A003132 -id:A003132 - cp's OEIS frontend

A108662 Numbers whose sum of squares of digits is a prime.

11, 12, 14, 16, 21, 23, 25, 27, 32, 38, 41, 45, 49, 52, 54, 56, 58, 61, 65, 72, 78, 83, 85, 87, 94, 101, 102, 104, 106, 110, 111, 113, 119, 120, 126, 131, 133, 137, 140, 146, 159, 160, 162, 164, 166, 168, 173, 179, 186, 191, 195, 197, 199, 201, 203, 205, 207, 210

Offset: 1

Zak Seidov, Jun 16 2005

If m is in the sequence, then so are 10*m and any anagram (even with adding zeros between digits) of m. E.g., 12 is a term, hence 21, 102, 120, 201, 10020 all are here.

A sequence of primitive terms is of interest. It starts with 11, 12, 14, 16, 23, 25, 27, 38, 45, 49, 56, 58, 78, 111, 113, 119, 126, 133, 137, 146, 159, 166, 168, 179, 199. Note that digits are in nondecreasing order. - Zak Seidov, Dec 31 2013

			23 is in the sequence because 2^2 + 3^2 = 13 is a prime.

Harvey Dale and Zak Seidov, Table of n, a(n) for n = 1..10000 [First 1000 terms from Harvey Dale]

Cf. A003132, A009994, A052034, A234021.

Select[Range[300],PrimeQ[Total[IntegerDigits[#]^2]]&] (* Harvey P. Dale, May 25 2012 *)

isok(n) = isprime(norml2(digits(n))); \\ Michel Marcus, Jan 09 2019

A072081 Numbers divisible by the square of the sum of their digits in base 10.

Original entry on oeis.org

1, 10, 20, 50, 81, 100, 112, 162, 200, 243, 324, 392, 400, 405, 500, 512, 605, 648, 810, 972, 1000, 1053, 1100, 1120, 1134, 1183, 1215, 1296, 1400, 1620, 1701, 1900, 1944, 2000, 2025, 2106, 2156, 2240, 2268, 2300, 2401, 2430, 2511, 2592, 2704, 2800, 2916

Offset: 1

Labos Elemer, Jun 14 2002

If k is a term, then 10 * k is a term. There are an infinite number of terms that are not divisible by 10. The numbers m = 24 * 10^(42 * k - 40) +1, k >= 1, are divisible by 7^2 = digsum(m)^2. Also, the numbers s = 491 * 10^(42 * k - 8) + 3, k >= 1, are divisible by 17^2 = digsum(s)^2. - Marius A. Burtea, Mar 19 2020

The numbers 2^A095412(n), n >= 5, are terms. - Marius A. Burtea, Apr 02 2020

			k=9477, sumdigits(9477)=27, q=9477=27*27*13.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A003132, A005349, A003634.

[k:k in [1..3000]| k mod &+Intseq(k)^2 eq 0]; // Marius A. Burtea, Mar 19 2020

sud[x_] := Apply[Plus, IntegerDigits[x]] Do[s=sud[n]^2; If[IntegerQ[n/s], Print[n]], {n, 1, 10000}]
Select[Range[3000],Divisible[#,Total[IntegerDigits[#]]^2]&] (* Harvey P. Dale, May 04 2011 *)

for(n=1,10^4,s=sumdigits(n);if(!(n%s^2),print1(n,", "))) \\ Derek Orr, Apr 29 2015

def ok(n): return n and n%sum(di for di in map(int, str(n)))**2 == 0
print([k for k in range(3000) if ok(k)]) # Michael S. Branicky, Jan 10 2025

A076313 a(n) = floor(n/10) - (n mod 10).

