A075110 -id:A075110 - cp's OEIS frontend

A061345 Powers of odd primes. Alternatively, 1 and the odd prime powers (p^k, p an odd prime, k >= 1).

1, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239

Offset: 0

N. J. A. Sloane, Jun 08 2001

Let a(n)=p^e, then tau(a(n)^2) = tau(p^(2*e)) = 2*e+1 = 2*(e+1)-1 = tau(2*a(n))-1 where tau=A000005. - Juri-Stepan Gerasimov, Jul 14 2011

For n > 0, also the allowed indices of a Cossidente-Penttila graph. - Eric W. Weisstein, Feb 24 2025

Robert Israel, Table of n, a(n) for n = 0..10000
L. J. Corwin, Irreducible polynomials over the integers which factor mod p for every p, Unpublished Bell Labs Memo, Sep 07 1967 [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Odd Prime Power.

Cf. A061346, A000961, A005408, A061344, A075109, A075110.

[1] cat [n: n in [3..300 by 2] | IsPrimePower(n)]; // Bruno Berselli, Feb 25 2016

select(t -> nops(ifactors(t)[2])<=1, [seq(2*i+1,i=0..1000)]); # Robert Israel, Jun 11 2014
# alternative:
A061345 := proc(n)
    option remember;
    local k ;
    if n = 0 then
        1;
    else
        for k from procname(n-1)+2 by 2 do
            if nops(numtheory[factorset](k)) = 1 then
                return k ;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jun 25 2016
isOddPrimepower := n -> type(n, 'primepower') and not type(n, 'even'):
A061345List := up_to -> select(isOddPrimepower, [`$`(1..up_to)]):
A061345List(240); # Peter Luschny, Feb 02 2023

t={1};k=0;Do[If[k==1,AppendTo[t,a1]];k=0;Do[c=Sqrt[a^2+b^2];If[IntegerQ[c]&&GCD[a,b,c]==1,k++;a1=a;b1=b;c1=c;],{b,4,a^2/2,2}],{a,3,260,2}];t (* Vladimir Joseph Stephan Orlovsky, Jan 29 2012 *)
Select[2 Range@ 130 - 1, PrimeNu@# < 2 &] (* Robert G. Wilson v, Jun 12 2014 *)
Join[{1}, Select[Range[1, 200, 2], PrimePowerQ]] (* Eric W. Weisstein, Feb 23 2025 *)

is(n)=my(p); if(isprimepower(n,&p), p>2, n==1) \\ Charles R Greathouse IV, Jun 08 2016

from sympy import primepi, integer_nthroot
def A061345(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x,k)[0])-1 for k in range(1,x.bit_length())))
    return bisection(f,n+1,n+1) # Chai Wah Wu, Feb 03 2025

a(n) = A061344(n)-1.

Intersection of A000961 (prime powers) and A005408 (odd numbers). - Robert Israel, Jun 11 2014

More terms from Larry Reeves (larryr(AT)acm.org), Jun 12 2001

A045532 Concatenate n with n-th prime.

Original entry on oeis.org

12, 23, 35, 47, 511, 613, 717, 819, 923, 1029, 1131, 1237, 1341, 1443, 1547, 1653, 1759, 1861, 1967, 2071, 2173, 2279, 2383, 2489, 2597, 26101, 27103, 28107, 29109, 30113, 31127, 32131, 33137, 34139, 35149, 36151, 37157, 38163, 39167, 40173, 41179, 42181

Offset: 1

Jeff Burch

Triangular numbers are 1653, 32131, 79401, ... - Ali Adams, Feb 04 2020

The next such terms are 173340627863131 and 1454987833022905581. - Giovanni Resta, Feb 04 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A075110.

Cf. A085451. [Reinhard Zumkeller, Jun 30 2010]

a045532 n = read $ show n ++ show (a000040 n) :: Integer
-- Reinhard Zumkeller, Jul 08 2014

[Seqint(Intseq(NthPrime(n)) cat Intseq(n)): n in [1..45]]; // Vincenzo Librandi, Apr 13 2019

Table[FromDigits[Join[IntegerDigits[n], IntegerDigits[Prime[n]]]], {n, 40}] (* Vincenzo Librandi, Apr 13 2019 *)
#[[1]]*10^IntegerLength[#[[2]]]+#[[2]]&/@Table[{n,Prime[n]},{n,50}] (* Harvey P. Dale, Oct 11 2024 *)

a(n) = eval(Str(n,prime(n))); \\ Michel Marcus, Apr 13 2019, simplified by M. F. Hasler, Feb 05 2020

from sympy import prime
def a(n): return int(str(n) + str(prime(n)))
print([a(n) for n in range(1, 43)]) # Michael S. Branicky, Dec 23 2021

a(n) = n*10^(A004216(A000040(n))+1) + A000040(n). - Reinhard Zumkeller, Sep 03 2002

A084669 Primes which are a concatenation of prime(n) and n.

