A007942 -id:A007942 - cp's OEIS frontend

A360736 Number of prime divisors of A007942(n) = decimal concatenation of sequence (n, n-1, ..., 2, 1, 2, ..., n-1, n) counted with multiplicity.

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

Offset: 1

Bernard Schott, Mar 18 2023

For n <= 1530, only a(13) = 1 (answer to Smaradanche problem 19).

First semiprimes appear in A007942 at indices 4, 17, 23 since a(4) = a(17) = a(23) = 2.

			a(4) = 2 since 4321234 = 2 * 2160617;
a(6) = 8 since 65432123456 = 2^6 * 7 * 146053847;
a(12) = 5 since 12111098765432123456789101112 = 2^3*60800821*24899126702236725259;
a(13) = 1 since 131211109876543212345678910111213 is prime.

M. Fleuren, Factoring of the Smarandache Mirror Sequence.
F. Smarandache, Only Problems, Not Solutions!, Mirror sequence, problem 19, page 20.

Cf. A001222, A007942, A110760, A361624.

from sympy import factorint
def A360736(n): return sum(factorint(int(''.join(map(str,range(n,1,-1)))+''.join(map(str,range(1,n+1))))).values()) # Chai Wah Wu, Mar 21 2023

a(n) = A001222(A007942(n)).

a(36)-a(54) from Amiram Eldar, Mar 19 2023

A361624 Number of distinct prime factors in decimal concatenation of integer (n, n-1, ..., 2, 1, 2, ..., n-1, n) = A007942(n).

Original entry on oeis.org

0, 2, 3, 2, 5, 3, 3, 4, 3, 3, 3, 3, 1, 4, 6, 2, 2, 3, 4, 7, 4, 8, 2, 3, 4, 6, 5, 7, 5, 6, 6, 3, 5, 7, 4, 5, 8, 5, 6, 6, 3, 3, 7, 7, 7, 7, 10, 7, 6, 6, 7, 4, 5, 5, 7

Offset: 1

Bernard Schott, Mar 18 2023

a(n) < A360736(n) when n > 10 is a multiple of 4 or of 25, since, for these indices, A007942(n) is divisible by 2^2 or 5^2; but this inequality holds also, for other indices: for n = 6 (see example) and n = 39 where A007942(39) = 29 * 617^2 * 10185403128074353 * ...

			a(4) = 2 since 4321234 = 2 * 2160617;
a(6) = 3 since 65432123456 = 2^6 * 7 * 146053847.

M. Fleuren, Factoring of the Smarandache Mirror Sequence.
F. Smarandache, Only Problems, Not Solutions!, Mirror sequence, problem 19, page 20.

Cf. A001221, A007942, A110760, A360736.

from sympy import primefactors
def A361624(n): return len(primefactors(int(''.join(map(str,range(n,1,-1)))+''.join(map(str,range(1,n+1)))))) # Chai Wah Wu, Mar 21 2023

a(n) = A001221(A007942(n)).

a(36)-a(54) from Amiram Eldar, Mar 19 2023
a(42) corrected by Sean A. Irvine, Sep 26 2023
a(55) from Sean A. Irvine, Oct 16 2023

A074836 a(1) = 1, a(n) = the largest prime divisor of concatenation of sequence (n,n-1,..,2,1,2,..,n-1,n) A007942.

Original entry on oeis.org

1, 53, 353, 2160617, 479, 146053847, 18533465459, 18588601, 444444443, 10987654321234567891, 41824630287288128897, 24899126702236725259, 131211109876543212345678910111213, 59722151033999393, 1081735780698166140181, 40378532802777469135803086419727527803285379

Offset: 1

Jason Earls, Sep 09 2002

			a(4) is 2160617 because 4321234 = 2.'2160617'.

Sean A. Irvine, Table of n, a(n) for n = 1..55
M. Fleuren, Smarandache Mirror Sequence.

Cf. A007942.

Table[FactorInteger[FromDigits[Join[Flatten[IntegerDigits/@ Range[ n,1,-1]],Flatten[IntegerDigits/@Range[2,n]]]]][[-1,1]],{n,2,15}] (* Harvey P. Dale, Sep 05 2015 *)

Edited by Charles R Greathouse IV, Apr 28 2010
More terms from Harvey P. Dale, Sep 05 2015
a(1)=1 inserted by Sean A. Irvine, Jan 30 2025

A110760 a(n) = number of divisors of the concatenation of n,n-1,...3,2,1,2,3,...,n-1,n.

Original entry on oeis.org

1, 6, 8, 4, 32, 28, 8, 16, 8, 8, 8, 16, 2, 16, 64, 6, 4, 8, 16, 384, 16, 256, 4, 12, 24, 64, 32, 256, 32, 64, 64, 12, 32, 128, 16, 80, 256, 32, 96, 96, 8, 8, 128, 256, 128, 128, 1024, 192, 64, 96, 128, 64, 32, 32, 128

Offset: 1

Amarnath Murthy, Aug 11 2005

			a(3) = tau(32123) = 8.

Cf. A000005, A007942, A110756, A110757, A110758, A110759.

a(n) = A000005(A007942(n)). - Michel Marcus, Mar 22 2023

More terms from Ryan Propper, Jul 21 2006
Corrected and extended by Tyler Busby, Feb 12 2023
a(45)-a(55) from Tyler Busby, Feb 26 2023

A007943 Concatenation of sequence (1,3,..,2n-1,2n,2n-2,..,2).