Original entry on oeis.org

0, -1, -2, -3, -4, -5, -6, -7, -8, -9, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, 8, 7, 6, 5, 4

Offset: 0

Reinhard Zumkeller, Oct 06 2002

For n<100 equal to the negated alternating digital sum of n (see A055017). - Hieronymus Fischer, Jun 17 2007

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A076314, A010879, A076309, A076310, A076311, A076312, A055017, A076314, A007953, A003132.

a076313 = uncurry (-) . flip divMod 10 -- Reinhard Zumkeller, Jun 01 2013

Table[Floor[n/10]-Mod[n,10],{n,0,100}] (* or *) LinearRecurrence[{1,0,0,0,0,0,0,0,0,1,-1},{0,-1,-2,-3,-4,-5,-6,-7,-8,-9,1},100] (* Harvey P. Dale, Nov 02 2022 *)

a(n)=n\10-n%10 \\ Charles R Greathouse IV, Jan 30 2012

From Hieronymus Fischer, Jun 17 2007: (Start)
a(n) = 11*floor(n/10)-n.
a(n) = (n-11*(n mod 10))/10.
a(n) = 11*A002266(A004526(n))-n=11*A004526(A002266(n))-n.
a(n) = (n-11*A010879(n))/10.
a(n) = (n-11*A000035(n)-22*A010874(A004526(n)))/10.
a(n) = (n-11*A010874(n)-55*A000035(A002266(n)))/10.
G.f.: x*(-8*x^10+11*x^9-1)/((1-x^10)*(1-x)^2). (End)

A080151 Let m = Wonderful Demlo number A002477(n); a(n) = sum of digits of m.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 64, 81, 82, 85, 90, 97, 106, 117, 130, 145, 162, 163, 166, 171, 178, 187, 198, 211, 226, 243, 244, 247, 252, 259, 268, 279, 292, 307, 324, 325, 328, 333, 340, 349, 360, 373, 388, 405, 406, 409, 414, 421, 430, 441, 454, 469, 486, 487

Offset: 1

Eric W. Weisstein, Jan 31 2003

Record values in A003132. - Reinhard Zumkeller, Jul 10 2011

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Demlo Number
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,1,-1).

Cf. A080150, A002477, A080160, A080161, A080162.

a n=(div n 9)*81+(mod n 9)^2
          A080151=map a [1..] \\ Chernin Nadav, Mar 06 2014

f := n -> 9*n - 81*frac(1/9*n) + 81*frac(1/9*n)^2:
map(f, [$1..100]); # Robert Israel, Aug 05 2019

(* by direct counting *)
Repunit[n_] := (-1 + 10^n)/9; A080151[n_]:=Plus @@ IntegerDigits[Repunit[n]^2];
(* by the formula *)
A080151[n_] := (9^2)*(n/9 - FractionalPart[n/9] + FractionalPart[n/9]^2)
(* or alternatively *)
A080151[n_] := 81*(Floor[n/9]+ FractionalPart[n/9]^2) (* Enrique Pérez Herrero, Nov 22 2009 *)

vector(100, n, (n\9)*81+(n%9)^2) \\ Colin Barker, Mar 05 2014

a(n) = A007953(A002477(n)).
a(n) = sqrt( A080150(n) ).
a(n) = (9^2)*(n/9 - {n/9} + {n/9}^2) = 81*(floor(n/9) + {n/9}^2), where the symbol {n} means fractional part of n. - Enrique Pérez Herrero, Nov 22 2009
a(n) = A003132(A051885(n)). - Reinhard Zumkeller, Jul 10 2011
a(9*n + k) = 81*n + k^2, with k in range 0 to 9. - Enrique Pérez Herrero, Nov 05 2022
Empirical g.f.: x*(17*x^8 + 15*x^7 + 13*x^6 + 11*x^5 + 9*x^4 + 7*x^3 + 5*x^2 + 3*x + 1) / ((x-1)^2*(x^2+x+1)*(x^6+x^3+1)). - Colin Barker, Mar 05 2014
Empirical g.f. confirmed. - Robert Israel, Aug 05 2019

A376270 a(n) is the product of the leading digit of n and the sum of the squares of its digits.

