A003132 -id:A003132 - cp's OEIS frontend

A055017 Difference between sums of alternate digits of n starting with the last, i.e., (sum of ultimate digit of n, antepenultimate digit of n, ...) - (sum of penultimate digit of n, preantepenultimate digit of n, ...).

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, -3

Offset: 0

Henry Bottomley, May 31 2000

a(n) is a multiple of 11 iff n is divisible by 11.
Digital sum with alternating signs starting with a positive sign for the rightmost digit. - Hieronymus Fischer, Jun 18 2007
For n < 100, a(n) = (n mod 10 - floor(n/10)) = -A076313(n). - Hieronymus Fischer, Jun 18 2007

			a(123) = 3-2+1 = 2, a(9875) = 5-7+8-9 = -3.

Hieronymus Fischer, Table of n, a(n) for n = 0..10000

Cf. A225693 (alternating sum of digits).
Unsigned version differs from A040114 and A040115 when n=100 and from A040997 when n=101.
Cf. A076313, A076314, A007953, A003132.
Cf. A004086.
Cf. analogous sequences for bases 2-9: A065359, A065368, A346688, A346689, A346690, A346691, A346731, A346732 and also A373605 (for primorial base).

sumodigs := proc(n) local dg; dg := convert(n,base,10) ; add(op(1+2*i,dg), i=0..floor(nops(dg)-1)/2) ; end proc:
sumedigs := proc(n) local dg; dg := convert(n,base,10) ; add(op(2+2*i,dg), i=0..floor(nops(dg)-2)/2) ; end proc:
A055017 := proc(n) sumodigs(n)-sumedigs(n) ; end proc: # R. J. Mathar, Aug 26 2011

def A055017(n): return sum((-1 if i % 2 else 1)*int(j) for i, j in enumerate(str(n)[::-1])) # Chai Wah Wu, May 11 2022

"Recursive version for general bases"
"Set base = 10 for this sequence"
altDigitalSumRight: base
| s |
base = 1 ifTrue: [^self \\ 2].
(s := self // base) > 0
  ifTrue: [^(self - (s * base) - (s altDigitalSumRight: base))]
  ifFalse: [^self]
[by Hieronymus Fischer, Mar 23 2014]

From Hieronymus Fischer, Jun 18 2007, Jun 25 2007, Mar 23 2014: (Start)
a(n) = n + 11*Sum_{k>=1} (-1)^k*floor(n/10^k).
a(10n+k) = k - a(n), 0 <= k < 10.
G.f.: Sum_{k>=1} (x^k-x^(k+10^k)+(-1)^k*11*x^(10^k))/((1-x^(10^k))*(1-x)).
a(n) = n + 11*Sum_{k=10..n} Sum_{j|k,j>=10} (-1)^floor(log_10(j))*(floor(log_10(j)) - floor(log_10(j-1))).
G.f. expressed in terms of Lambert series: g(x) = (x/(1-x)+11*L[b(k)](x))/(1-x) where L[b(k)](x) = Sum_{k>=0} b(k)*x^k/(1-x^k) is a Lambert series with b(k) = (-1)^floor(log_10(k)) if k>1 is a power of 10, otherwise b(k)=0.
G.f.: (1/(1-x)) * Sum_{k>=1} (1+11*c(k))*x^k, where c(k) = Sum_{j>=2,j|k} (-1)^floor(log_10(j))*(floor(log_10(j))-floor(log_10(j-1))).
Formulas for general bases b > 1 (b = 10 for this sequence).
a(n) = Sum_{k>=0} (-1)^k*(floor(n/b^k) mod b).
a(n) = n + (b+1)*Sum_{k>=1} (-1)^k*floor(n/b^k). Both sums are finite with floor(log_b(n)) as the highest index.
a(n) = a(n mod b^k) + (-1)^k*a(floor(n/b^k)), for all k >= 0.
a(n) = a(n mod b) - a(floor(n/b)).
a(n) = a(n mod b^2) + a(floor(n/b^2)).
a(n) = (-1)^m*A225693(n), where m = floor(log_b(n)).
a(n) = (-1)^k*A225693(A004086(n)), where k = is the number of trailing 0's of n, formally, k = max(j | n == 0 (mod 10^j)).
a(A004086(A004086(n))) = (-1)^k*a(n), where k = is the number of trailing 0's in the decimal representation of n. (End)

A055013 Sum of 4th powers of digits of n.

