Giovanni Teofilatto - cp's OEIS frontend

A271422 Concatenation of prime(n) and its square.

24, 39, 525, 749, 11121, 13169, 17289, 19361, 23529, 29841, 31961, 371369, 411681, 431849, 472209, 532809, 593481, 613721, 674489, 715041, 735329, 796241, 836889, 897921, 979409, 10110201, 10310609, 10711449, 10911881, 11312769, 12716129, 13117161, 13718769

Offset: 1

Giovanni Teofilatto, Jul 13 2016

All concatenations are divisible by the n-th prime.
a(n)/prime(n) gives a number of the form concatenate[(10^k)_prime(n)] for some k.
Subsequence of A053061.
Except for 24 and 525, a(n) have final decimal digit 1 or 9.

			For n=4, prime(4) = 7 (the fourth prime number) and 7^2 = 49. These are concatenated to get a(4) = 749. - _Michael B. Porter_, Jul 16 2016

Cf. A001248 (primes squared), A030078 (primes cubed), A053061, A038800 (number of primes between 10n and 10n+9).

seq(n*(n+10^(1+ilog10(n^2))), n=select(k->isprime(k),[$1..137])); # Peter Luschny, Jul 16 2016

Table[FromDigits@ Flatten@ Map[IntegerDigits, {#, #^2}] &@ Prime@ n, {n, 33}] (* Michael De Vlieger, Jul 13 2016 *)

a(n) = eval(Str(prime(n), prime(n)^2)) \\ Felix Fröhlich, Jul 14 2016

a(n) = Concatenate [prime(n)_A001248(n)].

a(n) = p*(p+10^A055642(p^2)) with p = prime(n). - Peter Luschny, Jul 17 2016

A274321 Primes equal to a concatenation of a prime and a nonzero palindromic number.

Original entry on oeis.org

13, 17, 23, 29, 31, 37, 43, 47, 53, 59, 67, 71, 73, 79, 83, 97, 113, 131, 137, 139, 167, 173, 179, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 271, 277, 283, 293, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 389, 397, 419, 431, 433, 439

Offset: 1

Giovanni Teofilatto, Jun 18 2016

David A. Corneth, Table of n, a(n) for n = 1..4366

Cf. A000040, A002113.

lista() = {my(l = List(), pal = vector(199,i,a2113(i)), pri = vector(primepi(10000)), t=0); forprime(i=2,10000,t++; pri[t]=i); for(i = 2, #pal, for(j=1,#pri, p = conc(pal[i], pri[j]); if(#digits(p) < 6, if(isprime(p), listput(l, p))); p = conc(pri[j], pal[i]); if(#digits(p) < 6, if(isprime(p), listput(l, p)))));listsort(l,1);l}a2113(n)={my(d, i, r); r=vector(#digits(n-10^(#digits(n\11)))+#digits(n\11)); n=n-10^(#digits(n\11)); d=digits(n); for(i=1, #d, r[i]=d[i]; r[#r+1-i]=d[i]); sum(i=1, #r, 10^(#r-i)*r[i])}
a2113(n)={my(d, i, r); r=vector(#digits(n-10^(#digits(n\11)))+#digits(n\11)); n=n-10^(#digits(n\11)); d=digits(n); for(i=1, #d, r[i]=d[i]; r[#r+1-i]=d[i]); sum(i=1, #r, 10^(#r-i)*r[i])}
conc(a, b) = {a * 10^(#digits(b)) + b} \\ David A. Corneth, Jun 18 2016

ispal(n) = n && (eval(subst(Polrev(digits(n)), x, 10)) == n);
isconc(n) = {d = digits(n); for (k=1, #d, na = n\10^k; nb = n % 10^k; if ((n == eval(concat(Str(na), Str(nb)))) && ((isprime(na) && ispal(nb)) || (isprime(nb) && ispal(na))), return(1)););}
isok(n) = isprime(n) && isconc(n); \\ Michel Marcus, Jun 20 2016

A273906 Primes equal to the concatenation of two nonzero palindromic numbers.

