A056020 -id:A056020 - cp's OEIS frontend

A340407 a(n) gives the number of side-branches that will be passed, if A342369 is used to trace the Collatz tree backward starting at 6*n-2 with n > 0.

2, 5, 1, 4, 5, 1, 4, 2, 1, 2, 3, 1, 5, 5, 1, 6, 2, 1, 2, 5, 1, 3, 3, 1, 3, 2, 1, 2, 4, 1, 6, 10, 1, 5, 2, 1, 2, 3, 1, 5, 8, 1, 4, 2, 1, 2, 4, 1, 3, 3, 1, 3, 2, 1, 2, 18, 1, 5, 4, 1, 6, 2, 1, 2, 3, 1, 4, 4, 1, 5, 2, 1, 2, 7, 1, 3, 3, 1, 3, 2, 1, 2, 7, 1, 4, 9, 1, 4, 2, 1

Offset: 1

Thomas Scheuerle, Mar 24 2021

nonn

Recursion into A342369 means tracing the Collatz tree backward, starting at k = A342369(6*n-2), then k = A342369(k) until k is divisible by 3. At each A342369(k) = 3*m - 1, a new side-branch is connected which would start at 6*m-2. If A342369(k) reached a value divisible by three no further side-branches will be found.

This sequence is a rearrangement of A087088 such that all values at positions divisible by 3 are unchanged.

			n = 2:
6*n-2 = 10.
A342369(10) = 20. -> 7*3 - 1 -> A side-branch is connected.
A342369(20) = 13.
A342369(13) = 26. -> 9*3 - 1 -> A side-branch is connected.
A342369(26) = 17. -> 6*3 - 1 -> A side-branch is connected.
A342369(17) = 11. -> 4*3 - 1 -> A side-branch is connected.
A342369(11) = 7.
A342369(7) = 14. -> 5*3 - 1 -> A side-branch is connected.
A342369(14) = 9. -> divisible by 3 we stop here.
-> We found 5 connected side-branches, a(2) = 5.

Thomas Scheuerle, Table of n, a(n) for n = 1..5000
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A014682, A342369, A342261, A087088.

function a = A340407( max_n )
    for n = 1:max_n
        c = 0;
        s = 6*n -2;
        while mod(s,3) ~= 0
            s = A342369( s );
            if mod(s,3) == 2
                c = c+1;
            end
        end
        a(n) = c;
    end
end
function b = A342369( n )
    if mod(n,3) == 2
        b = (2*n - 1)/3;
    else
        b = 2*n;
    end
end

a(n) > 0.
a(3*n) = 1.
a(9*n - b) = 2, b = {1, 8} row 2 of A342261. ( a(A056020(n)) = 2 ).
a(27*n - b) = 3, b = {2, 4, 5, 16} row 3 of A342261.
a(81*n - b) = 4, b = {13, 14, 22, 34, 38, 52, 74, 77} row 4 of A342261.
a(3^k*n - b) = k, b = row k of A342261.
( Sum_{k=1..j} a(k) )/j  lim_{j->infinity} = 3 = Sum_{k=1..infinity} k*2^(k-1)/3^k.

A135027 Numbers k such that the sum of the digits of k^2 is 10. Multiples of 10 are omitted.

Original entry on oeis.org

8, 19, 35, 46, 55, 71, 145, 152, 179, 251, 332, 361, 449, 451, 548, 649, 4499, 20249, 20251, 24499, 100549, 114499, 316261

Offset: 1

Zak Seidov, Feb 10 2008

A subsequence of A056020. - R. J. Mathar, Feb 10 2008
Next term > 10000000. - R. J. Mathar, Oct 20 2009
If it exists, a(24) > 10^10. - Hugo Pfoertner, May 17 2021
If it exists, a(24) > 10^29. - Michael S. Branicky, May 30 2021

			Corresponding squares are 64, 361, 1225, 2116, 3025, 5041, 21025, 23104, 32041, 63001, 110224, 130321, 201601, 203401, 300304, 421201, 20241001, 410022001, 410103001, 600201001, 10110101401, 13110021001, 100021020121.
8^2 = 64 and 6+4 = 10. 316261^2 = 100021020121 and 1+0+0+0+2+1+0+2+0+1+2+1 = 10. - _Zak Seidov_, Aug 26 2009

Michael S. Branicky, Python program

Cf. A007953, A056020.

