José de Jesús Camacho Medina - cp's OEIS frontend

A341763 Numbers whose trajectory under iteration of sum of cubes of digits (map) produce a narcissistic number greater than nine.

2, 3, 5, 6, 7, 8, 9, 11, 12, 14, 15, 17, 18, 19, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 50, 51, 53, 54, 56, 57, 58, 59, 60, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 83, 84, 85

Offset: 1

José de Jesús Camacho Medina, Feb 19 2021

Conjecture: all multiples of 3 are terms of this sequence.

			For a(1) = 2:
2^3 = 8.
8^3 = 512.
5^3 + 1^3 + 2^3 = 134.
1^3 + 3^3 + 4^3 = 92.
9^3 + 2^3 = 737.
7^3 + 3^3 + 7^3 = 713.
7^3 + 1^3 + 3^3 = 371.
371 is a narcissistic number.

J. J. Camacho, Un Método Insospechado Para Encontrar Números Narcisistas (in Spanish)

Cf. A055012 (sum of cubes of digits), A005188 (narcissistic numbers).

(* A example with recurrence formula to test if the number belongs to this sequence *)
f[1] = 2;
f[n_] := Total[IntegerDigits[f[n - 1]]^3]
Table[Total[IntegerDigits[f[n]]^3], {n, 1, 10}]

A335598 Squares that remain squares when the repunit with the same number of digits is added.

Original entry on oeis.org

0, 25, 289, 2025, 13225, 100489, 198025, 319225, 466489, 4862025, 19758025, 42471289, 1975358025, 3199599225, 60415182025, 134885049289, 151192657225, 197531358025, 207612366025, 248956092025, 447136954489, 588186226489, 19753091358025, 31996727599225, 311995522926025, 1975308691358025

Offset: 1

José de Jesús Camacho Medina, Jun 14 2020

			0 is a term because 0 + 1 = 1. The result is another square.
25 is a term because 25 + 11 = 36. The result is another square.
289 is a term because 289 + 111 = 400. The result is another square.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002275, A048612, A273229.

f:= proc(d,q,m) local x,y;
if d < q/d then return NULL fi;
x:= ((d-q/d)/2)^2;
if x >= 10^m and x < 10^(m+1) then x else NULL fi;
end proc:
R:= 0:
for m from 1 to 20 do
  q:= (10^m-1)/9;
  V:= sort(convert(map(f, numtheory:-divisors(q),q,m-1),list));
  R:= R, op(V);
od:
R; # Robert Israel, Aug 21 2020

lista(limit)={for(k=0, sqrtint(limit), my(t=k^2); if(issquare(t + (10^if(t, 1+logint(t,10), 1)-1)/9), print1(t, ", ")))}
{ lista(10^12) } \\ Andrew Howroyd, Aug 11 2020

Name corrected by Robert Israel, Aug 26 2020

A329192 Fibonacci numbers with arithmetic mean of digits an integer (sum of digits = a multiple of number of digits).

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 55, 144, 987, 6765, 10946, 9227465, 225851433717, 8944394323791464, 160500643816367088, 83621143489848422977, 59425114757512643212875125, 30960598847965113057878492344, 127127879743834334146972278486287885163

Offset: 1

José de Jesús Camacho Medina, Nov 07 2019

			55 is a term as the arithmetic mean of digits is an integer: (5+5)/2 = 5.
144 is a term as the arithmetic mean of digits is an integer: (1+4+4)/3 = 3.
6765 is a term as the arithmetic mean of digits is an integer: (6+7+6+5)/4 = 6.

Cf. A000045, A004090, A060384, A069709, A164947.

Mathematica
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a(n) = A000045(A164947(n+1)).

A296378 Fibonacci numbers that can be expressed as a sum of two or more Fibonacci numbers such that no digit appears more than once in the equation.

Original entry on oeis.org

3, 5, 8, 13, 34

Offset: 1

José de Jesús Camacho Medina, Dec 11 2017

This sequence is finite because the Fibonacci sequence grows very fast, forcing some digit to repeat. The last term is a(5) = 34.

			a(1) =  3 =  2 +  1.
a(2) =  5 =  2 +  3.
a(3) =  8 =  3 +  5.
a(4) = 13 =  5 +  8.
a(5) = 34 = 21 +  8 + 5.

Cf. A010784, A263448.

