A056991 -id:A056991 - cp's OEIS frontend

A316481 Squares whose arithmetic mean of digits is 1 (i.e., the sum of digits equals the number of digits).

1, 1100401, 2220100, 100040004, 100100025, 100220121, 100400400, 101002500, 102030201, 102212100, 103002201, 104040000, 110250000, 121022001, 121220100, 123210000, 132020100, 144000000, 210221001, 225000000, 310112100, 324000000, 400040001, 400400100

Offset: 1

Jon E. Schoenfield, Jul 04 2018

Each term's number of digits is in A056991 (Numbers with digital root 1, 4, 7, or 9). For every term k in A056991, this sequence contains at least one k-digit term, with the exception of k=4. (See A316480.)

			1049^2 = 1100401, a 7-digit number whose digit sum is 1+1+0+0+4+0+1 = 7, so 1100401 is a term.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1543 terms from Jon E. Schoenfield)

Cf. A056991, A069711, A316480.

Intersection of A000290 and A061384. - Michel Marcus, Jul 06 2018

A316484 Squares whose arithmetic mean of digits is 4 (i.e., the sum of digits is 4 times the number of digits).

Original entry on oeis.org

4, 1681, 3364, 3481, 4624, 7225, 9025, 1054729, 1069156, 1073296, 1149184, 1168561, 1183744, 1227664, 1263376, 1288225, 1308736, 1329409, 1366561, 1517824, 1522756, 1545049, 1567504, 1585081, 1607824, 1630729, 1635841, 1677025, 1682209, 1705636, 1729225

Offset: 1

Jon E. Schoenfield, Jul 04 2018

Each term's number of digits is in A056991 (Numbers with digital root 1, 4, 7, or 9). For every term k in A056991, this sequence contains at least one k-digit term. (See A316480.)

			1027^2 + 1054729, a 7-digit number whose digit sum is 1+0+5+4+7+2+9 = 28 = 4*7, so 1054729 is a term.
10044^2 = 100881936, a 9-digit number whose digit sum is 1+0+0+8+8+1+9+3+6 = 36 = 4*9, so 100881936 is a term.

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

Cf. A056991, A069711, A316480.

f:= proc(n) local L;
  L:= convert(n^2,base,10);
  if convert(L,`+`)=4*nops(L) then n^2 fi
end proc:
map(f, [$1..2000]); # Robert Israel, Jul 05 2018

Select[Range[1500]^2, Mean[IntegerDigits[#]] == 4 &] (* Giovanni Resta, Jul 05 2018 *)

isok(n) = (n>0) && issquare(n) && (sumdigits(n) == 4*#digits(n)); \\ Michel Marcus, Jul 05 2018

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

A061100 Squares with digital root 4.

Original entry on oeis.org

4, 49, 121, 256, 400, 625, 841, 1156, 1444, 1849, 2209, 2704, 3136, 3721, 4225, 4900, 5476, 6241, 6889, 7744, 8464, 9409, 10201, 11236, 12100, 13225, 14161, 15376, 16384, 17689, 18769, 20164, 21316, 22801, 24025, 25600, 26896, 28561, 29929

Offset: 1

Amarnath Murthy, Apr 19 2001

			256 = 16^2, 2 + 5 + 6 = 13, 1 + 3 = 4;
1849 = 43^2, 1 + 8 + 4 + 9 = 22, 2 + 2 = 4.

Harry J. Smith, Table of n, a(n) for n = 1..1000
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.

Cf. A056991.

seq(seq((a+9*k)^2,a=[2,7]),k=0..20); # Robert Israel, Jun 13 2018

fdsQ[n_]:=NestWhile[Total[IntegerDigits[#]]&,n,#>9&]==4; Select[Range[ 200]^2,fdsQ] (* Harvey P. Dale, Dec 15 2011 *)

a(n)=(n\2*9-2*(-1)^n)^2 \\ Charles R Greathouse IV, Sep 21 2012

From Colin Barker, Feb 18 2013: (Start)
Conjecture:
a(n) = (16-72*n+81*n^2)/4 for n even;
a(n)=(25-90*n+81*n^2)/4 for n odd;
g.f.: -x*(4*x^4+45*x^3+64*x^2+45*x+4) / ((x-1)^3*(x+1)^2). (End)
Conjecture is true since x^2 == 4 (mod 9) if and only if x == 2 or 7 (mod 9). The odd-numbered terms are (2+9*k)^2 and the even-numbered terms are (7+9*k)^2. - Robert Israel, Jun 13 2018

More terms from Harry J. Smith, Jul 18 2009

A061101 Squares with digital root 7.

