A038395 -id:A038395 - cp's OEIS frontend

A091308 Primes in A038395 (concatenate first n odd numbers in reverse order).

31, 737169676563615957555351494745434139373533312927252321191715131197531, 12312111911711511311110910710510310199979593918987858381797775737169676563615957555351494745434139373533312927252321191715131197531

Offset: 1

M. Grete (martha.grete(AT)web.de), Feb 21 2004

Among the first 72 terms only three terms are prime SmBackConOdd(2), SmBackConOdd(37) and SmBackConOdd(62), found by Micha Fleuren.
The next term (a(4)) has 1172 digits. - Harvey P. Dale, Jul 12 2022
a(5) = A038395(2717) has 10313 digits. - Robert Israel, Apr 14 2024

M. Fleuren, Smarandache Back Concatenated Odds.
Mathematics StackExchange, Is it known whether or not OEIS sequence A091308 is finite?

Cf. A038395.

x:= 1: d:= 1: count:= 0: R:= NULL:
for i from 2 while count < 5 do
  x:= (2*i-1)*10^d + x;
  d:= d + ilog10(2*i-1)+1;
  if isprime(x) then count:= count+1; R:= R,x; fi;
od:
R; # Robert Israel, Apr 14 2024

Select[Table[FromDigits[Flatten[IntegerDigits/@Range[n,1,-2]]],{n,1,341,2}],PrimeQ] (* Harvey P. Dale, Jul 12 2022 *)

Definition corrected by XU Pingya, Mar 16 2017

A260802 Odd numbers x = 2n - 1 such that the concatenation of A019519(n) and A038395(n-1) is prime.

Original entry on oeis.org

3, 13, 19, 21, 67

Offset: 1

Abhiram R Devesh, Jul 31 2015

			a(1) = 3 since 13_1 is prime;
a(2) = 13 since 135791113_1197531 is prime;
a(3) = 19 since 135791113151719_1715131197531 is prime.

Cf. A019519, A038395.

import sympy
n=1
while n>0:
    s=str(n)
    for m in range(n-2,0,-2):
        s=str(m)+s+str(m)
    p=int(s)
    if sympy.isprime(p)==True:
        print(n)
    n=n+2

A109837 Smallest prime factor of the reverse concatenation of the first n odd numbers.

Original entry on oeis.org

31, 3, 17, 7, 3, 1579, 17, 3, 2677, 131, 3, 54049423, 47, 3, 487, 25541, 3, 7, 211, 3, 31, 622639, 3, 498508631, 7, 3, 1259, 193, 3, 7669, 16229, 3, 13, 887377, 3, 737169676563615957555351494745434139373533312927252321191715131197531

Offset: 2

Zak Seidov, Jul 04 2005

Sequence starts with offset 2 since A038395(1)=1 has no prime factor. - M. F. Hasler
Starting with 2nd term each 3rd term is a(3m)=3.
Sequence A089922={3,73,123,817,...} lists the largest odd number (2n-1) used in the terms a(n) which are primes or probable primes.

			n=6: a(6)=1579 because 131197531=1579*83089.

Sean A. Irvine, Table of n, a(n) for n = 2..103

Cf. A038395, A089922, A104644, A138965.

s=1;Do[news=FromDigits[Flatten[{IntegerDigits[n], IntegerDigits[s]}]];fi=FactorInteger[news][[1, 1]];Print[fi];s=news, {n, 3, 70, 2}]

t=1; for( n=2,99, print1( factor( eval( t=Str( 2*n-1,t)))[1,1],", ")) \\ M. F. Hasler, Apr 13 2008

a(n) = A020639(A038395(n)). - R. J. Mathar, Sep 16 2007

More terms from R. J. Mathar, Sep 16 2007
Edited by M. F. Hasler, Apr 14 2008
Edited by T. D. Noe, Oct 30 2008

A078274 a(n) is the number from which if odd-positioned digits are deleted then one gets the concatenation of first n odd numbers, while if even-positioned digits are deleted then one gets reverse concatenation of first n odd numbers. Position of least significant digit is considered to be 1.

Original entry on oeis.org

11, 1331, 153351, 17355371, 1937557391, 11315977951311, 113351719917151331, 1135517391111917351351, 11375175911311113917551371, 113951779115111331115917751391, 1231517991171115311351117917952311, 12335271911911173115511371119927152331

Offset: 1

Amarnath Murthy, Nov 25 2002

a(n) is formed by interleaving the digits of A019519(n) and A038395(n). - Sean A. Irvine, Jun 26 2025

			a(4) = 17355371: deleting alternate digits starting from the LSD gives 1357. Deleting the other digits gives 7531.

Cf. A078275, A078276, A078277, A078278.

Cf. A019519, A038395.

a(n) = {my(d, v=w=[1]); for(i=2, n, v=concat(v, d=digits(2*i-1)); w=concat(d, w)); fromdigits(vector(2*#v, i, if(i%2, v[1+i\2], w[i/2]))); } \\ Sean A. Irvine, Jun 26 2025

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003
Corrected and extended by Harvey P. Dale, Apr 11 2012
Original terms restored by Sean A. Irvine, Jun 26 2025

A089922 Largest odd number in the reverse concatenation of the first n consecutive odd numbers when that concatenation is prime.

Original entry on oeis.org

3, 73, 123, 817, 5433

Offset: 1

Cino Hilliard, Jan 11 2004

These numbers are rare. For the case of the reverse concatenation of the consecutive numbers 1,2,3,... I have only found 828180...54321. Certainly, we can reduce the trials by noting that 2/3 of the numbers formed this way are multiples of 3 for the consecutive numbers and 1/3 of the odd consecutive numbers is a multiple of 3 but I do not think that will help much using PARI.

The number corresponding to 54335431...7531 has 10313 digits. The next term, if it exists is greater than 13000. - Dmitry Kamenetsky, Feb 21 2009

			31 is the reverse concatenation of the consecutive odd numbers 1 and 3, and it is prime.
817815813...531 is the reverse concatenation of the odd numbers 1 to 817, and it is a probable prime.

Cf. original concatenation in A038395. - Dmitry Kamenetsky, Feb 21 2009

revprime2(n) = { y=1; forstep(x=3,n,2, y=concat(Str(x),Str(y)); z=eval(y); if(ispseudoprime(z),print(x, ",")) ) }

Edited by T. D. Noe, Oct 30 2008

Added a new term 5433. This number is a probable prime with 20 iterations of Miller-Rabin test. - Dmitry Kamenetsky, Feb 21 2009

cp's OEIS Frontend

A091308 Primes in A038395 (concatenate first n odd numbers in reverse order).

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A260802 Odd numbers x = 2n - 1 such that the concatenation of A019519(n) and A038395(n-1) is prime.

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A109837 Smallest prime factor of the reverse concatenation of the first n odd numbers.

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A078274 a(n) is the number from which if odd-positioned digits are deleted then one gets the concatenation of first n odd numbers, while if even-positioned digits are deleted then one gets reverse concatenation of first n odd numbers. Position of least significant digit is considered to be 1.

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A089922 Largest odd number in the reverse concatenation of the first n consecutive odd numbers when that concatenation is prime.

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PARI

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