A004159 -id:A004159 - cp's OEIS frontend

A061904 Numbers n such that the iterative cycle: n -> sum of digits of n^2 has only one distinct element.

1, 3, 6, 9, 10, 12, 15, 18, 21, 30, 39, 45, 48, 51, 60, 90, 100, 102, 105, 111, 120, 150, 180, 201, 210, 249, 300, 318, 321, 348, 351, 390, 450, 480, 501, 510, 549, 600, 900, 1000, 1002, 1005, 1011, 1020, 1050, 1101, 1110, 1149, 1200, 1500, 1761, 1800, 2001

Offset: 1

Asher Auel, May 17 2001

Since the only numbers invariant under this iteration are 1 and 9, n is contained in this sequence iff the sum of digits of n^2 is 1 or 9.

			6 -> 3+6 = 9 -> 8+1 = 9 thus 9 is the only element of the iterative cycle of 6. 12 -> 1+4+4 = 9 -> 8+1 = 9 ...

Cf. A007953, A004159, A061903 - A061910.

A061907 The iterative cycle: n -> sum of digits of n^2 has only four distinct elements.

Original entry on oeis.org

2, 11, 20, 101, 110, 134, 136, 163, 172, 197, 200, 217, 233, 242, 244, 262, 278, 287, 296, 298, 307, 313, 314, 316, 343, 359, 386, 397, 406, 413, 422, 424, 431, 433, 442, 458, 467, 469, 476, 478, 487, 514, 523, 541, 577, 583, 586, 593, 604, 613, 614, 622

Offset: 1

Asher Auel, May 17 2001

			a(2) = 4 since 2 -> 4 -> 1+6 = 7 -> 4+9 = 13 -> 1+6+9 = 16 -> 2+5+6 = 13, thus {4,7,13,16} are the distinct elements of the iterative cycle of 2.

Cf. A007953, A004159, A061903 - A061910.

A118470 Numbers k for which digitsum(k) + digitsum(k^2) + digitsum(k^3) = digitsum(k^4).

Original entry on oeis.org

0, 162, 171, 351, 468, 558, 1620, 1710, 2106, 3321, 3510, 4023, 4680, 5121, 5247, 5544, 5580, 5868, 8001, 10008, 10071, 10224, 10305, 10503, 10818, 11025, 11241, 11511, 12321, 12654, 12888, 13239, 14004, 14301, 15471, 15876, 16011, 16200, 16218, 17100

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), May 04 2006

If x is a term, then so is 10*x. - Michael S. Branicky, Dec 25 2021

			162 is a term because s(162) = 9, s(162^2) = 18, s(162^3) = 27, s(162^4) = 54 and 9 + 18 + 27 = 54.

J.W.L. (Jan) Eerland, Table of n, a(n) for n = 1..3721 (all terms < 10^7)

Cf. A007953, A004159, A004164, A055565.

Select[Range[0, 20000], Sum[i*(DigitCount[ # ][[i]] + DigitCount[ #^2][[i]] + DigitCount[ #^3][[i]]), {i, 1, 9}] == Sum[i*DigitCount[ #^4][[i]], {i, 1, 9}] &] (* Stefan Steinerberger, May 04 2006 *)
s[n_] := Plus @@ IntegerDigits@n; Select[ Range[0, 16217], s@# + s[ #^2] + s[ #^3] == s[ #^4] &] (* Robert G. Wilson v, May 04 2006 *)
Parallelize[While[True,If[Total[IntegerDigits[n]]+Total[IntegerDigits[n^2]]+Total[IntegerDigits[n^3]]==Total[IntegerDigits[n^4]],Print[n]];n++];n] (* J.W.L. (Jan) Eerland, Dec 25 2021 *)

is(n)=my(s=sumdigits); s(n)+s(n^2)+s(n^3) == s(n^4) \\ Anders Hellström, Sep 16 2015

select(isA118470(n)={sumdigits(n)+sumdigits(n^2)+sumdigits(n^3) == sumdigits(n^4)}, [0..1000]) \\ J.W.L. (Jan) Eerland, Dec 25 2021

def sd(n): return sum(map(int, str(n)))
def ok(n): return sd(n) + sd(n**2) + sd(n**3) == sd(n**4)
print([k for k in range(20000) if ok(k)]) # Michael S. Branicky, Dec 25 2021

More terms from Joshua Zucker, May 11 2006

A153753 Numbers k such that there are 18 digits in k^2 and for each factor f of 18 (1,2,3,6,9) the sum of digit groupings of size f is a square.

Original entry on oeis.org

324344373, 333306315, 333321861, 333359685, 333361029, 334363803, 369396732, 370397193, 407380269, 407381484, 444475035, 666636972, 666695028, 666701463, 702667239, 702671124, 702736170, 703667130, 704741610

Offset: 1

Doug Bell, Dec 31 2008

This sequence is a subsequence of both A153745 and A061910.