Original entry on oeis.org

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 113, 199, 211, 223, 227, 229, 233, 277, 311, 331, 337, 433, 443, 449, 499, 557, 577, 599, 661, 677, 733, 773, 811, 877, 881, 883, 887, 911, 977, 991, 997, 1013, 1019, 1117, 1151, 1171, 1181

Offset: 1

Giovanni Teofilatto, Jun 03 2016

The only palindrome in this sequence below 10^9 is 11 (per request of Giovanni Teofilatto). A004022 is a subsequence. - David A. Corneth, Jun 10 2016

If we have a concatenation of two palindromes A = A', B = B' which is palindromic, concat(A,B) =: A.B = (A.B)' = B'.A' = B.A, then A*(10^LB-1) = B*(10^LA-1) (LX = length of X) <=> A*R(LB) = B*R(LA), where R(n) = (10^n-1)/9. To have A.B prime we also must have gcd(A,B) = 1, thus A | R(LA) and B | R(LB). Such numbers are listed in A249647 (not A014950), the only palindromes there are of the form 1...1, 3...3 or 9...9. Thus the only palindromic terms in this sequence A273906 are the repunit primes A004022. - M. F. Hasler, Jun 10 2016

			The prime 1013 is a term since 101 and 3 are palindromic.
The prime 101 is not a term, since it is not a concatenation of two nonzero palindromic numbers.
The prime 131 is not a term because it is not a concatenation of two nonzero palindromic numbers.

David A. Corneth, Table of n, a(n) for n = 1..25912
David A. Corneth, PARI program

Cf. A000040, A002113, A004022, A014950, A096489, A249647, A273906.

Take[#, 62] &@ Select[Sort@ Map[FromDigits@ Flatten@ IntegerDigits@ # &, Tuples[#, 2]], PrimeQ] &@ Select[Range[10^3], Reverse@ # == # &@ IntegerDigits@ # &] (* Michael De Vlieger, Jun 03 2016 *)
nxtPal[n_]:=With[{c=Join[{2},Flatten[Table[{10*10^d,11*10^d},{d,0,10}]]]},SelectFirst[n+c,PalindromeQ]]; Take[Join[{11},Select[ #[[1]]*10^IntegerLength[ #[[2]]]+#[[2]]&/@ Flatten[{#,Reverse[#]}&/@Subsets[Join[Range[8],NestList[nxtPal,9,100]],{2}],1],PrimeQ]//Union],60] (* Harvey P. Dale, Dec 08 2024 *)

\\ See program link from David A. Corneth, Jun 10 2016.

a(n) = A096489(n+1), n=1..21. - R. J. Mathar, Jun 12 2016. (This is a pure accident, I think, since A096489 is a finite sequence. - N. J. A. Sloane, Jun 12 2016)

More terms from Michael De Vlieger, Jun 03 2016

A273892 Numbers starting with an even (decimal) digit.

Original entry on oeis.org

0, 2, 4, 6, 8, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219

Offset: 1

Giovanni Teofilatto, Jun 02 2016

For positive terms, a number is a term iff the first digit is even. Therefore, for k > 0, there are 4 * 10^(k - 1) terms having precisely k digits. - David A. Corneth, Jun 02 2016

Also numbers such that when the leftmost digit is moved to the unit's place the result is divisible by 2. - Stefano Spezia, Jul 08 2025

			21 is a term because 2 is an even number. - _Altug Alkan_, Jun 02 2016

Bruno Berselli, Table of n, a(n) for n = 1..1000

Cf. A004086, A005843.

[0] cat [n: n in [1..220] | IsEven(Intseq(n)[#Intseq(n)])]; // Bruno Berselli, Jun 15 2016

Select[Range[0, 219], EvenQ@ FromDigits@ Reverse@ IntegerDigits@ # &] (* or *) {0} ~ Join ~ Select[Range@ 219, EvenQ@ Floor[#/10^Floor@ Log10@ #] &] (* Michael De Vlieger, Jun 03 2016 *)

A004086(n) = eval(concat(Vecrev(Str(n)))); lista(nn) = for(n=0, nn, if(A004086(n) % 2 == 0, print1(n, ", "))); \\ Altug Alkan, Jun 02 2016

a(n)=if(n==1, return(0), n--; k = logint(9*n\4, 10)); n -= 4 * ((10^k - 1) / 9); n--; 2 * (n \ 10^k + 1)*10^k+n%10^k
is(n) = n==0||digits(n)[1]%2==0 \\ David A. Corneth, Jun 02 2016

A273245 Non-palindromic binary numbers whose reversal is a palindrome.