Original entry on oeis.org

0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1, 2, 5, 10, 17, 26, 37, 50, 65, 82, 8, 10, 16, 26, 40, 58, 80, 106, 136, 170, 27, 30, 39, 54, 75, 102, 135, 174, 219, 270, 64, 68, 80, 100, 128, 164, 208, 260, 320, 388, 125, 130, 145, 170, 205, 250, 305, 370, 445, 530, 216, 222, 240, 270, 312, 366

Offset: 0

Michel Marcus, Sep 18 2024

Alois P. Heinz, Table of n, a(n) for n = 0..10000
N. Bradley Fox et al., Elated Numbers, arXiv:2409.09863 [math.NT], 2024.

Cf. A000030, A003132.

b-elated function: A000120 (2), A376270 (10).

a:= n-> (l-> l[-1]*add(i^2, i=l))(convert(n, base, 10)):
seq(a(n), n=0..65);  # Alois P. Heinz, Sep 18 2024

a[n_]:=First[d=IntegerDigits[n]]Norm[d]^2; Array[a,66,0] (* Stefano Spezia, Sep 18 2024 *)

a(n) = if (n, my(d=digits(n)); d[1]*norml2(d), 0);

def a(n): return (d:=list(map(int, str(n))))[0] * sum(di*di for di in d)
print([a(n) for n in range(66)]) # Michael S. Branicky, Sep 18 2024

a(n) = A000030(n)*A003132(n).

A055015 Sum of 6th powers of digits of n.

Original entry on oeis.org

0, 1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1, 2, 65, 730, 4097, 15626, 46657, 117650, 262145, 531442, 64, 65, 128, 793, 4160, 15689, 46720, 117713, 262208, 531505, 729, 730, 793, 1458, 4825, 16354, 47385, 118378, 262873

Offset: 0

Henry Bottomley, May 31 2000

The only fixed points (n = 0, 1 and 548834) are listed in row 6 of A252648. - M. F. Hasler, Apr 12 2015

Michel Marcus, Table of n, a(n) for n = 0..999
Index entries for Colombian or self numbers and related sequences

Cf. A003132.
Cf. A003132, A055012, A055013, A005514.
Cf. A007953, A055017, A076313, A076314.

[0] cat [&+[d^6: d in Intseq(n)]: n in [1..40]]; // Bruno Berselli, Feb 01 2013

for n from 0 to 3 do seq(n^6+j^6, j=0..9 ); od; # Zerinvary Lajos, Nov 06 2006

Table[Sum[DigitCount[n][[i]] i^6, {i, 9}], {n, 0, 40}] (* Bruno Berselli, Feb 01 2013 *)

A055015(n)=sum(i=1,#n=digits(n),n[i]^6) \\ M. F. Hasler, Apr 12 2015

a(n) = Sum_{k>0} (floor(n/10^k) - 10*floor(n/10^(k+1)))^6. - Hieronymus Fischer, Jun 25 2007

a(10n+k) = a(n) + k^6, 0 <= k < 10. - Hieronymus Fischer, Jun 25 2007

A257588 If n = abcd... in decimal, a(n) = |a^2 - b^2 + c^2 - d^2 + ...|.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 1, 0, 3, 8, 15, 24, 35, 48, 63, 80, 4, 3, 0, 5, 12, 21, 32, 45, 60, 77, 9, 8, 5, 0, 7, 16, 27, 40, 55, 72, 16, 15, 12, 7, 0, 9, 20, 33, 48, 65, 25, 24, 21, 16, 9, 0, 11, 24, 39, 56, 36, 35, 32, 27, 20, 11, 0, 13, 28, 45, 49

Offset: 0

N. J. A. Sloane, May 10 2015

a(n) = 0 iff n is in A352535. - Bernard Schott, Jul 08 2022

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A257587, A257796.