Original entry on oeis.org

16, 25, 169, 196, 484, 529, 961, 1024, 1600, 1681, 2401, 2500, 3364, 3481, 4489, 4624, 5776, 5929, 7225, 7396, 8836, 9025, 10609, 10816, 12544, 12769, 14641, 14884, 16900, 17161, 19321, 19600, 21904, 22201, 24649, 24964, 27556, 27889, 30625

Offset: 1

Amarnath Murthy, Apr 19 2001

			1681=41^2, 1+6+8+1 = 16, 1+6 =7, 4624=68^2, 4+6+2+4 = 16, 1+6 =7.

Harry J. Smith, Table of n, a(n) for n = 1..1000
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).

Cf. A056991.

seq(seq((9*i+j)^2, j=4..5), i=0..100); # Robert Israel, Jan 31 2017

a(n)=(n\2*9-4*(-1)^n)^2 \\ Charles R Greathouse IV, Sep 21 2012

Conjecture: a(n)=(9*n-8)^2/4 for n even. a(n)=(9*n-1)^2/4 for n odd. G.f.: x*(16+9*x+112*x^2+9*x^3+16*x^4)/((1-x)^3*(1+x)^2). - Colin Barker, Apr 21 2012

Conjecture is true, because x^2 == 7 (mod 9) if and only if x == 4 or 5 (mod 9). - Robert Israel, Jan 31 2017

More terms from Harry J. Smith, Jul 18 2009

A234429 Numbers which are the digital sum of the square of some prime.

Original entry on oeis.org

4, 7, 9, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154, 157, 160, 163, 166, 169, 172, 175, 178

Offset: 1

Antonio Graciá Llorente, Dec 26 2013

A123157 sorted and duplicates removed.

Cf. A067180, A067523, A123157, A056991.

Cf. A062397, A226803, A226802, A165492, A165459 A165493, A229058, A165502, A165503, A165504.

terms(nn) = {v = []; forprime (p = 1, nn, v = concat(v, sumdigits(p^2));); vecsort(v,,8);} \\ Michel Marcus, Jan 08 2014

From Robert G. Wilson v, Sep 28 2014: (Start)
Except for 3, all primes are congruent to +-1 (mod 3). Therefore, (3n +- 1)^2 = 9n^2 +- 6n + 1 which is congruent to 1 (mod 3).
2, 11, 101, ... (A062397);
5, 149, 1049, ... (A226803);
only 3;
19, 71, 179, 251, 449, 20249, 24499, 100549, ... (A226802);
7, 29, 47, 61, 79, 151, 349, 389, 461, 601, 1051, 1249, 1429, ... (A165492);
13, 23, 31, 41, 59, 103, 131, 139, 211, 229, 239, 347, 401, ... (A165459);
17, 37, 53, 73, 89, 107, 109, 127, 181, 199, 269, 271, 379, ... (A165493);
43, 97, 191, 227, 241, 317, 331, 353, 421, 439, 479, 569, 619, 641, ...;
67, 113, 157, 193, 257, 283, 311, 337, 373, 409, 419, 463, ... (A229058);
163, 197, 233, 307, 359, 397, 431, 467, 487, 523, 541, 577, 593, 631, ...;
83, 137, 173, 223, 263, 277, 281, 367, 443, 457, 547, 587, ... (A165502);
167, 293, 383, 563, 607, 617, 733, 787, 823, 859, 877, 941, 967, 977, ...;
433, 613, 683, 773, 827, 863, 1063, 1117, 1187, 1223, 1567, ... (A165503);
313, 947, 983, 1303, 1483, 1609, 1663, 1933, 1973, 1987, 2063, 2113, ...;
887, 1697, 1723, 1867, 1913, 2083, 2137, 2417, 2543, 2633, ... (A165504);
883, 937, 1367, 1637, 2213, 2447, 2683, 2791, 2917, 3313, 3583, 3833, ...;
1667, 2383, 2437, 2617, 2963, 4219, 4457, 5087, 5281, 6113, 6163, ...;
...  Also see A229058. (End)
Conjectures from Chai Wah Wu, Apr 16 2025: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 5.
G.f.: x*(2*x^4 - x^3 - x^2 - x + 4)/(x - 1)^2. (End)

a(36) from Michel Marcus, Jan 08 2014
a(37)-a(54) from Robert G. Wilson v, Sep 28 2014
More terms from Giovanni Resta, Aug 15 2019

A244713 Positive numbers primitively represented by the binary quadratic form (1, 1, -2).