A281745 Numbers k with the property that the square root of the product of the digits of k is equal to the sum of the square roots of its digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 44, 149, 194, 228, 282, 333, 419, 491, 822, 914, 941, 11199, 11444, 11919, 11991, 14144, 14414, 14441, 19119, 19191, 19911, 41144, 41414, 41441, 44114, 44141, 44411, 91119, 91191, 91911, 99111, 11111449, 11111494, 11111944, 11114149

Offset: 1

José de Jesús Camacho Medina, Jan 28 2017

			    1 is a term because     sqrt(1) = sqrt(1);
   44 is a term because   sqrt(4*4) = sqrt(4) + sqrt(4);
  941 is a term because sqrt(9*4*1) = sqrt(9) + sqrt(4) + sqrt(1).

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

Cf. A007953, A007954, A034710.

Select[Range[10^6], Sqrt[Times @@ #] == Total[Sqrt@ #] &@ IntegerDigits@ # &] (* Michael De Vlieger, Feb 02 2017 *)

isok(n) = my(d = vecsort(digits(n))); sqrt(prod(k=1, #d, d[k])) == sum(k=1, #d, sqrt(d[k])); \\ Michel Marcus, Jan 29 2017

More terms from Jon E. Schoenfield, Jan 30 2017

A281724 Numbers k with the property that (sum of digits of k) times (number of digits of k) equals (sum of {each digit of k} raised to the power {number of digits in k}).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 22, 102, 120, 201, 210

Offset: 1

José de Jesús Camacho Medina, Jan 28 2017

Numbers with the property that q-p = 0, where p = (sum of digits of k) times (number of digits of k) (A110805) and q = (sum of {each digit of k} raised to the power {number of digits in k}) (A101337).
From David Consiglio, Jr., Jan 29 2017: (Start)
The sequence is finite and 210 is the last term. Proof:
1. Let Y = length(k).
2. The maximum value of digit_sum(k) is 9Y.
3. Since p = Y*digit_sum(k), p has a maximum value of 9Y^2.
4. 2^Y > 9Y^2 for all Y > 9. Therefore q > p for all numbers longer than 9 digits that contain any digits > 1.
   a. Example: 1,000,000,002. q = 1^5 + 8*0^5 + 2^10 = 1025. The largest p value for a 10-digit number would be for 9,999,999,999 which has p = 10*(9*10) = 900. Since q > all possible p values at this size, any term of this sequence must either have fewer than 10 digits or contain only 0's and 1's as digits.
5. Now we can consider only numbers with 0 and 1 digits.
6. Let Z = number of 1 digits in a number k.
7. q = Z because q = Z*1^Y + (Y-Z)*0^Y = Z.
8. p = YZ because the sum of the digits is equal to the number of 1's.
9. Z = YZ only in the case of Y = 1. Thus, the only term of this sequence that contains only 0's and 1's has a length of only 1 digit. Thus, k = 1 is in this sequence.
10. Therefore a candidate number must have fewer than 10 digits if it contains a digit 2 or larger, and must have fewer than 2 digits if it does not. All numbers in this range have been checked, and no additional values of k with q = p have been found.
Thus the sequence is finite. (End)

			    9 is a term because             9^1 = 1*(9);
   20 is a term because       2^2 + 0^2 = 2*(2 + 0);
  210 is a term because 2^3 + 1^3 + 0^3 = 3*(2 + 1 + 0).

Cf. A101337, A110805.

W = Array[Total[IntegerDigits[#]^IntegerLength[#]]&, 100]-Table[IntegerLength[n] * Total[IntegerDigits[n]], {n, 100}]; Flatten[Position[W,0]]

A281549 Numbers n such that abs(n - 4^k) is prime for k = 1..10.

Original entry on oeis.org

525, 635, 3335, 22547, 132725, 242073, 590663, 598793, 795143, 963815, 1910433, 2506647, 5833497, 13032533, 13336685, 22163097, 22291055, 27730755, 36069767, 39412953, 41006745, 42520623, 44594457, 47869577, 63735185, 73770483, 76852937, 89703543, 100998105

Offset: 1

José de Jesús Camacho Medina, Jan 23 2017

nonn

			For n = 525, the primes are {521, 509, 461, 269, 499, 3571, 15859, 65011, 261619, 1048051}.

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A281047.

AA =  Table[PrimeQ[Abs[n - 4^k]], {n, 1, 150000}, {k, 1, 10}];
Flatten[Position[AA, {True, True, True, True, True, True, True, True, True, True}]]

is(n)=for(k=1, 10, if(!isprime(abs(n-4^k)), return(0))); 1

Extended by Charles R Greathouse IV, Jan 24 2017

a(25)-a(29) from Chai Wah Wu, Jan 25 2017

A281047 Numbers n such that abs(n - 4^k) is prime for k = 1..8.

