A016038 -id:A016038 - cp's OEIS frontend

A135550 Number of bases b, 1 < b < n-1, in which n is a palindrome, allowing leading zeros when testing if a number is a palindrome.

0, 0, 0, 0, 1, 1, 2, 1, 3, 2, 4, 0, 5, 1, 3, 3, 5, 2, 6, 0, 6, 4, 2, 1, 8, 2, 4, 4, 6, 1, 8, 2, 6, 3, 4, 2, 10, 1, 3, 3, 9, 1, 8, 1, 4, 5, 4, 0, 11, 2, 6, 4, 6, 0, 8, 4, 8, 4, 2, 1, 14, 1, 4, 6, 8, 5, 7, 2, 7, 3, 6, 1, 14, 2, 3, 5, 4, 2, 9, 0, 12, 5, 4, 1, 14, 5, 3, 2, 7, 1, 13, 4, 6, 4, 2, 2

Offset: 0

John P. Linderman, Feb 26 2008, Feb 28 2008

Every integer n is a palindrome when expressed in unary, or in base n-1 (where it will be 11). So here we assume 1 < b < n-1.

Here 4 = 100 counts as a palindrome in base 2, since 00100 is palindromic.

Paul Tek, Table of n, a(n) for n = 0..10000
John P. Linderman, Description of A135549-A135551 and A016038
John P. Linderman, Perl program [Use the command: LEADING0S=1 palin.pl]

Cf. A135549, A135551, A016038.

A087155 Primes having nontrivial palindromic representation in some (at least one) base.

Original entry on oeis.org

5, 7, 13, 17, 23, 29, 31, 37, 41, 43, 59, 61, 67, 71, 73, 83, 89, 97, 101, 107, 109, 113, 127, 131, 151, 157, 173, 181, 191, 193, 197, 199, 211, 227, 229, 233, 239, 241, 251, 257, 271, 277, 281, 307, 313, 331, 337, 349, 353, 373, 379, 383, 397, 401, 409, 419

Offset: 1

Randy L. Ekl, Oct 18 2003

Number of terms < 10^n: 2, 18, 129, 1010, 8392, ..., . - Robert G. Wilson v, Jun 19 2014
Every whole number has single-digit representation in all large bases and all greater than 2 have representation 11 in the base one less than itself. Other palindromic representations are the nontrivial ones. - James G. Merickel, Jul 25 2015
Primes not in A016038. - Robert Israel, Jul 27 2015

			17 is in the list because 17_2 = 10001 and 17_4 = 101, two nontrivial palindromic representations. 19 is not in the list because 19 is not a multidigit palindrome in any base other than base 18.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A016038.

filter:= proc(n) local b,L;
if not isprime(n) then return false fi;
for b from 2 to floor(sqrt(n)) do
  L:= convert(n,base,b);
  if L = ListTools:-Reverse(L) then return true fi;
od:
false
end proc:
select(filter, [2*i+1 $ i=1..1000]); # Robert Israel, Jul 27 2015

palindromicBases[n_] := Module[{p}, Table[p = IntegerDigits[n, b]; If[p == Reverse[p], {b, p}, Sequence @@ {}], {b, 2, n - 2}]]; Select[ Prime@ Range@ 300, palindromicBases[#] !={}&] (* Robert G. Wilson v, May 06 2014 *)

q=1; forprime(m=3,500,count=0; for(b=2,m-1, w=b+1; k=0; i=m; while(i>0,k=k*w+i%b; i=floor(i/b)); l=0; j=k; while(j>0,l=l*w+j%w; j=floor(j/w)); if(l==k,count=count+1; if(count>1,print1(m,", "); q=b; m=nextprime(m+1); q=1; b=1,q=b),)))

Title, comments and example changed to agree with convention on single-digit numbers and incorporate 'nontrivial' concept by James G. Merickel, Jul 25 2015

A138329 List of strictly non-palindromic twin primes {p, p+2}.

Original entry on oeis.org

137, 139, 4337, 4339, 8291, 8293, 9419, 9421, 10937, 10939, 13757, 13759, 19427, 19429, 20981, 20983, 36011, 36013, 38327, 38329, 43397, 43399, 59441, 59443, 71327, 71329, 74717, 74719, 76871, 76873, 90437, 90439, 91571, 91573, 117239

Offset: 1

Karl Hovekamp, Mar 14 2008

The strictly non-palindromic twin primes are a part of the normal twin primes. See the list of twin primes A077800 and A016038 for the strictly non-palindromic numbers.

