cf. A002113, A002385, A084317, A084092, A192137, A192138, A192140, A192141.

A192139=[1]; forprime(p=1,M=1e7, p==A004086(p) && A192139=setunion(A192139,vector(log(M)\log(p),k,p^k))) \\ M. F. Hasler, May 11 2015

Missing term 625 inserted and more terms added by M. F. Hasler, May 11 2015

A192138 Palindromic concatenation of prime divisors of numbers from A192137.

Original entry on oeis.org

2, 3, 2, 5, 7, 2, 3, 11, 2, 5, 3, 2, 313, 7, 2, 323, 3, 101, 313, 717, 11, 5, 2, 343, 131, 151, 353, 181, 191, 323, 373, 3, 383, 2, 737, 313, 747, 3113, 7, 313, 353, 373, 383, 343, 1331, 31113, 767, 353, 313, 2, 323, 5, 373, 3223, 797, 727, 3, 383, 757, 787, 3553

Offset: 1

Jaroslav Krizek, Jun 24 2011

			a(13) = 313 because A192137(13) = 39 = 3 * 13.

Cf. A002113, A002385, A084317, A084092, A192137, A192139, A192140, A192141.

f[n_] := FromDigits[Flatten[IntegerDigits /@ FactorInteger[n][[;; , 1]]]]; Select[f /@ Range[2, 1900], PalindromeQ] (* Amiram Eldar, Aug 06 2024 *)

More terms from Amiram Eldar, Aug 06 2024

A192140 Palindromic numbers m such that their concatenation of prime divisors are also palindromic numbers.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 101, 121, 131, 151, 181, 191, 313, 343, 353, 373, 383, 727, 747, 757, 787, 797, 919, 929, 1331, 10001, 10201, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14641, 14741, 15451, 15551, 16061, 16361

Offset: 1

Jaroslav Krizek, Jun 24 2011

The corresponding values of palindromic concatenation in A192141.
Superset of A002385 (palindromic primes) and A084092 (prime power decimal palindromes).
Subset of A002113 (palindromic numbers) and A192137.

			Concatenation of prime divisors of number 747 = 3^2 * 83 is 383 (palindromic number).

Cf. A002113, A002385, A084317, A084092, A192137, A192138, A192139, A192141.

f[n_] := FromDigits[Flatten[IntegerDigits /@ FactorInteger[n][[;; , 1]]]]; Select[Range[2, 20000], And @@ (PalindromeQ /@ {#, f[#]}) &] (* Amiram Eldar, Aug 06 2024 *)

More terms from Amiram Eldar, Aug 06 2024

A192141 Palindromic concatenation of prime divisors of numbers from A192140.

Original entry on oeis.org

2, 3, 2, 5, 7, 2, 3, 11, 101, 11, 131, 151, 181, 191, 313, 7, 353, 373, 383, 727, 383, 757, 787, 797, 919, 929, 11, 73137, 101, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 11, 14741, 15451, 15551, 16061, 16361, 16561, 16661

Offset: 1

Jaroslav Krizek, Jun 24 2011

			a(21) = 383 because A192140(21) = 747 = 3 * 83.

Cf. A002113, A002385, A084317, A084092, A192137, A192138, A192139, A192140.

f[n_] := FromDigits[Flatten[IntegerDigits /@ FactorInteger[n][[;; , 1]]]]; f /@ Select[Range[2, 20000], And @@ (PalindromeQ /@ {#, f[#]}) &] (* Amiram Eldar, Aug 06 2024 *)

More terms from Amiram Eldar, Aug 06 2024

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

A084093 Prime power decimal palindromes that are not prime.

Original entry on oeis.org

1, 4, 8, 9, 121, 343, 1331, 10201, 14641, 94249, 1030301, 104060401, 900075181570009, 10022212521222001, 12124434743442121, 12323244744232321, 12341234943214321, 1022321210249420121232201, 1210024420147410244200121

Offset: 1

Reinhard Zumkeller, May 11 2003

A084092 without A002385.

a(n) = A072037(n-1) for n > 1. - Georg Fischer, Oct 19 2018

			a(9)=14641=11^4; a(10)=94249=307^2=A000040(63)^2;
a(11)=1030301=101^3=A000040(26)^3.

Donovan Johnson, Table of n, a(n) for n = 1..27 (terms < 10^30)
Eric Weisstein's World of Mathematics, Palindromic Prime.
Eric Weisstein's World of Mathematics, Prime Power.

Cf. A002385, A072037, A084092.

a(13)-a(17) from Donovan Johnson, Feb 22 2008

Missing term 104060401 added by Donovan Johnson, Jul 02 2011

cp's OEIS Frontend

A282845 Number of ways to write n as an ordered sum of 6 prime power palindromes (A084092).

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A192137 Numbers m such that their concatenation of prime divisors are palindromic numbers.

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A192139 Powers p^m, m >= 0, of palindromic primes p (A002385).

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A192138 Palindromic concatenation of prime divisors of numbers from A192137.

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A192140 Palindromic numbers m such that their concatenation of prime divisors are also palindromic numbers.

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A192141 Palindromic concatenation of prime divisors of numbers from A192140.

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

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A084093 Prime power decimal palindromes that are not prime.

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