s={};Do[If[Mod[n,10]>0&&10==Total[IntegerDigits[n^2]],AppendTo[s,n]], {n,10^8}];s (* Zak Seidov, Aug 26 2009 *)

is(n) = sumdigits(n^2)==10 && n%10 > 0 \\ Felix Fröhlich, May 17 2021

def A007953(n):
    a=0
    sh=n
    while sh > 0:
        a += sh % 10
        sh //= 10
    return a
def isA135027(n):
    if n % 10 == 0:
        return False
    else:
        return A007953(n**2) == 10
for n in range(70000):
    if isA135027(n):
        print(n)
# R. J. Mathar, Oct 20 2009

# See linked program to go to large numbers
def ok(n): return n%10 != 0 and sum(map(int, str(n*n))) == 10
print(list(filter(ok, range(316262)))) # Michael S. Branicky, May 30 2021

A056528 Sum of digits of square of sum of digits of square.

Original entry on oeis.org

1, 7, 9, 13, 13, 9, 16, 1, 9, 1, 7, 9, 13, 13, 9, 16, 10, 9, 1, 7, 9, 13, 13, 9, 16, 10, 9, 10, 16, 9, 13, 13, 9, 16, 1, 9, 10, 16, 9, 13, 13, 9, 16, 10, 9, 1, 16, 9, 13, 13, 9, 16, 10, 9, 1, 16, 9, 13, 13, 9, 16, 10, 18, 10, 16, 9, 13, 13, 9, 16, 1, 9, 10, 16, 9, 13, 13, 9, 16, 1, 9

Offset: 1

Henry Bottomley, Jun 19 2000

			a(2)=7 because sum of digits of square of 2 is 4 and sum of digits of square of 4 is 1+6=7

Cf. A004159 for sum of digits of square, A056020 where iteration settles to 1, A056020 where iteration settles to 9, A056527 where iteration settles to 13 and 16. See also A056529.

Array[Total[IntegerDigits[(Total[IntegerDigits[#^2]])^2]]&,90] (* Harvey P. Dale, Jan 17 2012 *)
Table[Nest[Total[IntegerDigits[#^2]]&,n,2],{n,90}] (* Harvey P. Dale, Mar 07 2018 *)

a(n)=A004159(A004159(n))

A056529 Number of iterations of sum of digits of square to reach 1, 9, 13 or 16.

Original entry on oeis.org

0, 3, 1, 2, 2, 1, 1, 2, 0, 1, 3, 1, 0, 1, 1, 0, 3, 1, 2, 3, 1, 1, 1, 2, 1, 3, 2, 3, 1, 1, 1, 2, 2, 1, 2, 2, 3, 1, 1, 2, 1, 2, 2, 3, 1, 2, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 1, 1, 1, 3, 3, 3, 1, 2, 2, 1, 2, 1, 2, 2, 3, 2, 2, 2, 2, 2, 1, 2, 2, 3, 2, 2, 1, 2, 3, 2, 3, 1, 3, 2, 3, 2, 1, 2, 2, 3, 2, 1, 3, 1, 1, 1, 1

Offset: 1

Henry Bottomley, Jun 19 2000

			a(2)=3 because successive iterations give 2 -> 4 -> 7 -> 13 -> 16 -> 13 etc.

Cf. A004159 for sum of digits of square, A056528 for two iterations, A056020 where iteration settles to 1, A056020 where iteration settles to 9, A056527 where iteration settles to 13 and 16.

A326614 Smallest Euler-Jacobi pseudoprime to base n.

Original entry on oeis.org

9, 561, 121, 341, 781, 217, 25, 9, 91, 9, 133, 91, 85, 15, 1687, 15, 9, 25, 9, 21, 221, 21, 169, 25, 217, 9, 121, 9, 15, 49, 15, 25, 545, 33, 9, 35, 9, 39, 133, 39, 21, 451, 21, 9, 481, 9, 65, 49, 25, 49, 25, 51, 9, 55, 9, 55, 25, 57, 15, 481, 15, 9, 529, 9, 33, 65, 33, 25, 35, 69, 9

Offset: 1

Richard N. Smith, Jul 14 2019

nonn

a(n) = 9 for n == 1 or 8 mod 9 (see A056020).