Original entry on oeis.org

0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 1, 2, 17, 82, 257, 626, 1297, 2402, 4097, 6562, 16, 17, 32, 97, 272, 641, 1312, 2417, 4112, 6577, 81, 82, 97, 162, 337, 706, 1377, 2482, 4177, 6642, 256, 257, 272, 337, 512, 881, 1552, 2657, 4352, 6817

Offset: 0

Henry Bottomley, May 31 2000

Fixed points are listed in A052455, row 4 of A252648. See also A061210. - M. F. Hasler, Apr 12 2015

Seiichi Manyama, Table of n, a(n) for n = 0..10000
K. Chikawa, K. Iséki, and T. Kusakabe, On a problem by H. Steinhaus, Acta Arithmetica 7 (1962), 251-252. - _Don Knuth_, Sep 07 2015
Index entries for Colombian or self numbers and related sequences

Cf. A003132, A055012.
Cf. A007953, A055017, A076313, A076314.
Cf. A052455, A252648; A061210.

[0] cat [&+[d^4: d in Intseq(n)]: n in [1..50]]; // Bruno Berselli, Feb 01 2013

A055013 := proc(n)
        add(d^4,d=convert(n,base,10)) ;
end proc:
seq(A055013(n),n=0..20) ; # R. J. Mathar, Nov 07 2011

Table[Sum[DigitCount[n][[i]] i^4, {i, 9}], {n, 0, 50}] (* Bruno Berselli, Feb 01 2013 *)
Table[Total[IntegerDigits[n]^4],{n,0,50}] (* Harvey P. Dale, Jul 28 2019 *)

a(n)=round(normlp(n,4)^4) \\ Quite slow. - M. F. Hasler, Apr 12 2015

A055013(n)=sum(i=1,#n=digits(n),n[i]^4) \\ M. F. Hasler, Apr 12 2015

a(n) = sum{k>0, (floor(n/10^k)-10*floor(n/10^(k+1)))^4}. - Hieronymus Fischer, Jun 25 2007

a(10n+k) = a(n)+k^4, 0<=k<10. - Hieronymus Fischer, Jun 25 2007

A031177 Unhappy numbers: numbers having period-8 2-digitized sequences.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 29, 30, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 83

Offset: 1

Eric W. Weisstein

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digitaddition
Eric Weisstein's World of Mathematics, Unhappy Number

Complement of happy numbers A007770. Cf. A056527.

Cf. A003132, A103369.

a031177 n = a031177_list !! (n-1)
a031177_list = filter ((/= 1) . a103369) [1..]
-- Reinhard Zumkeller, Aug 24 2011

from itertools import count, islice
def A031177_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        m = n
        while m not in {1,37,58,89,145,42,20,4,16}:
            m = sum((0, 1, 4, 9, 16, 25, 36, 49, 64, 81)[ord(d)-48] for d in str(m))
        if m > 1:
            yield n
A031177_list = list(islice(A031177_gen(),20)) # Chai Wah Wu, Aug 02 2023

A039943 Every integer eventually goes to one of these under the "x goes to sum of squares of digits of x" map.

Original entry on oeis.org

0, 1, 4, 16, 20, 37, 42, 58, 89, 145

Offset: 0

N. J. A. Sloane

The subset of the first three terms also satisfies the current definition. An alternate definition would be: Periodic points of A003132. - M. F. Hasler, May 24 2009

Following I. Ja. Tanatar (Moscow), one can easily prove that, for a given x, there exists an iteration of the map f(x) given in the definition which reaches 1 or 89. Indeed, it is easy to see that if x has at least 3 digits, then f(x) < x. Therefore there exists an iteration of f with not more than 2 digits. For two-digit numbers the property is verified directly. See Kordemsky. - Vladimir Shevelev, May 06 2013

B. A. Kordemsky, Matematicheskaja Smekalka, Moscow, 1955, pp. 305 and 557 (in Russian).

Arthur Porges, A set of eight numbers, Amer. Math. Monthly 52 (1945), 379-382.
Arthur Porges, A set of eight numbers, Amer. Math. Monthly, 52 (1945), 379-382. [Annotated scanned copy]
Eric W. Weisstein, MathWorld: Happy Number
Wikipedia, Periodic point

Cf. A000216, A003621.