Original entry on oeis.org

10, 100, 110, 1000, 1010, 1100, 1110, 10000, 10010, 10100, 11000, 11100, 11110, 100000, 100010, 100100, 101000, 101010, 110000, 110110, 111000, 111100, 111110, 1000000, 1000010, 1000100, 1001000, 1010000, 1010100, 1011010, 1100000, 1100110, 1101100, 1110000, 1111000, 1111100

Offset: 1

Giovanni Teofilatto, May 18 2016

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

Cf. A006995, A273329, A272670 (this sequence written in base 10), A057890 (and divided by 2).

# see A272670 for the other subroutines
for n from 1 to 400 do
    if isA272670(n) then
        printf("%d,",A007088(n)) ;
    end if;
end do: # R. J. Mathar, May 20 2016

A273245_list = [int(m) for m in (bin(n)[2:] for n in range(1,10**4)) if m != m[::-1] and m.rstrip('0') == m[::-1].lstrip('0')] # Chai Wah Wu, May 21 2016

Consists of binary palindromes (A006995) times nontrivial powers of 2, that is, 2^i, i>0. - N. J. A. Sloane, May 19 2016

a(n) = A007088(A272670(n)). - R. J. Mathar, May 20 2016

A273239 Non-palindromic numbers whose reversal is a palindrome.

Original entry on oeis.org

10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 200, 220, 300, 330, 400, 440, 500, 550, 600, 660, 700, 770, 800, 880, 900, 990, 1000, 1010, 1100, 1110, 1210, 1310, 1410, 1510, 1610, 1710, 1810, 1910, 2000, 2020, 2120, 2200, 2220, 2320, 2420, 2520, 2620, 2720, 2820, 2920

Offset: 1

Giovanni Teofilatto, May 18 2016

Subsequence of A118959. - Altug Alkan, May 18 2016

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A118959, A273245.

More terms from Altug Alkan, May 18 2016

A272550 Lexicographically earliest increasing sequence of primes such that odd-indexed terms have final digit 1 and even-indexed terms have final digit 9.

Original entry on oeis.org

11, 19, 31, 59, 61, 79, 101, 109, 131, 139, 151, 179, 181, 199, 211, 229, 241, 269, 271, 349, 401, 409, 421, 439, 461, 479, 491, 499, 521, 569, 571, 599, 601, 619, 631, 659, 661, 709, 751, 769, 811, 829, 881, 919, 941, 1009, 1021, 1039, 1051, 1069, 1091, 1109

Offset: 1

Giovanni Teofilatto, May 11 2016

a(n) + a(n+1) = 0 (mod 10) for all n >= 1.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

a:= proc(n) option remember; local p, d;
      if n=1 then p:= 11
    else p:= a(n-1); d:= `if`(n::odd, 1, 9);
         while irem(p, 10)<>d do p:=nextprime(p) od
      fi; p
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 11 2016

a[1] = 11; a[n_] := a[n] = Block[{d, q = a[n-1]}, d=10-Mod[q,10]; While[ Mod[q = NextPrime@ q, 10] != d]; q]; Array[a, 30] (* Giovanni Resta, May 11 2016 *)

A267348 Decimal equivalents of terms of A266926 interpreted as binary numbers.

Original entry on oeis.org

0, 1, 2, 6, 54, 3510, 14380470, 241264265751990, 67909853583655146508751957430, 5380372916045726369002105219892285499516304666683458153910, 33773148168338125039096320085837383261496919374684668572527108632210618661283323381212228218472784834977109705977270

Offset: 1

Giovanni Teofilatto, Jan 13 2016

For n>1, a(n+1)/a(n) is an integer. Therefore, after 2, a(n) is divisible by 6.