Karl Hovekamp, Palindromzahlen in adischen Zahlensystemen, 2004

K. Hovekamp, Palindromic numbers in different bases.
K. Hovekamp, Strictly non-palindromic prime twins up to 1 billion.
Wikipedia, Strictly non-palindromic numbers.

Cf. A016038, A047811, A016038, A077800, A099609.

Twin primes, where both numbers {p} and {p+2} are strictly non-palindromic.

A138348 Lesser of twin primes such that both twin primes have no bases b, 1 < b < p-1, in which p is a palindrome.

Original entry on oeis.org

137, 4337, 8291, 9419, 10937, 13757, 19427, 20981, 36011, 38327, 43397, 59441, 71327, 74717, 76871, 90437, 91571, 117239, 120941, 121019, 167021, 181787, 191561, 196871, 197597, 221717, 228881, 239387, 240881, 271277, 279119, 289031

Offset: 1

Robert G. Wilson v, Mar 09 2008

Also primes in A016038 which are 2 less than their immediate successors.

Prime index of A138348: {33, 592, 1040, 1165, 1328, 1627, 2201, 2359, 3826, 4046, 4524, 6009, 7060, 7367, 7557, 8756, 8852, ...

Robert G. Wilson v, Table of n, a(n) for n = 1..95

Cf. A001359, A016038.

palindromicBases[n_] := Module[{p}, Table[p = IntegerDigits[n, b]; If[p == Reverse[p], {b, p}, Sequence @@ {}], {b, 2, n - 2}]]; lst = {}; Do[ If[ Length@ palindromicBases@ Prime@ n == 0, AppendTo[lst, Prime@n]], {n, 22189}]; lst[[ # ]] & /@ Select[ Range@ Length@ lst - 1, lst[[ # ]] + 2 == lst[[ # + 1]] &]
f[n_] := Block[{k = 2}, While[id = IntegerDigits[n, k]; id != Reverse@ id, k++ ]; k]; lst = {2}; Do[p = Prime@ n; If[ f@p == p - 1, AppendTo[lst, p]; Print@p], {n, 128149}]; lst[[ # ]] & /@ Select[Range@11284, lst[[ # ]] + 2 == lst[[ # + 1]] &]
nbQ[n_]:=NoneTrue[Table[IntegerDigits[n,b],{b,2,n-2}],#==Reverse[#]&] && NoneTrue[ Table[IntegerDigits[n+2,b],{b,2,n}],#==Reverse[#]&]; Select[ Select[Partition[Prime[Range[26000]],2,1],#[[2]]-#[[1]]==2&][[All,1]],nbQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 03 2021 *)

A138358 List of triples of strictly non-palindromic primes without an ordinary prime in between.

Original entry on oeis.org

137, 139, 149, 1433, 1439, 1447, 4337, 4339, 4349, 5297, 5303, 5309, 8287, 8291, 8293, 13049, 13063, 13093, 30293, 30307, 30313, 36007, 36011, 36013, 43391, 43397, 43399

Offset: 1

Karl Hovekamp, Mar 16 2008

Up to 10^9 there are 2992 triples of strictly non-palindromic primes if the quadruples and quintuples are not counted.
For quadruples of this kind, see A138359.
For quintuples of this kind, see A138360.

			Primes:
...
113 is palindromic in base 8
127 is palindromic in base 2 and base 9
131 is palindromic in base 10
137 is strictly non-palindromic
139 is strictly non-palindromic
149 is strictly non-palindromic
151 is palindromic in base 3 and base 10
157 is palindromic in base 7 and base 12
...
So {137, 139, 149} is the first triple of strictly non-palindromic primes.

Karl Hovekamp, Palindromzahlen in adischen Zahlensystemen, 2004

K. Hovekamp, Palindromic numbers in different bases.
K. Hovekamp, Strictly non-palindromic prime 5-tuple, quadruple and triple up to 1 billion
Wikipedia, Strictly non-palindromic numbers.

Cf. A138359, A138360, A138329, A016038, A047811, A016038.

A small fraction of the primes are strictly non-palindromic. Notice that all strictly non-palindromic numbers >6 are prime! (see: A016038) Triples of these strictly non-palindromic primes, without any normal prime in between, are listed here.

A138359 List of quadruples of strictly non-palindromic primes without an ordinary prime in between them.

Original entry on oeis.org

44449, 44453, 44483, 44491, 120811, 120817, 120823, 120829, 315037, 315047, 315059, 315067, 583069, 583087, 583127, 583139, 617411, 617429, 617447, 617453, 1553423, 1553429, 1553437, 1553467, 1712329, 1712339, 1712353, 1712369

Offset: 1

Karl Hovekamp, Mar 16 2008

For triples of this kind, see A138358.