Original entry on oeis.org

53, 239, 6719, 7321, 15737, 30763, 38977, 41981, 44987, 587107, 661121, 751133, 1051177, 1229201, 1297211, 1303213, 1327217, 1823281, 1913293, 1999303, 2131321, 2179327, 2207329, 2239333, 2371351, 2689391, 2699393, 3067439

Offset: 1

Zak Seidov, Jun 29 2003

			a(2)=239 because prime[9]=23 and concatenation of 23 and 9 is 239 which is prime.

Cf. A084667.

Subsequence of primes of A075110.

[p: n in [1..500] | IsPrime(p) where p is Seqint(Intseq(n) cat Intseq(NthPrime(n)))]; // Bruno Berselli, Sep 15 2015

Select[Table[FromDigits[Flatten[{IntegerDigits[Prime[n]], IntegerDigits[n]}]], {n, 1, 500}], PrimeQ]

A138821 Concatenation of n-th Fibonacci number and n-th prime.

Original entry on oeis.org

12, 13, 25, 37, 511, 813, 1317, 2119, 3423, 5529, 8931, 14437, 23341, 37743, 61047, 98753, 159759, 258461, 418167, 676571, 1094673, 1771179, 2865783, 4636889, 7502597, 121393101, 196418103, 317811107, 514229109, 832040113, 1346269127

Offset: 1

Omar E. Pol, Apr 05 2008, Apr 07 2008

			a(10)=5529 because A000045(10)=55 and A000040(10)=29.

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

Cf. A000040, A000045, A045532, A075110.

[Seqint(Intseq(NthPrime(n)) cat Intseq(Fibonacci(n))): n in [1..50]]; // Vincenzo Librandi, Jul 24 2019

f:= proc(n) local p;
  p:= ithprime(n);
  combinat:-fibonacci(n)*10^(1+ilog10(p))+p
end proc:
seq(f(n),n=1..100); # Robert Israel, Dec 15 2014

With[{nn=40},FromDigits[Flatten[IntegerDigits/@#]]&/@ Thread[ {Fibonacci[ Range[nn]],Prime[Range[nn]]}]] (* Harvey P. Dale, Dec 23 2011 *)

A138822 Concatenation of n-th prime and n-th Fibonacci number.

Original entry on oeis.org

21, 31, 52, 73, 115, 138, 1713, 1921, 2334, 2955, 3189, 37144, 41233, 43377, 47610, 53987, 591597, 612584, 674181, 716765, 7310946, 7917711, 8328657, 8946368, 9775025, 101121393, 103196418, 107317811, 109514229, 113832040, 1271346269

Offset: 1

Omar E. Pol, Apr 05 2008

			a(10)=2955 because A000040(10)=29 and A000045(10)=55.

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

Cf. A000040, A000045, A045532, A075110, A138821.

[Seqint(Intseq(Fibonacci(n)) cat Intseq(NthPrime(n))): n in [1..50]]; // Vincenzo Librandi, Jul 24 2019

With[{nn=50},FromDigits[Flatten[IntegerDigits[#]]]&/@Thread[ {Prime[ Range[ nn]],Fibonacci[Range[nn]]}]] (* Harvey P. Dale, Jul 12 2014 *)

A084670 Numbers k such that concatenation of prime(k) and k is prime.

Original entry on oeis.org

3, 9, 19, 21, 37, 63, 77, 81, 87, 107, 121, 133, 177, 201, 211, 213, 217, 281, 293, 303, 321, 327, 329, 333, 351, 391, 393, 439, 481, 503, 507, 519, 543, 547, 551, 561, 579, 581, 599, 621, 639, 657, 663, 667, 711, 721, 727, 743, 793, 813, 819, 827, 829, 831, 837

Offset: 1

Zak Seidov, Jun 29 2003

			9 is a term because prime(9) = 23 and concatenation of 23 and 9 is prime

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

Cf. A075110, A084669.

Select[Range[1000],PrimeQ[Prime[#]10^IntegerLength[#]+#]&] (* Harvey P. Dale, Apr 09 2022 *)

is(k) = ispseudoprime(eval(Str(prime(k), k))); \\ Jinyuan Wang, Apr 10 2020

from sympy import isprime, prime
def aupto(lim):
  return [k for k in range(1, lim+1) if isprime(int(str(prime(k))+str(k)))]
print(aupto(837)) # Michael S. Branicky, Mar 09 2021

More terms from Jinyuan Wang, Apr 10 2020

A139114 Concatenation of n-th Fibonacci number and n.

Original entry on oeis.org

11, 12, 23, 34, 55, 86, 137, 218, 349, 5510, 8911, 14412, 23313, 37714, 61015, 98716, 159717, 258418, 418119, 676520, 1094621, 1771122, 2865723, 4636824, 7502525, 12139326, 19641827, 31781128, 51422929, 83204030, 134626931, 217830932

Offset: 1

Omar E. Pol, Apr 09 2008

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

Cf. A000027, A000045, A045532, A075110, A138821, A138822, A139113.