Cf. A003132 (the iterated map), A003621, A039943, A031176, A007770, A000216 (orbit of 2), A000218 (orbit of 3), A080709 (orbit of 4, the only nontrivial limit cycle), A000221 (orbit of 5), A008460 (orbit of 6), A008462 (orbit of 8), A008463 (orbit of 9), A139566 (orbit of 15), A122065 (orbit of 74169).

a039943 n = a039943_list !! n
a039943_list = [0,1,4,16,20,37,42,58,89,145]
-- Reinhard Zumkeller, Mar 16 2013

lst = {}; Do[a = NestWhile[Plus @@ (IntegerDigits@#^2) &, n, Unequal, All]; If[FreeQ[lst, a], AppendTo[lst, a]], {n, 10^4}] (* Robert G. Wilson v, Jan 19 2006 *)
Union[Table[NestWhile[Total[IntegerDigits[#]^2]&,n,Unequal,All],{n,0,100}]] (* Harvey P. Dale, Nov 26 2013 *)

A000216 Take sum of squares of digits of previous term, starting with 2.

Original entry on oeis.org

2, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37

Offset: 1

N. J. A. Sloane

As the orbit of 2 under A003132, this could also have offset 0. Merges into A080709 right after the first term: a(n+1) = A080709(n) for all n >= 1. - M. F. Hasler, Apr 27 2018

R. Honsberger, Ingenuity in Mathematics, Random House, 1970, p. 83.
P. Kiss, A generalization of a problem in number theory, Math. Sem. Notes Kobe Univ., 5 (1977), no. 3, 313-317. MR 0472667 (57 #12362).

Vincenzo Librandi, Table of n, a(n) for n = 1..100
H. G. Grundmann, Semihappy Numbers, J. Int. Seq. 13 (2010), 10.4.8, Theorem 1.
P. Kiss, A generalization of a problem in number theory, [Hungarian], Mat. Lapok, 25 (No. 1-2, 1974), 145-149.
Arthur Porges, A set of eight numbers, Amer. Math. Monthly 52 (1945), 379-382.
A. Porges, A set of eight numbers, Amer. Math. Monthly, 52 (1945), 379-382. [Annotated scanned copy]
H. J. J. te Riele, Iteration of number-theoretic functions, Nieuw Archief v. Wiskunde, (4) 1 (1983), 345-360.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 1).

Cf. A003132 (the iterated map), A003621, A039943, A099645, A031176, A007770, A000218 (starting with 3), A080709 (starting with 4), A000221 (starting with 5), A008460 (starting with 6), A008462 (starting with 8), A008463 (starting with 9), A139566 (starting with 15), A122065 (starting with 74169). - M. F. Hasler, May 24 2009

a000216 n = a000216_list !! (n-1)
a000216_list = iterate a003132 2 -- Reinhard Zumkeller, Aug 24 2011

[2] cat &cat[[4, 16, 37, 58, 89, 145, 42, 20]: n in [0..17]]; // Vincenzo Librandi, Jan 29 2013

NestList[Total[IntegerDigits[#]^2]&, 2, 80] (* Vincenzo Librandi, Jan 29 2013 *)

A000216(n)=[42, 20, 4, 16, 37, 58, 89, 145, 2][n%8+8^(n<2)] \\ M. F. Hasler, May 24 2009, edited Apr 27 2018

Periodic with period 8.

A055014 Sum of 5th powers of digits of n.

Original entry on oeis.org

0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 1, 2, 33, 244, 1025, 3126, 7777, 16808, 32769, 59050, 32, 33, 64, 275, 1056, 3157, 7808, 16839, 32800, 59081, 243, 244, 275, 486, 1267, 3368, 8019, 17050, 33011, 59292, 1024, 1025, 1056, 1267, 2048

Offset: 0

Henry Bottomley, May 31 2000

Fixed points are listed in A052464. - M. F. Hasler, Apr 12 2015

Seiichi Manyama, Table of n, a(n) for n = 0..10000
K. Chikawa, K. Iséki, T. Kusakabe, and K. Shibamura, Computation of cyclic parts of Steinhaus problem for power 5, Acta Arithmetica 7 (1962), 253-254. [From _Don Knuth_, Sep 07 2015]
Index entries for Colombian or self numbers and related sequences

Cf. A003132, A055012, A055013.
Cf. A007953, A055017, A076313, A076314.
Cf. A052464.