			324344373^2 = 105199272296763129;
1+0+5+1+9+9+2+7+2+2+9+6+7+6+3+1+2+9 = 81 = 9^2;
10+51+99+27+22+96+76+31+29 = 441 = 21^2;
105+199+272+296+763+129 = 1764 = 42^2;
105199+272296+763129 = 1140624 = 1068^2;
105199272+296763129 = 401962401 = 20049^2.

Cf. A004159, A061910, A153745.

A159879 Numbers n such that digit sum of n^2 is 2*(digit sum of n).

Original entry on oeis.org

0, 2, 11, 20, 27, 36, 54, 72, 74, 81, 92, 101, 108, 110, 128, 135, 144, 153, 162, 171, 191, 200, 209, 218, 225, 227, 252, 254, 261, 270, 317, 326, 344, 353, 360, 371, 387, 405, 416, 425, 504, 506, 515, 540, 605, 641, 684, 711, 720, 722, 731, 740, 767, 774, 801

Offset: 1

Zak Seidov, Apr 25 2009

A007953(n^2) = 2*A007953(n), sod(n^2) = 2*sod(n).

a(n) == {0 or 2} (mod 9). - Robert G. Wilson v, May 27 2009

Zak Seidov, Table of n, a(n) for n=1..3001

A007953 Digital sum (i.e., sum of digits) of n. A004159 Sum of digits of n^2.

fQ[n_] := Plus @@ IntegerDigits[n^2] == 2 Plus @@ IntegerDigits@n; Select[ Range[0, 809], fQ@# &] (* Robert G. Wilson v, May 27 2009 *)

Indices in b-file corrected by N. J. A. Sloane, Aug 31 2009

A166550 Numbers n with property that n^2 and n-th prime have the same sum of digits.

Original entry on oeis.org

4, 8, 11, 22, 34, 59, 61, 85, 179, 229, 260, 266, 352, 385, 391, 403, 418, 440, 491, 565, 595, 619, 724, 770, 832, 844, 961, 965, 980, 1012, 1039, 1069, 1075, 1099, 1108, 1127, 1139, 1148, 1211, 1217, 1390, 1468, 1585, 1589, 1649, 1747, 1780, 1789, 1795, 1799

Offset: 1

Zak Seidov, Oct 16 2009

A007605(n)=A004159(n).

			4^2=16, prime(4)=7, 1+6=7
8^2=64, prime(8)=19, 6+4=1+9
11^2=121, prime(11)=31, 1+2+1=3+1
22^2=484, prime(22)=79, 4+8+4=7+9
34^2=1156, prime(22)=139, 1+1+5+6=1+3+9.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A117224 Numbers for which the square and the cube have the same digital sum, A007605 Sum of digits of n-th prime, A004159 Sum of digits of n^2.

Position[Table[Plus@@IntegerDigits[Prime[n]]-Plus@@IntegerDigits[n^2],{n,1,3000}],0]//Flatten
Select[Range[2000],Total[IntegerDigits[#^2]]==Total[IntegerDigits[ Prime[ #]]]&] (* Harvey P. Dale, May 22 2015 *)

A185076 a(n) is the least number k such that (sum of digits of k^2) + (number of digits of k^2) = n, or 0 if no such k exists.

Original entry on oeis.org

0, 1, 0, 10, 2, 100, 11, 1000, 4, 3, 6, 8, 19, 35, 7, 16, 34, 106, 13, 41, 24, 17, 37, 107, 323, 43, 124, 317, 67, 113, 63, 114, 134, 343, 83, 133, 367, 1024, 167, 374, 264, 314, 386, 1043, 313, 583, 1303, 3283, 707, 1183, 3316, 836, 1333, 3286, 10133

Offset: 1

Carmine Suriano, Feb 23 2011

a(n) < sqrt(10^(n-1)). 0 < a(2m) <= 10^(m-1) with the upper bound reached for 1<=m<=4. - Chai Wah Wu, Mar 15 2023

			a(7)=11 since 7 = sumdigits(121) + numberdigits(121) = 4 + 3.

Chai Wah Wu, Table of n, a(n) for n = 1..185

Cf. A004159, A185679.

Table[k=1; While[d=IntegerDigits[k^2]; n>Length[d] && n != Total[d] + Length[d], k++]; If[Length[d] >= n, k=0]; k, {n, 50}]

from itertools import count
def A185076(n):
    for k in count(1):
        if n == (t:=len(s:=str(k**2)))+sum(map(int,s)):
            return k
        if t >= n:
            return 0 # Chai Wah Wu, Mar 15 2023

n = A004159(a(n)) + A185679(a(n)).

A215618 Sum of digits of n^2 is a square.