Original entry on oeis.org

12, 1342, 135642, 13578642, 13579108642, 135791112108642, 1357911131412108642, 13579111315161412108642, 135791113151718161412108642, 1357911131517192018161412108642

Offset: 1

R. Muller

M. Le, Perfect Powers in the Smarandache Permutation Sequence, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, 148-149.

Harvey P. Dale, Table of n, a(n) for n = 1..184
F. Smarandache, Only Problems, Not Solutions!

Cf. A007942.

Table[FromDigits[Join[Flatten[IntegerDigits/@Range[1,2n+1,2]],Flatten[ IntegerDigits/@ Range[2n+2,2,-2]]]],{n,0,10}] (* Harvey P. Dale, Jul 30 2021 *)

A272617 Concatenation of the numbers from n down to 1 with numbers from 1 to n.

Original entry on oeis.org

11, 2112, 321123, 43211234, 5432112345, 654321123456, 76543211234567, 8765432112345678, 987654321123456789, 1098765432112345678910, 11109876543211234567891011, 121110987654321123456789101112, 1312111098765432112345678910111213, 14131211109876543211234567891011121314

Offset: 1

José de Jesús Camacho Medina, May 03 2016

Conjecture: a(1) is the only prime number.

No other prime terms up to a(8000). - Giovanni Resta, May 07 2016

			a(1) = concatenate("1", "1") = 11.
a(2) = concatenate("2", "1", "1", "2") = 2112.
a(3) = concatenate("3", "2", "1", "1", "2", "3") = 321123.

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A259937, A007942.

FromDigits@Flatten@IntegerDigits@Join[Reverse@#, #] & /@ Table[Range@n, {n, 20}]

a(n)={fromdigits(concat(concat(Vecrev(vector(n,i,digits(i)))), concat(vector(n,i,digits(i)))))} \\ Andrew Howroyd, Dec 23 2019

a(n) = A000422(n) concatenated with A007908(n).

Terms a(11) and beyond from Andrew Howroyd, Dec 23 2019

A261135 Decimal value a(n) of the binary number b(n) obtained by starting from n, sequentially concatenating all binary numbers down to 1 and then sequentially concatenating all binary numbers from 2 up to n.

Original entry on oeis.org

1, 22, 475, 20188, 1472229, 112441134, 8415721847, 1234163177336, 336405959636873, 95454487901599898, 26891544907776231851, 7513814442828887530172, 2084725487959034609380301, 574954374994698424018451678, 157752074714160807772190133743, 86177704805459827544499089423856

Offset: 1

Umut Uludag, Aug 10 2015

			a(1) = binary_to_decimal(1) = 1;
a(2) = binary_to_decimal(10110) = 22;
a(3) = binary_to_decimal(111011011) = 475;
a(4) = binary_to_decimal(100111011011100) = 20188.

Cf. A007942 for a base-10 version.

Cf. A173427 for an inverted ordering of repeated binary numbers.

Table[d = IntegerDigits[#, 2] & /@ Range@ n; FromDigits[#, 2] &@
Flatten[{Flatten@ Reverse@ d, Flatten@ Rest@ d}, 1], {n, 16}] (* Michael De Vlieger, Aug 20 2015 *)

a(n) = binary_to_decimal(concatenate(binary(n), binary(n-1), binary(n-2), ..., 11, 10, 1, 10, 11, ..., binary(n-2), binary(n-1), binary(n)))

a(11)-a(16) from Michael De Vlieger, Aug 20 2015

A287294 Concatenation of sequence (2n-1,2n-3,..,3,1,3,..,2n-3,2n-1).

Original entry on oeis.org

1, 313, 53135, 7531357, 975313579, 1197531357911, 13119753135791113, 151311975313579111315, 1715131197531357911131517, 19171513119753135791113151719, 211917151311975313579111315171921, 2321191715131197531357911131517192123

Offset: 1

XU Pingya, May 22 2017

For n <= 1400, a(2) = 313, a(10) = 19171513119753135791113151719, a(12) = 2321191715131197531357911131517192123 and a(110) = 219217..313..217219 are primes.

F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.
Eric Weisstein's World of Mathematics, Smarandache Sequences

Cf. A007942.

Block[{nn = 12, s}, s = IntegerDigits@ Range[1, 2 nn - 1, 2]; Table[FromDigits@ Flatten@ Join[Reverse[Rest@ #], #] &@ Take[s, n], {n, nn}]] (* Michael De Vlieger, May 23 2017 *)

cp's OEIS Frontend

A360736 Number of prime divisors of A007942(n) = decimal concatenation of sequence (n, n-1, ..., 2, 1, 2, ..., n-1, n) counted with multiplicity.

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A361624 Number of distinct prime factors in decimal concatenation of integer (n, n-1, ..., 2, 1, 2, ..., n-1, n) = A007942(n).

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A074836 a(1) = 1, a(n) = the largest prime divisor of concatenation of sequence (n,n-1,..,2,1,2,..,n-1,n) A007942.

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A110760 a(n) = number of divisors of the concatenation of n,n-1,...3,2,1,2,3,...,n-1,n.

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A007943 Concatenation of sequence (1,3,..,2n-1,2n,2n-2,..,2).

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A272617 Concatenation of the numbers from n down to 1 with numbers from 1 to n.

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A261135 Decimal value a(n) of the binary number b(n) obtained by starting from n, sequentially concatenating all binary numbers down to 1 and then sequentially concatenating all binary numbers from 2 up to n.

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A287294 Concatenation of sequence (2n-1,2n-3,..,3,1,3,..,2n-3,2n-1).

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