A052034 -id:A052034 - cp's OEIS frontend

A109181 a(n) = A003132(A052034(n)).

2, 13, 17, 37, 73, 2, 11, 11, 59, 59, 131, 83, 131, 163, 17, 89, 11, 19, 59, 19, 67, 43, 67, 139, 139, 17, 97, 41, 113, 53, 61, 101, 37, 53, 61, 101, 73, 109, 131, 67, 139, 107, 179, 149, 109, 137, 83, 163, 139, 131, 179, 163, 211, 11, 83, 11, 19, 83, 131, 11, 83, 47, 67, 103, 11, 19, 59, 47, 107, 43, 67, 107, 179, 47, 127, 167, 199, 131, 67, 163

Offset: 1

Zak Seidov, Jun 21 2005

For the primes p see A052034.

			q=13 is a term because 13 = 2^2 + 3^2 and merging digits 2 and 3 makes p=23, which is a prime.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A003132, A052034.

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 add(nn[j]^2,j=1..L) else end if end proc: seq(a(n),n=1..1200); # Emeric Deutsch, Jan 08 2008

a(n) = A003132(A052034(n)). - Zak Seidov, Dec 30 2013

More terms from Emeric Deutsch and Alvin Hoover Belt, Jan 08 2008

A083444 Erroneous duplicate of A052034.

Original entry on oeis.org

23, 41, 61, 83, 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

dead

A003132 Sum of squares of digits of n.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 1, 2, 5, 10, 17, 26, 37, 50, 65, 82, 4, 5, 8, 13, 20, 29, 40, 53, 68, 85, 9, 10, 13, 18, 25, 34, 45, 58, 73, 90, 16, 17, 20, 25, 32, 41, 52, 65, 80, 97, 25, 26, 29, 34, 41, 50, 61, 74, 89, 106, 36, 37, 40, 45, 52, 61, 72, 85, 100, 117, 49

Offset: 0

N. J. A. Sloane

It is easy to show that a(n) < 81*(log_10(n)+1). - Stefan Steinerberger, Mar 25 2006
It is known that a(0)=0 and a(1)=1 are the only fixed points of this map. For more information about iterations of this map, see A007770, A099645 and A000216 ff. - M. F. Hasler, May 24 2009
Also known as the "Happy number map", since happy numbers A007770 are those whose trajectory under iterations of this map ends at 1. - M. F. Hasler, Jun 03 2025

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Hugo Steinhaus, One Hundred Problems in Elementary Mathematics, Dover New York, 1979, republication of English translation of Sto Zadań, Basic Books, New York, 1964. Chapter I.2, An interesting property of numbers, pp. 11-12 (available on Google Books).

T. D. Noe, Table of n, a(n) for n = 0..10000
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29; [preprint from author's web page, PostScript format].
Arthur Porges, A set of eight numbers, Amer. Math. Monthly 52 (1945), 379-382.
B. M. Stewart, Sums of functions of digits, Canad. J. Math., 12 (1960), 374-389.
Robert Walker, Self Similar Sloth Canon Number Sequences
Index entries for Colombian or self numbers and related sequences

Cf. A052034, A052035.
Cf. A007953, A055017, A076313, A076314.
Concerning iterations of this map, see A003621, A039943, A099645, A031176, A007770, A000216 (starting with 2), A000218 (starting with 3), A080709 (starting with 4, this is the only nontrivial limit cycle), 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
Cf. A080151, A051885 (record values and where they occur).
Cf. A257588, A332919.

a003132 0 = 0
a003132 x = d ^ 2 + a003132 x' where (x', d) = divMod x 10
-- Reinhard Zumkeller, May 10 2015, Aug 07 2012, Jul 10 2011

[0] cat [&+[d^2: d in Intseq(n)]: n in [1..80]]; // Bruno Berselli, Feb 01 2013

A003132 := proc(n) local d; add(d^2,d=convert(n,base,10)) ; end proc: # R. J. Mathar, Oct 16 2010

Table[Sum[DigitCount[n][[i]]*i^2, {i, 1, 9}], {n, 0, 40}] (* Stefan Steinerberger, Mar 25 2006 *)
Total/@(IntegerDigits[Range[0,80]]^2) (* Harvey P. Dale, Jun 20 2011 *)

A003132(n)=norml2(digits(n)) \\ M. F. Hasler, May 24 2009, updated Apr 12 2015

def A003132(n): return sum(int(d)**2 for d in str(n)) # Chai Wah Wu, Apr 02 2021

a(n) = n^2 - 20*n*floor(n/10) + 81*(Sum_{k>0} floor(n/10^k)^2) + 20*Sum_{k>0} floor(n/10^k)*(floor(n/10^k) - floor(n/10^(k+1))). - Hieronymus Fischer, Jun 17 2007
a(10n+k) = a(n)+k^2, 0 <= k < 10. - Hieronymus Fischer, Jun 17 2007
a(n) = A007953(A048377(n)) - A007953(n). - Reinhard Zumkeller, Jul 10 2011