For quintuples of this kind, see A138360.

			Primes:
...
44417 palindromic in bases 50, 106, 135 and 141
44449 strictly non-palindromic
44453 strictly non-palindromic
44483 strictly non-palindromic
44491 strictly non-palindromic
44497 palindromic in base 67 and base 206
...
So {44449, 44453, 44483, 44491} is the first quadruple of strictly non-palindromic primes.

Karl Hovekamp, Palindromzahlen in adischen Zahlensystemen, 2004

K. Hovekamp, Palindromic numbers in different bases.
K. Hovekamp, Strictly non-palindromic prime 5-tuple, quadruple and triple up to 1 billion
Wikipedia, Strictly non-palindromic numbers.

Cf. A138358, A138360, A138329, A016038, A047811, A016038.

A138360 Quintuples of 5 consecutive strictly non-palindromic primes.

Original entry on oeis.org

3253177, 3253219, 3253223, 3253231, 3253241, 20189111, 20189119, 20189123, 20189137, 20189167, 22122937, 22122979, 22122983, 22123021, 22123043, 61309069, 61309081, 61309091, 61309093, 61309097, 89073521, 89073533, 89073583, 89073599, 89073613

Offset: 1

Karl Hovekamp, Mar 16 2008

The quintuples T(n,1), T(n,2), .. T(n,5), n>=1, in this array are 5 consecutive primes (consecutive in A000040) which are also members of A016038.
Notice that all strictly non-palindromic numbers >6 are prime! (See A016038.) Quintuples of these strictly non-palindromic primes, without any normal prime in between, are listed here.
Up to 1 billion there are only 5 quintuples of strictly non-palindromic primes. May be that there are no more quintuples of this kind. Up to 1 billion there are no n-tuples of strictly non-palindromic primes with n>5.

			Primes:
...
3253153 palindromic in bases 203, 356, 495, 1316, 1442, 1504 and 1648
3253177 strictly non-palindromic
3253219 strictly non-palindromic
3253223 strictly non-palindromic
3253231 strictly non-palindromic
3253241 strictly non-palindromic
3253253 palindromic in bases 653, 768, 910 and 1001
...
So {3253177, 3253219, 3253223, 3253231, 3253241} is the first quintuple of the strictly non-palindromic primes.

Karl Hovekamp, Palindromzahlen in adischen Zahlensystemen, 2004

K. Hovekamp, Palindromic numbers in different bases.
K. Hovekamp, Strictly non-palindromic prime 5-tuple, quadruple and triple up to 1 billion
Wikipedia, Strictly non-palindromic numbers.

Cf. A138358 (triples), A138359 (4-tuples), A138329, A016038, A047811, A016038.

More terms from Mauro Fiorentini, Jan 03 2016

A375201 a(n) is the sum of the bases b with 1 < b < n-1 in which n is a palindrome.

Original entry on oeis.org

0, 2, 0, 2, 3, 2, 7, 0, 5, 3, 6, 6, 10, 6, 13, 0, 12, 12, 10, 3, 23, 4, 20, 10, 22, 4, 23, 7, 22, 12, 20, 6, 41, 6, 22, 12, 38, 5, 37, 6, 31, 24, 31, 0, 56, 6, 40, 22, 45, 0, 51, 20, 43, 30, 28, 4, 82, 6, 35, 34, 53, 26, 63, 11, 52, 22, 56, 7, 91, 10, 42, 38, 55, 10, 87, 0, 91, 34, 52, 5, 112, 29

Offset: 4

Robert Israel, Oct 15 2024

If n = s*t is composite with s <= t-2, then n = s * (t-1) + s is a two-digit palindrome in base t-1, while s^2 = (s-1)^2 + 2 * (s-1) + 1 is a palindrome in base s-1. Thus a(n) >= sqrt(n)-1 for composite n > 6. On the other hand, there may be infinitely many primes for which a(n) = 0 (see A016038).

			a(10) = 7 because 10 = 101_3 = 22_4 is a palindrome in bases 3 and 4, and 3 + 4 = 7.

Robert Israel, Table of n, a(n) for n = 4..10000

Cf. A016038, A135549, A375350.

ispali:= proc(x, b) local F; F:= convert(x, base, b);
  andmap(t -> F[t] = F[-t], [$1.. nops(F)/2])
end proc:
f:= proc(k) convert(select(b -> ispali(k, b), [$2..k-2]), `+`) end proc:
map(f, [$4 .. 100]);

from sympy.ntheory import is_palindromic
def a(n): return sum(b for b in range(2, n-2) if is_palindromic(n, b))
print([a(n) for n in range(4, 86)]) # Michael S. Branicky, Oct 15 2024

A344512 a(n) is the least number larger than 1 which is a self number in all the bases 2 <= b <= n.

