A056992 -id:A056992 - cp's OEIS frontend

A251754 Digital root of A027444(n) = n + n^2 + n^3, n>=1. Repeat(3, 5, 3, 3, 2, 6, 3, 8, 9).

3, 5, 3, 3, 2, 6, 3, 8, 9, 3, 5, 3, 3, 2, 6, 3, 8, 9, 3, 5, 3, 3, 2, 6, 3, 8, 9, 3, 5, 3, 3, 2, 6, 3, 8, 9, 3, 5, 3, 3, 2, 6, 3, 8, 9, 3, 5, 3, 3, 2, 6, 3, 8, 9, 3, 5, 3, 3, 2, 6, 3, 8, 9, 3, 5, 3, 3, 2, 6, 3, 8, 9, 3, 5, 3, 3, 2, 6, 3, 8, 9, 3, 5, 3, 3, 2, 6, 3, 8, 9, 3, 5, 3, 3, 2, 6, 3, 8, 9, 3

Offset: 1

Peter M. Chema, Dec 07 2014

Periodic with cycle of length 9: {3, 5, 3, 3, 2, 6, 3, 8, 9}.

a(n) also arises from the decimal expansion of 117775463/333333333 = 0.repeat(353326389).

			For a(11) = 5 because 11+11^2+11^3 = 1463, and 1+4+6+3 = 14.  Result is 5, which is the digital root of 14.

Eric Weisstein's World of Mathematics, Digital Root.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A027444, A010888, A056992, A073636.

&cat [[3,5,3,3,2,6,3,8,9]^^10]; // Vincenzo Librandi, Jul 18 2016

PadRight[{}, 120, {3, 5, 3, 3, 2, 6, 3, 8, 9}] (* Vincenzo Librandi, Jul 18 2016 *)

a(n) = sum of digits of (n+n^2+n^3), reduced to digital root.
a(n) = A010888(A027444(n)), and sequence may start at n=0.
a(n) = A010888(A010888(n) + A056992(n) + A073636(n)).
G.f.: x*(9*x^8 + 8*x^7 + 3*x^6 + 6*x^5 + 2*x^4 + 3*x^3 + 3*x^2 + 5*x + 3)/(1 - x^9). - Chai Wah Wu, Jul 17 2016

Edited: name specified, digital root link added, a comment rewritten and moved to formula section. - Wolfdieter Lang, Jan 05 2015

A251755 Digital root of n + n^2.

Original entry on oeis.org

0, 2, 6, 3, 2, 3, 6, 2, 9, 9, 2, 6, 3, 2, 3, 6, 2, 9, 9, 2, 6, 3, 2, 3, 6, 2, 9, 9, 2, 6, 3, 2, 3, 6, 2, 9, 9, 2, 6, 3, 2, 3, 6, 2, 9, 9, 2, 6, 3, 2, 3, 6, 2, 9, 9, 2, 6, 3, 2, 3, 6, 2, 9, 9, 2, 6, 3, 2, 3, 6, 2, 9, 9, 2, 6, 3, 2, 3, 6, 2, 9, 9, 2, 6, 3, 2, 3, 6, 2, 9, 9, 2, 6, 3, 2, 3, 6, 2, 9, 9, 2, 6, 3, 2, 3

Offset: 0

Peter M. Chema, Dec 07 2014

Positive integers give a cycle of period 9: {2, 6, 3, 2, 3, 6, 2, 9, 9}, which may be expressed as a decimal expansion of 87745433/333333333. Note that a(-n)=a(n-1), and negative integers give a mirrored period cycle, generating the cycle in reverse. Sequence is palindromic.

a(n) equals the digital root sum of A010888 and A056992.

			For a(7) = 2 because 7+7^2 = 56, and 5+6 = 11, yielding result of digital root of 2 (1+1).
For a(-3) = 6 because -3+(-3)^2 = -6, with digital root of 6.