Original entry on oeis.org

1, 4, 7, 10, 13, 16, 18, 19, 22, 25, 27, 28, 31, 34, 37, 40, 43, 45, 46, 49, 52, 54, 55, 58, 61, 64, 67, 70, 72, 73, 76, 79, 81, 82, 85, 88, 91, 94, 97, 99, 100, 103, 106, 108, 109, 112, 115, 118, 121, 124, 126, 127, 130, 133, 135, 136, 139, 142, 145, 148, 151

Offset: 1

Peter Luschny, Jul 04 2014

nonn

Discriminant = 9.

Ray Chandler, Table of n, a(n) for n = 1..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A002476, A007645. A subsequence of A056991 and A242660.

Reap[For[n = 1, n < 1000, n++, r = Reduce[x^2 + x y - 2 y^2 == n, {x, y}, Integers]; If[r =!= False, If[AnyTrue[{x, y} /. {ToRules[r /. C[1] -> 0]}, CoprimeQ @@ # &], Sow[n]]]]][[2, 1]] (* Jean-François Alcover, Oct 31 2016 *)

Conjectures from Colin Barker, Oct 31 2016: (Start)
a(n) = a(n-1)+a(n-11)-a(n-12) for n>12.
G.f.: (1 +2*x)*(1 +x +x^2)*(1 +x^3 +x^7) / ((1 -x)^2*(1 +x +x^2 +x^3 +x^4 +x^5 +x^6 +x^7 +x^8 +x^9 +x^10)).
(End)

A316487 Squares whose arithmetic mean of digits is 7 (i.e., the sum of digits is 7 times the number of digits).

Original entry on oeis.org

2778889, 4695889, 5678689, 5697769, 5938969, 6568969, 6589489, 6848689, 6895876, 7974976, 7997584, 8779369, 9878449, 9966649, 299739969, 377796969, 396686889, 458687889, 467986689, 487658889, 488984769, 496977849, 538889796, 557998884, 559984896, 569967876

Offset: 1

Jon E. Schoenfield, Jul 04 2018

Each term's number of digits is in A056991 (Numbers with digital root 1, 4, 7, or 9). For every term k in A056991, this sequence contains at least one k-digit term, with the exception of k=1 and k=4. (See A316480.)

			17313^2 = 299739969, a 9-digit number whose digit sum is 2+9+9+7+3+9+9+6+9 = 63 = 7*9, so 299739969 is a term.
43474^2 = 1889988676, a 10-digit number whose digit sum is 1+8+8+9+9+8+8+6+7+6 = 70 = 7*10, so 1889988676 is a term.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Cf. A056991, A069711, A316480.

Intersection of A000290 and A061424. - Michel Marcus, Jul 06 2018

A358702 a(n) is the least k > 0 such that the sum of the decimal digits of k^2 is n, or 0 if no such k exists.

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 4, 0, 3, 8, 0, 0, 7, 0, 0, 13, 0, 24, 17, 0, 0, 43, 0, 0, 67, 0, 63, 134, 0, 0, 83, 0, 0, 167, 0, 264, 314, 0, 0, 313, 0, 0, 707, 0, 1374, 836, 0, 0, 1667, 0, 0, 2236, 0, 3114, 4472, 0, 0, 6833, 0, 0, 8167, 0, 8937, 16667, 0, 0, 21886, 0, 0, 29614

Offset: 1

Hugo Pfoertner, Dec 04 2022

The nonzero terms are A067179.

Cf. A056991, A231897 (similar for binary weight).

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

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A316481 Squares whose arithmetic mean of digits is 1 (i.e., the sum of digits equals the number of digits).

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A316484 Squares whose arithmetic mean of digits is 4 (i.e., the sum of digits is 4 times the number of digits).

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A061099 Squares with digital root 1.

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A061100 Squares with digital root 4.

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A061101 Squares with digital root 7.

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A234429 Numbers which are the digital sum of the square of some prime.

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A244713 Positive numbers primitively represented by the binary quadratic form (1, 1, -2).

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A316487 Squares whose arithmetic mean of digits is 7 (i.e., the sum of digits is 7 times the number of digits).

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