#[[1]]*10^IntegerLength[#[[2]]]+#[[2]]&/@Table[{Fibonacci[n],n},{n,40}] (* Harvey P. Dale, Nov 03 2016 *)

A253910 Concatenation of n-th prime and n-th nonprime.

Original entry on oeis.org

21, 34, 56, 78, 119, 1310, 1712, 1914, 2315, 2916, 3118, 3720, 4121, 4322, 4724, 5325, 5926, 6127, 6728, 7130, 7332, 7933, 8334, 8935, 9736, 10138, 10339, 10740, 10942, 11344, 12745, 13146, 13748, 13949, 14950, 15151, 15752, 16354, 16755, 17356, 17957, 18158, 19160, 19362, 19763, 19964, 21165, 22366, 22768, 22969

Offset: 1

Omar E. Pol, Feb 06 2015

Concatenate A000040(n) and A018252(n).

			a(5) = 119 because the 5th prime is 11 and the 5th nonprime is 9.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A018252, A038530, A045532, A075110, A106582, A138821, A138822, A253911.

import Data.Function (on)
a253911 n = a253911_list !! (n-1)
a253911_list = map read $
   zipWith ((++) `on` show) a018252_list a000040_list :: [Integer]
-- Reinhard Zumkeller, Feb 09 2015

nprime(n)=c=0;k=1;while(k,if(!isprime(k),c++);if(c==n,return(k));k++)
vector(50,n,eval(concat(Str(prime(n)),Str(nprime(n))))) \\ Derek Orr, Feb 06 2015

A253911 Concatenation of n-th nonprime and n-th prime.

Original entry on oeis.org

12, 43, 65, 87, 911, 1013, 1217, 1419, 1523, 1629, 1831, 2037, 2141, 2243, 2447, 2553, 2659, 2761, 2867, 3071, 3273, 3379, 3483, 3589, 3697, 38101, 39103, 40107, 42109, 44113, 45127, 46131, 48137, 49139, 50149, 51151, 52157, 54163, 55167, 56173, 57179, 58181, 60191, 62193, 63197, 64199, 65211, 66223, 68227, 69229

Offset: 1

Omar E. Pol, Feb 06 2015

Concatenate A018252(n) and A000040(n).

			a(5) = 911 because the 5th nonprime is 9 and the 5th prime is 11.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A018252, A038530, A045532, A075110, A106582, A138821, A138822, A253910.

import Data.Function (on)
a253911 n = a253911_list !! (n-1)
a253911_list = map read $
   zipWith ((++) `on` show) a018252_list a000040_list :: [Integer]
-- Reinhard Zumkeller, Feb 09 2015

cncat[{a_,b_}]:=FromDigits[Flatten[IntegerDigits/@{a,b}]]; Module[ {nn=100,np,len},np = Select[Range[nn],!PrimeQ[#]&];len=Length[np];cncat/@Thread[{np,Prime[Range[len]]}]] (* Harvey P. Dale, Oct 17 2020 *)

nprime(n)=c=0; k=1; while(k, if(!isprime(k), c++); if(c==n, return(k)); k++)
vector(50, n, eval(concat(Str(nprime(n)), Str(prime(n))))) \\ Derek Orr, Feb 06 2015

A139113 Concatenation of n and n-th Fibonacci number.

Original entry on oeis.org

11, 21, 32, 43, 55, 68, 713, 821, 934, 1055, 1189, 12144, 13233, 14377, 15610, 16987, 171597, 182584, 194181, 206765, 2110946, 2217711, 2328657, 2446368, 2575025, 26121393, 27196418, 28317811, 29514229, 30832040, 311346269, 322178309

Offset: 1

Omar E. Pol, Apr 09 2008

Cf. A000027, A000045, A045532, A075110, A138821, A138822, A139114.

Table[FromDigits[Join[IntegerDigits[n],IntegerDigits[Fibonacci[n]]]],{n,40}] (* Harvey P. Dale, Dec 31 2015 *)

cp's OEIS Frontend

A061345 Powers of odd primes. Alternatively, 1 and the odd prime powers (p^k, p an odd prime, k >= 1).

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A045532 Concatenate n with n-th prime.

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A084669 Primes which are a concatenation of prime(n) and n.

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Magma

Mathematica

A138821 Concatenation of n-th Fibonacci number and n-th prime.

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Magma

Maple

Mathematica

A138822 Concatenation of n-th prime and n-th Fibonacci number.

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Magma

Mathematica

A084670 Numbers k such that concatenation of prime(k) and k is prime.

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Mathematica

PARI

Python

Extensions

A139114 Concatenation of n-th Fibonacci number and n.

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Mathematica