Cf. A225693, A003132, A352535.

a257588 = abs . f 1 where
   f _ 0 = 0
   f s x = s * d ^ 2 + f (negate s) x' where (x', d) = divMod x 10
-- Reinhard Zumkeller, May 10 2015

a:= n-> (l-> abs(add(l[i]^2*(-1)^i, i=1..nops(l))))(convert(n, base, 10)):
seq(a(n), n=0..70);  # Alois P. Heinz, Mar 24 2022

Array[Abs@ Total@ MapIndexed[(2 Boole@ EvenQ[First[#2]] - 1) (#1^2) &, IntegerDigits[#]] &, 70] (* Michael De Vlieger, Feb 27 2022 *)

a(n) = my(d=digits(n)); abs(sum(k=1, #d, (-1)^k*d[k]^2)); \\ Michel Marcus, Feb 27 2022

def A257588(n):
    return abs(sum((int(d)**2*(-1)**j for j,d in enumerate(str(n)))))
# Chai Wah Wu, May 10 2015

A258881 a(n) = n + the sum of the squared digits of n.

Original entry on oeis.org

0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 11, 13, 17, 23, 31, 41, 53, 67, 83, 101, 24, 26, 30, 36, 44, 54, 66, 80, 96, 114, 39, 41, 45, 51, 59, 69, 81, 95, 111, 129, 56, 58, 62, 68, 76, 86, 98, 112, 128, 146, 75, 77, 81, 87, 95, 105, 117, 131, 147, 165, 96, 98, 102

Offset: 0

M. F. Hasler, Jul 19 2015

Carlos Rivera, Puzzle 776. Ten consecutive integers such that..., The Prime Puzzles and Problems Connection, March 2015.

Cf. A003132, A062028, A259391, A259567, A033936, A076161 (indices of primes), A329179 (indices of squares).

Total[Flatten@ {#, IntegerDigits[#]^2}] & /@ Range[0, 61] (* Michael De Vlieger, Jul 20 2015 *)
Table[n+Total[IntegerDigits[n]^2],{n,0,100}] (* Harvey P. Dale, Nov 27 2022 *)

PARI
```
A258881(n)=n+norml2(digits(n))
```

def ssd(n): return sum(int(d)**2 for d in str(n))
def a(n): return n + ssd(n)
print([a(n) for n in range(63)]) # Michael S. Branicky, Jan 30 2021

A001273 Smallest number that takes n steps to reach 1 under iteration of sum-of-squares-of-digits map (= smallest "happy number" of height n).

Original entry on oeis.org

1, 10, 13, 23, 19, 7, 356, 78999

Offset: 0

N. J. A. Sloane

Subsequent terms are too large to display in full.
a(8) = 3789 * 10^973 - 1 (3788 followed by 973 9's).
a(9) = 78889 * 10^((a(8) - 305)/81) - 1 (78888 followed by (421 * 10^973 - 34)/9 9's, specified by Warut Roonguthai for UPINT3).
a(10) = 259 * 10^((a(9) - 93)/81) - 1.
a(11) = 179 * 10^((a(10) - 114)/81) - 1.
a(12) = 47 * 10^((a(11) - 52)/81) - 1.
From Ya-Ping Lu, Jul 26 2025: (Start)
a(13) = 137 * 10^((a(12) - 46)/81) - 1.
a(14) = 1128 * 10^((a(13) - 55)/81) - 1.
a(15) = 58 * 10^((a(14) - 74)/81) - 1.
a(16) = 228 * 10^((a(15) - 57)/81) - 1. (End)

Richard K. Guy, Unsolved Problems in Number Theory, Sect. E34. (2nd ed. UPINT2 = 1994, 3rd ed. UPINT3 = 2004)

Tianxin Cai and Xia Zhou, On The Heights of Happy Numbers, Rocky Mountain J. Math., Vol. 38, No. 6 (2008), 1921-1926.
H. G. Grundman and E. A. Teeple, Heights of happy numbers and cubic happy numbers, Fib Quart. 41 (4) (2003) 301-306.
Hans Havermann, Big and Happy
Gabriel Lapointe, On finding the smallest happy numbers of any heights, arXiv:1904.12032 [math.NT], 2019.
May Mei and Andrew Read-McFarland, Numbers and the Heights of their Happiness, arXiv:1511.01441 [math.NT], 2015.