[0] cat [&+[d^5: d in Intseq(n)]: n in [1..45]]; // Bruno Berselli, Feb 01 2013

A055014 := proc(n)
   add(d^5,d=convert(n,base,10)) ;
end proc: # R. J. Mathar, Jul 08 2012

Total/@(IntegerDigits[Range[50]]^5)  (* Harvey P. Dale, Jan 22 2011 *)
Table[Sum[DigitCount[n][[i]] i^5, {i, 9}], {n, 0, 45}] (* Bruno Berselli, Feb 01 2013 *)

A055014(n)=sum(i=1, #n=digits(n), n[i]^5) \\ M. F. Hasler, Apr 12 2015

a(n) = Sum_{k>=1} (floor(n/10^k) - 10*floor(n/10^(k+1)))^5. - Hieronymus Fischer, Jun 25 2007

a(10n+k) = a(n) + k^5, 0 <= k < 10. - Hieronymus Fischer, Jun 25 2007

A031176 Periods of sum of squares of digits iterated until the sequence becomes periodic.

Original entry on oeis.org

1, 1, 8, 8, 8, 8, 8, 1, 8, 8, 1, 8, 8, 1, 8, 8, 8, 8, 8, 1, 8, 8, 8, 1, 8, 8, 8, 8, 1, 8, 8, 1, 1, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 1, 8, 8, 8, 8, 1, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 1, 8, 1, 8, 8, 8, 8, 8, 8, 8, 8, 1, 8, 8, 1, 8, 8, 8, 1, 8, 8, 8, 8, 1, 8, 8, 1, 8, 8, 1, 8, 8

Offset: 0

Eric W. Weisstein

Eric Weisstein's World of Mathematics, Digit addition.

Cf. A003132, A099645. [From R. J. Mathar, May 31 2009]

The value a(0)=1 was added by R. J. Mathar, May 31 2009

A052034 Primes such that the sum of the squares of their digits is also a prime.

Original entry on oeis.org

11, 23, 41, 61, 83, 101, 113, 131, 137, 173, 179, 191, 197, 199, 223, 229, 311, 313, 317, 331, 337, 353, 373, 379, 397, 401, 409, 443, 449, 461, 463, 467, 601, 641, 643, 647, 661, 683, 719, 733, 739, 773, 797, 829, 863, 883, 911, 919, 937, 971, 977, 991, 997, 1013

Offset: 1

Patrick De Geest, Dec 15 1999

Primes p such that the sum of the squared digits of p is a prime q. For the values of q see A109181.

			p = 23 is in the sequence because q = 2^2 + 3^2 = 13 is a prime.
9431 -> 9^2 + 4^2 + 3^2 + 1^2 = 107 (which is prime).

Clifford A. Pickover, A Passion for Mathematics, John Wiley & Sons, Inc., 2005, p. 89.
Charles W. Trigg, Journal of Recreational Mathematics, Vol. 20(2), 1988.

Zak Seidov, Table of n, a(n) for n = 1..10000
Mike Mudge, Morph code, Hands On Numbers Count, Personal Computer World, May 1997, p. 290.

Cf. A003132, A052035, A091367, A108662, A109181.

a:=proc(n) local nn, L: nn:=convert(n,base,10): L:=nops(nn): if isprime(n)= true and isprime(add(nn[j]^2,j=1..L))=true then n else end if end proc: seq(a(n),n=1..1000); # Emeric Deutsch, Jan 08 2008

Select[Prime[Range[250]],PrimeQ[Total[IntegerDigits[#]^2]]&]  (* Harvey P. Dale, Dec 19 2010 *)

from sympy import isprime, primerange
def ok(p): return isprime(sum(int(d)**2 for d in str(p)))
def aupto(limit): return [p for p in primerange(1, limit+1) if ok(p)]
print(aupto(1013)) # Michael S. Branicky, Nov 23 2021

Edited by N. J. A. Sloane, Dec 15 2007 and again on Dec 05 2008 at the suggestion of Zak Seidov

A000218 Take sum of squares of digits of previous term; start with 3.

Original entry on oeis.org

3, 9, 81, 65, 61, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37

Offset: 1

N. J. A. Sloane

Could also have offset 0, considered as the orbit of 3 under A003132, i.e., n-fold application of A003132 on the initial value 3. - M. F. Hasler, Apr 27 2018

R. Honsberger, Ingenuity in Math., Random House, 1970, p. 83.

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Arthur Porges, A set of eight numbers, Amer. Math. Monthly 52 (1945), 379-382.
A. Porges, A set of eight numbers, Amer. Math. Monthly, 52 (1945), 379-382. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 1).