Richard N. Smith, Table of n, a(n) for n = 1..5000
Wikipedia, Euler-Jacobi pseudoprime

Cf. A047713, A048950, A090086 (least Fermat pseudoprime to base n), A298756 (least strong pseudoprime to base n).

ejpspQ[n_,b_] := CoprimeQ[n,b] && CompositeQ[n] && Mod[b^((n - 1)/2) - JacobiSymbol[b, n], n] == 0; leastEJpsp[b_] := Module[{k=9}, While[!ejpspQ[k, b], k+=2]; k]; Array[leastEJpsp, 100] (* Amiram Eldar, Jul 15 2019 *)

isok(k, n) = ((k%2==1) && (gcd(k, n)==1) && Mod(n, k)^((k-1)/2)==kronecker(n, k) && !isprime(k));
a(n) = my(k=2); while (! isok(k, n), k++); k; \\ Michel Marcus, Jul 15 2019

A061099 Squares with digital root 1.

Original entry on oeis.org

1, 64, 100, 289, 361, 676, 784, 1225, 1369, 1936, 2116, 2809, 3025, 3844, 4096, 5041, 5329, 6400, 6724, 7921, 8281, 9604, 10000, 11449, 11881, 13456, 13924, 15625, 16129, 17956, 18496, 20449, 21025, 23104, 23716, 25921, 26569, 28900, 29584

Offset: 1

Amarnath Murthy, Apr 19 2001

			289 = 17^2, 2+8+9 = 19, 1+9 = 1, 1369 = 37^2, 1+3+6+9 = 19, 1+9 = 1.

Harry J. Smith, Table of n, a(n) for n = 1..1001
Amarnath Murthy & Charles Ashbacher, Fabricating a perfect square with a given valid digit sum, in Generalized Partitions and New Ideas On Number Theory and Smarandache Sequences, pp 154-156.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Squares of A056020.

Cf. A056991.

a(n)=(n\2*9-(-1)^n)^2 \\ Charles R Greathouse IV, Sep 20 2012

From Colin Barker, Apr 21 2012: (Start)
a(n) = (9*n+2)^2/4 for n even; a(n)=(9*n+7)^2/4 for n odd.
G.f.: x*(1+63*x+34*x^2+63*x^3+x^4)/((1-x)^3*(1+x)^2). (End)
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5). - Wesley Ivan Hurt, Apr 21 2021

More terms from Harry J. Smith, Jul 17 2009

A235399 Numbers which are the digital sum of the cube of some prime.

Original entry on oeis.org

8, 9, 10, 17, 19, 26, 28, 35, 37, 44, 46, 53, 55, 62, 64, 71, 73, 80, 82, 89, 91, 98, 100, 107, 109, 116, 118, 125, 127, 134, 136, 143, 145, 152, 154, 161, 163, 170, 172, 179, 181, 188, 190, 197, 199, 206, 208, 215, 217, 224, 226, 233, 235, 242, 244, 251, 253

Offset: 1

Antonio Graciá Llorente, Jan 09 2014

A235398 sorted and duplicates removed.

Cf. A235398, A004164, A007953, A123157.

Total[IntegerDigits[#]] & /@ (Prime[Range[5000000]]^3) // Union (* The program generates the first 39 terms of the sequence. To generate more, increase the Range constant. *) (* Harvey P. Dale, Sep 19 2021 *)

list(maxx)={v=List();n=2; while(n<=maxx,q=n^3;summ=sumdigits(q);
if(setsearch(v,summ)<1,listput(v,summ));n=nextprime(n+1));vecsort(v,,8) ;} \\ Bill McEachen, Jan 29 2014

Conjecture: for n > 4, a(n) = a(n-2) + 9 = A056020(n).

Conjecture: a(n) = (1/4)*(-1)^n*(9*(-1)^n*(2*n-1) + 5), n >= 3. - Bill McEachen, Feb 13 2021

a(37)-a(57) from Lars Blomberg, Feb 10 2016

A266957 Numbers m such that 9*m+10 is a square.

Original entry on oeis.org

-1, 6, 10, 31, 39, 74, 86, 135, 151, 214, 234, 311, 335, 426, 454, 559, 591, 710, 746, 879, 919, 1066, 1110, 1271, 1319, 1494, 1546, 1735, 1791, 1994, 2054, 2271, 2335, 2566, 2634, 2879, 2951, 3210, 3286, 3559, 3639, 3926, 4010, 4311, 4399, 4714, 4806, 5135, 5231, 5574

Offset: 1

Bruno Berselli, Jan 07 2016

Equivalently, numbers of the form h*(9*h+2)-1, where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ...

Also, integer values of k*(k+2)/9 minus 1.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A132355.
Cf. similar sequences listed in A266956.
Cf. A056020: square roots of 9*a(n)+10.