Cf. A003132, A007770, A018785, A176762.

f = lambda h: sum(int(d)**2 for d in str(h)); a = 356; n_mx = 19
for n in range(7, n_mx+1):
    b = a%81; a1 = max(a%(2*3**(3*(n_mx+1-n))), b); t = max(a1//81-6,0); h = 1
    while f((h+1)*10**t - 1) != a1:
        h += 1; s = str(h)
        if '0' in s: p0 = s.index('0'); c = 10**(len(s)-p0); h = h//c*c + int(s[p0-1])*(c-1)//9
    c9 = str(h).count('9'); hc = h//(10**c9); a = (hc+1)*10**((a1-f(hc))//81)-1
    print('a(',n,') =', hc+1,'x 10 ^ ( ( a(',n-1,') -', f(hc),') / 81) - 1')  # Ya-Ping Lu, Jul 26 2025

For n >= 7, a(n) = k(n)*10^((a(n-1)-A003132(k(n)-1))/81)-1, where k(n) = 79, 3789, 78889, 259, 179, 47, 137, 1128, 58, 228, 19, 34, 145 for n = 7, 8,.., 19. - Ya-Ping Lu, Jul 27 2025

a(7), a(8) from Jud McCranie, Sep 15 1994
a(9)-a(12) from Hans Havermann, May 02 2010
Edited by Hans Havermann, May 03 2010, May 04 2010

A175396 Numbers whose sum of squares of digits is a square.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 34, 40, 43, 50, 60, 68, 70, 80, 86, 90, 100, 122, 148, 184, 200, 212, 221, 236, 244, 263, 269, 296, 300, 304, 326, 340, 362, 366, 400, 403, 418, 424, 430, 442, 447, 474, 481, 488, 500, 600, 608, 623, 629, 632, 636, 663

Offset: 1

Ctibor O. Zizka, Apr 30 2010

Previous name: Numbers n such that Sum_{i=1..r, x(i)^2} is a perfect square, where x(i) = digits of n. r=1+floor(log_10 n).

			34 is a term: 3^2 + 4^2 = 25 = 5^2.
122 is a term: 1^2 + 2^2 + 2^2 = 9 = 3^2.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Cf. A000290, A003132, A084561.

Select[Range[0, 666], IntegerQ[Sqrt[Plus @@ (IntegerDigits[#]^2)]] &] (* Ivan Neretin, Aug 03 2015 *)

isok(n) = {my(digs = digits(n)); issquare(sum(i=1, #digs, digs[i]^2))} \\ Michel Marcus, Jun 02 2013

from math import isqrt
def ok(n): s = sum(int(i)**2 for i in str(n)); return isqrt(s)**2 == s
print(*[k for k in range(664) if ok(k)], sep = ', ')  # Ya-Ping Lu, Jul 07 2025

Corrected and extended by Neven Juric, Jul 12 2010

Simpler definition by Michel Marcus, Jun 02 2013

cp's OEIS Frontend

A108662 Numbers whose sum of squares of digits is a prime.

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A072081 Numbers divisible by the square of the sum of their digits in base 10.

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A076313 a(n) = floor(n/10) - (n mod 10).

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A080151 Let m = Wonderful Demlo number A002477(n); a(n) = sum of digits of m.

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A376270 a(n) is the product of the leading digit of n and the sum of the squares of its digits.

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A055015 Sum of 6th powers of digits of n.

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A257588 If n = abcd... in decimal, a(n) = |a^2 - b^2 + c^2 - d^2 + ...|.

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