Original entry on oeis.org

4, 13, 13, 13, 287, 287, 2971, 2971, 27163, 27163, 90163, 90163, 5940609, 5940609, 6069129, 6069129, 276404649, 276404649

Offset: 2

Amiram Eldar, May 21 2021

Since the sequence of base-b self numbers for odd b is the sequence of the odd numbers (A005408) (Joshi, 1973), all the terms beyond a(2) are odd numbers.
For the corresponding sequence with only even bases, see A344513.
a(20) > 1.5*10^10, if it exists.

			a(2) = 4 since the least binary self number after 1 is A010061(2) = 4.
a(3) = 13 since the least binary self number after 1 which is also a self number in base 3 is A010061(4) = 13.

Vijayshankar Shivshankar Joshi, Contributions to the theory of power-free integers and self-numbers, Ph.D. dissertation, Gujarat University, Ahmedabad (India), October, 1973.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.

Eric Weisstein's World of Mathematics, Self Number.
Wikipedia, Self number.
Index entries for Colombian or self numbers and related sequences

Cf. A003052, A010061, A010064, A010067, A010070, A339211, A339212, A339213, A339214, A339215, A342729, A344513.

Similar sequences: A016038, A217705, A225427, A226320, A228768, A258107.

s[n_, b_] := n + Plus @@ IntegerDigits[n, b]; selfQ[n_, b_] := AllTrue[Range[n, n - (b - 1) * Ceiling @ Log[b, n], -1], s[#, b] != n &]; a[2] = 4; a[b_] := a[b] = Module[{n = a[b - 1]}, While[! AllTrue[Range[2, b], selfQ[n, #] &], n++]; n]; Array[a, 10, 2]

a(2*n+1) = a(2*n) for n >= 2.

A344513 a(n) is the least number larger than 1 which is a self number in all the even bases b = 2*k for 1 <= k <= n.

Original entry on oeis.org

4, 13, 287, 294, 6564, 90163, 1136828, 3301262, 276404649, 5643189146

Offset: 1

Amiram Eldar, May 21 2021

Joshi (1973) proved that for all odd b the sequence of base-b self numbers is the sequence of odd numbers (A005408). Therefore, in this sequence the bases are restricted to even values. For the corresponding sequence with both odd and even bases, see A344512.

			a(1) = 4 since the least binary self number after 1 is A010061(2) = 4.
a(2) = 13 since the least binary self number after 1 which is also a self number in base 2*2 = 4 is A010061(4) = A010064(4) = 13.

Vijayshankar Shivshankar Joshi, Contributions to the theory of power-free integers and self-numbers, Ph.D. dissertation, Gujarat University, Ahmedabad (India), October, 1973.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.

Eric Weisstein's World of Mathematics, Self Number.
Wikipedia, Self number.
Index entries for Colombian or self numbers and related sequences

Cf. A003052, A010061, A010064, A010067, A010070, A339211, A339212, A339213, A339214, A339215, A342729, A344512.

Similar sequences: A016038, A217705, A225427, A226320, A228768, A258107.

s[n_, b_] := n + Plus @@ IntegerDigits[n, b]; selfQ[n_, b_] := AllTrue[Range[n, n - (b - 1) * Ceiling @ Log[b, n], -1], s[#, b] != n &]; a[1] = 4; a[n_] := a[n] = Module[{k = a[n - 1]}, While[! AllTrue[Range[1, n], selfQ[k, 2*#] &], k++]; k]; Array[a, 7]

cp's OEIS Frontend

A135550 Number of bases b, 1 < b < n-1, in which n is a palindrome, allowing leading zeros when testing if a number is a palindrome.

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A087155 Primes having nontrivial palindromic representation in some (at least one) base.

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A138329 List of strictly non-palindromic twin primes {p, p+2}.

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A138348 Lesser of twin primes such that both twin primes have no bases b, 1 < b < p-1, in which p is a palindrome.

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A138358 List of triples of strictly non-palindromic primes without an ordinary prime in between.

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A138359 List of quadruples of strictly non-palindromic primes without an ordinary prime in between them.

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A138360 Quintuples of 5 consecutive strictly non-palindromic primes.

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Extensions

A375201 a(n) is the sum of the bases b with 1 < b < n-1 in which n is a palindrome.

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Maple

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A344512 a(n) is the least number larger than 1 which is a self number in all the bases 2 <= b <= n.