[n: n in [-1..6000] | IsSquare(9*n+10)];

[(18*(n-1)*n+5*(2*n-1)*(-1)^n-3)/8: n in [1..50]];

Select[Range[-1, 6000], IntegerQ[Sqrt[9 # + 10]] &]
Table[(18 (n - 1) n + 5 (2 n - 1) (-1)^n - 3)/8, {n, 1, 50}]

for(n=-1, 6000, if(issquare(9*n+10), print1(n, ", ")))

vector(50, n, n; (18*(n-1)*n+5*(2*n-1)*(-1)^n-3)/8)

from gmpy2 import is_square
[n for n in range(-1,6000) if is_square(9*n+10)]

[(18*(n-1)*n+5*(2*n-1)*(-1)**n-3)/8 for n in range(1, 60)]

[n for n in (-1..6000) if is_square(9*n+10)]

[(18*(n-1)*n+5*(2*n-1)*(-1)^n-3)/8 for n in (1..50)]

G.f.: x*(-1 + 7*x + 6*x^2 + 7*x^3 - x^4)/((1 + x)^2*(1 - x)^3).
a(n) = a(-n+1) = (18*(n-1)*n + 5*(2*n-1)*(-1)^n - 3)/8.
a(n) = A132355(n) + 1.

A319293 Numbers of the form 27^i(9j +- 1).

Original entry on oeis.org

1, 8, 10, 17, 19, 26, 27, 28, 35, 37, 44, 46, 53, 55, 62, 64, 71, 73, 80, 82, 89, 91, 98, 100, 107, 109, 116, 118, 125, 127, 134, 136, 143, 145, 152, 154, 161, 163, 170, 172, 179, 181, 188, 190, 197, 199, 206, 208, 215, 216, 217, 224, 226, 233, 235, 242, 244

Offset: 1

Jianing Song, Sep 16 2018

nonn

{+-a(n)} gives all nonzero cubes modulo all powers of 3, that is, cubes over 3-adic integers. So this sequence is closed under multiplication.

Jianing Song, Table of n, a(n) for n = 1..9231 (all terms <= 40000)

A056020 is a proper subsequence.
Perfect powers over 3-adic integers:
Squares: positive: A055047; negative: A055048 (negated);
Cubes: this sequence.

isA319293(n)= n\27^valuation(n, 27)%9==1||n\27^valuation(n, 27)%9==8

from sympy import integer_log
def A319293(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 n+x-sum(((m:=x//27**i)-1)//9+(m-8)//9+2 for i in range(integer_log(x,27)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Feb 14 2025

a(n) = 13*n/3 + O(log(n)).

A135029 Numbers k such that k and k^2 both have digit sum 10. Multiples of 10 are omitted.

Original entry on oeis.org

19, 46, 55, 145, 361, 451, 20251

Offset: 1

Zak Seidov, Feb 10 2008

A subsequence of A056020. - R. J. Mathar, Feb 10 2008
No further terms < 3*10^12 of the form 9*k+1, k=1,2,3,... - Lars Blomberg, Jun 28 2011
No further terms < 10^30. - Charles R Greathouse IV, Jun 29 2011

			Corresponding squares are 361, 2116, 3025, 21025, 130321, 203401, 410103001.

Cf. A056020.

dsum(n)=my(s=n%10);while(n\=10,s+=n%10);s
do(L)=my(E,F,G,H,t);for(a=1,L-1, for(b=0,a, for(c=0,b, for(d=0,c, for(e=0,d, E=10^a+10^b+10^c+10^d+10^e; for(f=0,e,F=E+10^f; for(g=0,f, G=F+10^g; for(h=0,g, H=G+10^h; for(i=0,h, t=H+10^i+1; if(dsum(t^2)==10, print1(t", ")))))))))));
do(30) \\ Charles R Greathouse IV, Jun 29 2011

cp's OEIS Frontend

A340407 a(n) gives the number of side-branches that will be passed, if A342369 is used to trace the Collatz tree backward starting at 6*n-2 with n > 0.

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A135027 Numbers k such that the sum of the digits of k^2 is 10. Multiples of 10 are omitted.

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A056528 Sum of digits of square of sum of digits of square.

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A056529 Number of iterations of sum of digits of square to reach 1, 9, 13 or 16.

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A326614 Smallest Euler-Jacobi pseudoprime to base n.

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Mathematica

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A061099 Squares with digital root 1.

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A235399 Numbers which are the digital sum of the cube of some prime.

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Mathematica

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Extensions

A266957 Numbers m such that 9*m+10 is a square.

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A319293 Numbers of the form 27^i(9j +- 1).