The term a(12) has 925 decimal digits; a(13) has 1850 decimal digits. - Michael De Vlieger, Jan 13 2016

Cf. A266926.

a = {0, 1}; Do[AppendTo[a, FromDigits@ Flatten@ Map[IntegerDigits@ # &, If[n < 2, Reverse@ a, a]]], {n, 9}]; FromDigits[IntegerDigits@ #, 2] & /@ a (* Michael De Vlieger, Jan 13 2016 *)

More terms from Michael De Vlieger, Jan 13 2016

A266926 a(0)=0, a(1)=1, a(2)=10; for n>2, a(n) = concat(a(1), ..., a(n-1)).

Original entry on oeis.org

0, 1, 10, 110, 110110, 110110110110, 110110110110110110110110, 110110110110110110110110110110110110110110110110, 110110110110110110110110110110110110110110110110110110110110110110110110110110110110110110110110

Offset: 0

Giovanni Teofilatto, Jan 06 2016

Decimal conversions: 0, 1, 2, 6, 54, 3510, 14380470, 241264265751990, 67909853583655146508751957430, ... . (See A267348.) - Michael De Vlieger, Jan 06 2016
After 10, a(n) is '110' repeated 2^(n-3) times. Therefore, for n>3, a(n) is the concatenation of a(n-1) with itself.
After 1, each term with the 0's omitted is a member of A136308.
The number of digits in a(n) is A098011(n+1).
The number of digits in a(n+2)/a(n+1) gives A103204 with 2 repeated.

			a(3) = concat(1, 10, 110) = 110110.
a(4) = concat(1, 10, 110, 110110) = 110110110110.

Cf. A000079, A098011, A103204, A136308, A267348.

[n le 2 select n*5^(n-1) else 110*(10^(3*2^(n-3))-1)/999: n in [0..8]]; // Bruno Berselli, Jan 29 2016

a = {0, 1}; Do[AppendTo[a, FromDigits@ Flatten@ Map[IntegerDigits@ # &, If[n < 2, Reverse@ a, a]]], {n, 8}]; a (* Michael De Vlieger, Jan 06 2016 *)

a(n) = 110*(10^(3*2^(n-3))-1)/999 for n>2. - Bruno Berselli, Jan 29 2016

Definition by Michael De Vlieger, Jan 06 2016

Edited by Editors of the OEIS, Jan 29 2016

A265128 Nonprimes excluding numbers of the forms 2*p and p^i where p is a prime and i is a positive integer.

Original entry on oeis.org

0, 1, 12, 15, 18, 20, 21, 24, 28, 30, 33, 35, 36, 39, 40, 42, 44, 45, 48, 50, 51, 52, 54, 55, 56, 57, 60, 63, 65, 66, 68, 69, 70, 72, 75, 76, 77, 78, 80, 84, 85, 87, 88, 90, 91, 92, 93, 95, 96, 98, 99, 100, 102, 104, 105, 108, 110, 111, 112, 114, 115, 116

Offset: 1

Giovanni Teofilatto, Dec 02 2015

Cf. A000961, A024619, A100484, A264828.

Select[Range@ 120, And[! PrimeQ@ #, Nand[EvenQ@ #, PrimeQ[#/2]], ! PrimePowerQ@ #] &] (* Michael De Vlieger, Dec 02 2015 *)

isok(n) = ! (isprime(n) || (! (n%2) && isprime(n/2)) || isprimepower(n)); \\ Michel Marcus, Dec 02 2015

cp's OEIS Frontend

User: Giovanni Teofilatto

A271422 Concatenation of prime(n) and its square.

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A274321 Primes equal to a concatenation of a prime and a nonzero palindromic number.

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A273906 Primes equal to the concatenation of two nonzero palindromic numbers.

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A273892 Numbers starting with an even (decimal) digit.

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A273245 Non-palindromic binary numbers whose reversal is a palindrome.

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A273239 Non-palindromic numbers whose reversal is a palindrome.

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A272550 Lexicographically earliest increasing sequence of primes such that odd-indexed terms have final digit 1 and even-indexed terms have final digit 9.

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A267348 Decimal equivalents of terms of A266926 interpreted as binary numbers.

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