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A002378, A010888, A056992.

a251755[n_Integer] := Module[{f},
  f[x_] := Last@NestWhileList[Plus @@ IntegerDigits[#] &, x, # > 9 &];
f /@ Table[i + i^2, {i, 0, n}]]; a251755[60] (* Michael De Vlieger, Dec 17 2014 *)

DR(n)=s=sumdigits(n);while(s>9,s=sumdigits(s));s
for(n=0,100,print1(DR(abs(n+n^2)),", ")) \\ Derek Orr, Dec 30 2014

a(n) = A010888(A002378(n)).

A251780 Digital root of A069778(n-1) = n^3 - n^2 + 1, n >= 1. Repeat(1, 6, 3, 7, 6, 6, 4, 6, 9).

Original entry on oeis.org

1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9

Offset: 1

Peter M. Chema, Dec 08 2014

Periodic with cycle of 9: {1, 6, 3, 7, 6, 6, 4, 6, 9}.

The decimal expansion of 54588823/333333333 = 0.repeat(163766469).

			For a(3) = 3 because 3^3 - 3^2 + 3  = 27 - 9 + 3 = 21 with digit sum 3 which is also the digital root of 21.

Eric Weisstein's World of Mathematics, Digital Root.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A069778, A251754, A010888, A056992, A073636.

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 1},{1, 6, 3, 7, 6, 6, 4, 6, 9},108] (* Ray Chandler, Jul 25 2016 *)

DR(n)=s=sumdigits(n);while(s>9,s=sumdigits(s));s
for(n=1,100,print1(DR(abs(n^2-n-n^3)),", ")) \\ Derek Orr, Dec 30 2014

a(n) = digital root of n^3 - n^2 + n.

More terms from Derek Orr, Dec 30 2014

Edited: name changed; formula, comment and example rewritten; digital root link added. - Wolfdieter Lang, Jan 05 2015

A303296 Digital roots of fourth powers A000583.

Original entry on oeis.org

1, 7, 9, 4, 4, 9, 7, 1, 9, 1, 7, 9, 4, 4, 9, 7, 1, 9, 1, 7, 9, 4, 4, 9, 7, 1, 9, 1, 7, 9, 4, 4, 9, 7, 1, 9, 1, 7, 9, 4, 4, 9, 7, 1, 9, 1, 7, 9, 4, 4, 9, 7, 1, 9, 1, 7, 9, 4, 4, 9, 7, 1, 9, 1, 7, 9, 4, 4, 9, 7, 1, 9, 1, 7, 9, 4, 4, 9, 7, 1, 9, 1, 7, 9, 4, 4, 9, 7, 1, 9

Offset: 1

Gaston Maire and students, Apr 21 2018

This sequence is related to A056992, the digital roots of the squares, and also presents a period of 9, in this case repeat [1, 7, 9, 4, 4, 9, 7, 1, 9].
a(n) = 9 if n is a multiple of 3.
Replace 4 with 7 and 7 with 4 in A056992. - Omar E. Pol, Apr 21 2018
a(n) is also the decimal expansion of 598165730/333333333. - Enrique Pérez Herrero, Nov 13 2021

I. Izmirli, On Some Properties of Digital Roots, Advances in Pure Mathematics, Vol. 4 No. 6 (2014), Article ID:47285.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Cf. A000583, A010888, A056992.

Table[FixedPoint[Total[IntegerDigits[#]] &, n^4], {n, 90}]

a(n) = (n^4-1)%9+1; \\ Michel Marcus, Apr 22 2018

a(n) = A010888(A000583(n)) = a(n - 9).

A073637 Digital root (cf. A010888) of prime(n)^3.

Original entry on oeis.org

8, 9, 8, 1, 8, 1, 8, 1, 8, 8, 1, 1, 8, 1, 8, 8, 8, 1, 1, 8, 1, 1, 8, 8, 1, 8, 1, 8, 1, 8, 1, 8, 8, 1, 8, 1, 1, 1, 8, 8, 8, 1, 8, 1, 8, 1, 1, 1, 8, 1, 8, 8, 1, 8, 8, 8, 8, 1, 1, 8, 1, 8, 1, 8, 1, 8, 1, 1, 8, 1, 8, 8, 1, 1, 1, 8, 8, 1, 8, 1, 8, 1, 8, 1, 1, 8, 8, 1, 8, 1, 8, 8, 1, 8, 1, 8, 8, 8, 1, 1

Offset: 1

Zak Seidov, Sep 01 2002, Oct 23 2009

Apart from a(2)=9 all other terms are either 1 or 8.

			a(3)=8 because p(3)=5 and 5^3=125 -> sum-of-digits = 8. a(4)=1 because p(3)=7 and 7^3=343 -> sum-of-digits = 10 -> sum-of-digits = 1.