Original entry on oeis.org

483, 525, 635, 2015, 2043, 3335, 3575, 4275, 10317, 14955, 17855, 22547, 49433, 51695, 52773, 79877, 125807, 132725, 176417, 242073, 314177, 451937, 463517, 590663, 598793, 707817, 795143, 798335, 963815, 994565, 1095465

Offset: 1

José de Jesús Camacho Medina, Jan 13 2017

nonn

			For n = 483, the primes are {479, 467, 419, 227, 541, 3613, 15901, 65053}.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
José Camacho Medina, Cadenas de Numeros Primos (in Spanish).

Select[Range[11*10^5],AllTrue[Abs[#-4^Range[8]],PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 28 2020 *)

is(n)=for(k=1,8,if(!isprime(abs(n-4^k)), return(0))); 1 \\ Charles R Greathouse IV, Jan 13 2017

a(n) >> n log^8 n. - Charles R Greathouse IV, Jan 13 2017

More terms from Charles R Greathouse IV, Jan 13 2017

A279610 a(n) = concatenate n consecutive integers, starting with the last number of the previous batch.

Original entry on oeis.org

1, 12, 234, 4567, 7891011, 111213141516, 16171819202122, 2223242526272829, 293031323334353637, 37383940414243444546, 4647484950515253545556, 565758596061626364656667, 67686970717273747576777879, 7980818283848586878889909192

Offset: 1

José de Jesús Camacho Medina, Dec 09 2016, and Paolo Iachia, Dec 15 2016

A variant of A053067. The first number of the concatenation a(n) is A152947(n) = (n-2)*(n-1)/2+1 and the last is (n-1)*n/2+1.
The fourth term, 4567, is a prime. When is the next prime, if there is another? - N. J. A. Sloane, Dec 16 2016
a(n) is the concatenation of the terms of the n-th row of A122797 when seen as a triangle. - Michel Marcus, Dec 17 2016

			a(4) is the concatenation of 4 numbers beginning with the last number (4) that was used to build a(3), so a(4) = 4 5 6 7 = 4567. Then a(5) is the concatenation of 5 numbers beginning with the last number of a(4), which is 7, so a(5) = 7 8 9 10 11 = 7891011. And so on.
For n = 3, n^2/2 - n/2 + 1 = 4; a(3) = 4 + 3*10^1 + 2*10^(1+1) = 234.

Chai Wah Wu, Table of n, a(n) for n = 1..200

Cf. A053067, A122797, A152947.

A subsequence of A035333. For primes in latter, see A052087.

Table[FromDigits[Flatten[IntegerDigits /@ Range[(n(n - 1))/2 + 1, (n(n + 1))/2 + 1 ]]], {n, 0, 20}]

from _future_ import division
def A279610(n):
    return int(''.join(str(d) for d in range((n-1)*(n-2)//2+1,n*(n-1)//2+2))) # Chai Wah Wu, Dec 17 2016

a(n) = n^2/2 - n/2 + 1 + Sum{k=1..n-1} ((n^2/2 - n/2 + 1 - k)*10^Sum{j=0..k-1} (floor(1+log_10(n^2/2 - n/2 + 1 - j)))).

A278816 Numbers that can be produced from their own digits by applying one or more of the eight operations {+, -, *, /, sqrt(), ^, !, concat11()}, with no operation used more than once, where "concat11()" means the operation of concatenating two single digits.

Original entry on oeis.org

0, 1, 2, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84

Offset: 1

José de Jesús Camacho Medina, Nov 28 2016

Extending the sequence will soon become impractical without a program to check every possible way to apply one or more of the eight operations. - Editors, Dec 03 2016

			    0 is a term because      sqrt(0) = 0;
   24 is a term because (2+sqrt(4))! = 24;
   25 is a term because          5^2 = 25;
  343 is a term because      (3+4)^3 = 343.

Matematico Fresnillense, Reto Matematico Interesante (In Spanish).

cp's OEIS Frontend

User: José de Jesús Camacho Medina

A341763 Numbers whose trajectory under iteration of sum of cubes of digits (map) produce a narcissistic number greater than nine.

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A335598 Squares that remain squares when the repunit with the same number of digits is added.

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Maple

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A329192 Fibonacci numbers with arithmetic mean of digits an integer (sum of digits = a multiple of number of digits).

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A296378 Fibonacci numbers that can be expressed as a sum of two or more Fibonacci numbers such that no digit appears more than once in the equation.

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A281745 Numbers k with the property that the square root of the product of the digits of k is equal to the sum of the square roots of its digits.

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Mathematica

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A281724 Numbers k with the property that (sum of digits of k) times (number of digits of k) equals (sum of {each digit of k} raised to the power {number of digits in k}).

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Mathematica

A281549 Numbers n such that abs(n - 4^k) is prime for k = 1..10.

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Mathematica

PARI

Extensions

A281047 Numbers n such that abs(n - 4^k) is prime for k = 1..8.

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