Cf. A003132 (the iterated map), A003621, A039943, A099645, A031176, A007770, A000216 (starting with 2), A080709 (starting with 4), A000221 (starting with 5), A008460 (starting with 6), A008462 (starting with 8), A008463 (starting with 9), A139566 (starting with 15), A122065 (starting with 74169). - M. F. Hasler, May 24 2009

a000218 n = a000218_list !! (n-1)
a000218_list = iterate a003132 3
-- Reinhard Zumkeller, Aug 24 2011

[3, 9, 81, 65, 61] cat &cat[[37, 58, 89, 145, 42, 20, 4, 16]: n in [0..17]]; // Vincenzo Librandi, Jan 29 2013

NestList[Total[IntegerDigits[#]^2]&, 3, 80] (* Vincenzo Librandi, Jan 29 2013 *)

A000218(n)=[89, 145, 42, 20, 4, 16, 37, 58, 3, 9, 81, 65, 61][n%8+8^(n<6)] \\ M. F. Hasler, May 24 2009, edited Apr 27 2018

Eventually periodic with period 8.

A099645 Number of iterations until n reaches a number in A039943 under "x goes to sum of squares of digits of x" map.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Nov 08 2004

Length of transient when the f[n]=Sum[digit^2 of n] function is iterated.

In A031176 including cycle lengths[=c] of this iteration only c=1 and c=8 occur. A007770 lists cases of c=1, the happy numbers.

			n=99999999999: iteration-list={99999999999,891,146,53,34,25,29,85,89,145,42,20,[4,16,37,58,89,145,42,20],4,...}. Lengths of transient=12, of cycle=8.

Hugo Steinhaus: "Sto zadan" (1958), "One Hundred Problems in Elementary Mathematics" (1964), problem 2. - M. F. Hasler, May 24 2009

T. D. Noe, Table of n, a(n) for n = 1..1000
Arthur Porges, A set of eight numbers, Amer. Math. Monthly 52 (1945), 379-382. - _M. F. Hasler_, May 24 2009

Cf. A007700, A031176.
Cf. A000216, A003132, A003621
Cf. A039943, A031176, A007770, A000216 (orbit of 2), A000218 (orbit of 3), A080709 (orbit of 4), A000221 (orbit of 5), A008460 (orbit of 6), A008462 (orbit of 8), A008463 (orbit of 9), A139566 (orbit of 15), A122065 (orbit of 74169). - M. F. Hasler, May 24 2009

a099645 = length . takeWhile (`notElem` a039943_list) . iterate a003132
a099645_list = map a099645 [1..]
-- Reinhard Zumkeller, Aug 24 2011

fu[x_] :=Apply[Plus, IntegerDigits[x]^2];hs=20; (* transient lengths are obtained by: *) a[n_] :=-1+Min[Flatten[Position[NestList[fu, n, Length[Union[NestList[fu, n, hs]]]] -Last[NestList[fu, n, Length[Union[NestList[fu, n, hs]]]]], 0]]]; Table[a[n], {n, 1, 256}]

A099645(n)={ local( c=0, S=Set([1,4,16,37,58,89,145,42,20])); while( !setsearch(S,n), n=A003132(n); c++); c} \\ M. F. Hasler, May 24 2009

Terms checked using the given PARI code. However, according to the domain of A003132 and the definition of A039943 (which both include 0), an initial a(0)=0 should be added here, too. - M. F. Hasler, May 24 2009

cp's OEIS Frontend

A055017 Difference between sums of alternate digits of n starting with the last, i.e., (sum of ultimate digit of n, antepenultimate digit of n, ...) - (sum of penultimate digit of n, preantepenultimate digit of n, ...).

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A055013 Sum of 4th powers of digits of n.

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A031177 Unhappy numbers: numbers having period-8 2-digitized sequences.

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A039943 Every integer eventually goes to one of these under the "x goes to sum of squares of digits of x" map.

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A000216 Take sum of squares of digits of previous term, starting with 2.

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A055014 Sum of 5th powers of digits of n.

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A031176 Periods of sum of squares of digits iterated until the sequence becomes periodic.

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