Cf. A010888, A021596, A056992, A010888, A038194, A166923.

n=3; su[x_] := Sum[IntegerDigits[x][[i]], {i, Length[IntegerDigits[x]]}]; Table[su[su[su[su[Prime[x]^n]]]], {x, 100}]
Table[If[(m9=Mod[Prime[n]^3,9])==0,9,m9],{n,200}]

Edited by N. J. A. Sloane, Oct 29 2009

A340536 Digital root of 2*n^2.

Original entry on oeis.org

2, 8, 9, 5, 5, 9, 8, 2, 9, 2, 8, 9, 5, 5, 9, 8, 2, 9, 2, 8, 9, 5, 5, 9, 8, 2, 9, 2, 8, 9, 5, 5, 9, 8, 2, 9, 2, 8, 9, 5, 5, 9, 8, 2, 9, 2, 8, 9, 5, 5, 9, 8, 2, 9, 2, 8, 9, 5, 5, 9, 8, 2, 9, 2, 8, 9, 5, 5, 9, 8, 2, 9, 2, 8, 9, 5, 5, 9, 8, 2, 9, 2, 8, 9, 5, 5, 9, 8, 2, 9

Offset: 1

Radaev Nikita, Jan 10 2021

Period 9: repeat [2, 8, 9, 5, 5, 9, 8, 2, 9].

This sequence is also right if digital root of factor in "2*n^2" is 2 (for example, "11*n^2", "20*n^2", "29*n^2", etc.)

			For n=5, 2*5^2 = 50 and digital root is 5, so a(5) = 5.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Cf. A001105, A010888, A056992.

dr(n) = if(n, (n-1)%9+1); \\ A010888
a(n) = dr(2*n^2); \\ Michel Marcus, Jan 21 2021

a(n) = A010888(A001105(n)).

A375166 Nonsquares congruent to {0, 1, 4, 7} modulo 9.

Original entry on oeis.org

7, 10, 13, 18, 19, 22, 27, 28, 31, 34, 37, 40, 43, 45, 46, 52, 54, 55, 58, 61, 63, 67, 70, 72, 73, 76, 79, 82, 85, 88, 90, 91, 94, 97, 99, 103, 106, 108, 109, 112, 115, 117, 118, 124, 126, 127, 130, 133, 135, 136, 139, 142, 145, 148, 151, 153, 154, 157, 160, 162

Offset: 1

Stefano Spezia, Aug 05 2024

Squares are congruent to {0, 1, 4, 7} modulo 9, but the reverse is not always true since there are nonsquares that have the same congruence property. See Beiler.

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See p. 140.

Intersection of A000037 and A056991.

Cf. A000290, A056992, A070433.

Select[Range[0,162], !IntegerQ[Sqrt[#]] && MemberQ[{0,1,4,7}, Mod[#,9]] &]

from itertools import count, islice
from sympy.ntheory.primetest import is_square
def A375166_gen(): # generator of terms
    for i in count(0,9):
        for j in (0,1,4,7):
            if not is_square(i+j): yield i+j
A375166_list = list(islice(A375166_gen(),40)) # Chai Wah Wu, Jun 05 2025

cp's OEIS Frontend

A251754 Digital root of A027444(n) = n + n^2 + n^3, n>=1. Repeat(3, 5, 3, 3, 2, 6, 3, 8, 9).

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A251755 Digital root of n + n^2.

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A251780 Digital root of A069778(n-1) = n^3 - n^2 + 1, n >= 1. Repeat(1, 6, 3, 7, 6, 6, 4, 6, 9).

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A303296 Digital roots of fourth powers A000583.

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A073637 Digital root (cf. A010888) of prime(n)^3.

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A340536 Digital root of 2*n^2.

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A375166 Nonsquares congruent to {0, 1, 4, 7} modulo 9.

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