Original entry on oeis.org

1, 2, 3, 6, 9, 11, 12, 13, 14, 15, 18, 21, 22, 23, 31, 39, 41, 45, 48, 51, 58, 59, 67, 68, 76, 77, 85, 86, 94, 95, 101, 102, 103, 104, 105, 111, 112, 113, 121, 122, 131, 139, 148, 157, 158, 166, 175, 176, 184, 185, 193, 194, 201, 202, 203, 211, 212, 221, 229

Offset: 1

Zak Seidov, Aug 17 2012

No trailing zeros allowed.

			a(59)=229 because 229^2=52441 and 5+2+4+4+1=16=4^2.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A004159 (sum of digits of n^2).

sdn2Q[n_]:=Module[{idn2=IntegerDigits[n^2]},Last[idn2]>0&&IntegerQ[Sqrt[ Total[idn2]]]]; Select[Range[300],sdn2Q] (* Harvey P. Dale, Aug 27 2013 *)

A224977 n^2 minus sum of digits of n^2.

Original entry on oeis.org

0, 0, 0, 0, 9, 18, 27, 36, 54, 72, 99, 117, 135, 153, 180, 216, 243, 270, 315, 351, 396, 432, 468, 513, 558, 612, 657, 711, 765, 828, 891, 945, 1017, 1071, 1143, 1215, 1278, 1350, 1431, 1512, 1593, 1665, 1746, 1827, 1917, 2016, 2106, 2196, 2295, 2394, 2493

Offset: 0

Kevin L. Schwartz and Christian N. K. Anderson, Apr 21 2013

a(n) mod 9 = 0 for all n.

			a(0) = 0^2 - 0 = 0.
a(12) = 144 - (1+4+4) = 135.

Christian N. K. Anderson, Table of n, a(n) for n = 0..9999
Christian N. K. Anderson, Ulam spiral of a(n) in red, with other multiples of 9 in gray.

Cf. A067552.

Derived from A000290 and A004159.

Table[n^2 - Sum[DigitCount[n^2][[i]]i, {i, 9}], {n, 50}] (* Alonso del Arte, Apr 21 2013 *)
#^2-Total[IntegerDigits[#^2]]&/@Range[0,50] (* Harvey P. Dale, May 10 2017 *)

library(gmp); digsum<-function(x) sum(as.numeric(unlist(strsplit(as.character(x),split=""))))
ans=rep(0,100); n=0
while(n<=100) ans[(n=n+1)]=n^2-digsum(n^2); ans

a(n) = A000290(n) - A004159(n).

A268135 Numbers n such that the digit sum of n^2 is a divisor of the digit sum of n.

Original entry on oeis.org

1, 9, 10, 18, 19, 45, 46, 55, 90, 99, 100, 145, 149, 180, 189, 190, 198, 199, 289, 351, 361, 369, 379, 388, 450, 451, 459, 460, 468, 495, 496, 549, 550, 558, 559, 568, 585, 595, 639, 729, 739, 775, 838, 855, 900, 954, 955, 990, 999, 1000, 1049, 1098, 1099, 1179, 1188, 1189, 1198

Offset: 1

Melvin Peralta, Jan 26 2016

Because A058369 (with offset 1) is a subsequence, this sequence is infinite.
Conjecture: The relative complement of A058369 with respect to this sequence is infinite. That is, there are infinitely many n such that the digit sum of n^2 is a proper divisor of the digit sum of n.
If the digit sum of n^2 is a proper divisor of the digit sum of n, then this property holds for 10*n as well, i.e. the digit sum of n = 149*10^k has as a proper divisor the digit sum of n^2 for all k > 0. Are there infinitely many n that are not a multiple of 10 such that the digit sum of n^2 is a proper divisor of the digit sum of n? The first few such numbers are: 149, 549, 1049, 14499, 19499, 55679, 59499, 64499, 73499, 118499, 144999, 145949, 179249, 244949, 244998, 334679, 347855, 473499, 548735, 549549, 549639, 556965, 837855, ... - Chai Wah Wu, Mar 16 2016

			Digit sum of 149^2 = 7. Digit sum of 149 = 14. Since 7 is a divisor of 14, 149 is in the sequence.

Indranil Ghosh, Table of n, a(n) for n = 1..10001

Cf. A007953, A004159, A058369.

Select[Range[200], Mod[Total[IntegerDigits[#]], Total[IntegerDigits[#^2]]] == 0 &]

isok(n) = (sumdigits(n) % sumdigits(n^2)) == 0; \\ Michel Marcus, Jan 27 2016

More terms from Michel Marcus, Jan 27 2016

cp's OEIS Frontend

A061904 Numbers n such that the iterative cycle: n -> sum of digits of n^2 has only one distinct element.

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A061907 The iterative cycle: n -> sum of digits of n^2 has only four distinct elements.

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A118470 Numbers k for which digitsum(k) + digitsum(k^2) + digitsum(k^3) = digitsum(k^4).

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A153753 Numbers k such that there are 18 digits in k^2 and for each factor f of 18 (1,2,3,6,9) the sum of digit groupings of size f is a square.

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A159879 Numbers n such that digit sum of n^2 is 2*(digit sum of n).

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A166550 Numbers n with property that n^2 and n-th prime have the same sum of digits.

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A185076 a(n) is the least number k such that (sum of digits of k^2) + (number of digits of k^2) = n, or 0 if no such k exists.

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A215618 Sum of digits of n^2 is a square.

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A224977 n^2 minus sum of digits of n^2.

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