More terms from Stefan Steinerberger, Mar 25 2006
Terms checked using the given PARI code, M. F. Hasler, May 24 2009
Replaced the Maple program with a version which works also for arguments with >2 digits, R. J. Mathar, Oct 16 2010
Added ref to Porges. Steinhaus also treated iterations of this function in his Polish book Sto zadań, but I don't have access to it. - Don Knuth, Sep 07 2015

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

Original entry on oeis.org

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

A091366 Primes p such that the sum of the cubes of the digits of p is prime.

Original entry on oeis.org

11, 101, 113, 131, 139, 151, 193, 199, 223, 227, 241, 263, 269, 281, 283, 311, 337, 353, 359, 373, 421, 449, 461, 463, 487, 557, 577, 593, 599, 641, 643, 661, 733, 757, 821, 823, 827, 829, 883, 887, 919, 953, 991, 997, 1013, 1031, 1039, 1051, 1093, 1103, 1123

Offset: 1

Chuck Seggelin, Jan 03 2004

Apparently, in most cases the sum of the digits of such primes is also prime, see A091365 for the exceptions.

I conjecture the contrary: the relative density of numbers in this sequence with prime digit sum is 0. - Charles R Greathouse IV, Sep 08 2010

			a(1)=11 because 1^3 + 1^3 = 2 which is prime. a(10)=227 because 2^3 + 2^3 + 7^3 = 359 which is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A046704 (primes whose digits sum to a prime) A052034 (primes whose digits squared sum to a prime) A091365 (primes whose digits cubed sum to a prime but whose digits do not sum to a prime).

Select[Prime[Range[2, 200]], PrimeQ[Total[IntegerDigits[#]^3]]&] (* Vincenzo Librandi, Apr 13 2013 *)

is(n)=my(v);if(!isprime(n),return(0));v=eval(Vec(Str(n)));isprime(sum(i=1,#v,v[i]^3)) \\ Charles R Greathouse IV, Sep 08 2010

a(44) = 997 inserted by Charles R Greathouse IV, Sep 08 2010

A091367 Primes p such that the sum of the digits raised to the 4th power is prime.

Original entry on oeis.org

11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 113, 131, 179, 191, 197, 223, 269, 311, 313, 331, 353, 379, 397, 401, 443, 461, 601, 607, 641, 661, 719, 739, 809, 883, 911, 937, 971, 1013, 1019, 1031, 1033, 1091, 1097, 1103, 1109, 1181, 1301, 1303, 1367, 1433

Offset: 1

Chuck Seggelin, Jan 03 2004

			a(1) = 11 because 1^4 + 1^4 = 2 which is prime.
a(10) = 89 because 8^4 + 9^4 = 10657 which is prime.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A046704 (primes whose digits sum to a prime), A052034 (primes whose digits squared sum to a prime), A091366 (primes whose digits cubed sum to a prime).

upto=500;Select[Prime[Range[upto]],PrimeQ[Total[IntegerDigits[#]^4]]&] (* Paolo Xausa, Nov 23 2021 *)

from sympy import isprime, primerange
def ok(p): return isprime(sum(int(d)**4 for d in str(p)))
def aupto(limit): return [p for p in primerange(1, limit+1) if ok(p)]
print(aupto(1433)) # Michael S. Branicky, Nov 23 2021

A052035 Palindromic primes whose sum of squared digits is also prime.

Original entry on oeis.org

11, 101, 131, 191, 313, 353, 373, 797, 919, 10301, 11311, 12721, 13331, 13931, 14341, 14741, 16361, 17971, 18181, 19391, 30103, 30703, 33533, 71317, 71917, 74747, 75557, 76367, 77977, 79397, 90709, 93139, 93739, 95959, 96769, 97379

Offset: 1

Patrick De Geest, Dec 15 1999

From Bernard Schott, Oct 20 2021: (Start)
Except for 11, all terms have an odd number of digits.
Except for terms of the form 10^k+1, k >= 2, the middle digit is always odd; the unique known term of the form 10^k+1 for 2 <= k <= 100000 is 101 (see comment in A000533). (End)

			373 -> 3^2 + 7^2 + 3^2 = 67, which is prime.

Charles W. Trigg, Journal of Recreational Mathematics, Vol. 20(2), 1988.

Michel Marcus, Table of n, a(n) for n = 1..1215
Mike Mudge, Morph code, Hands On Numbers Count, Personal Computer World, May 1997, p. 290.

Cf. A000533, A002385, A052034, A003132.

Select[Prime@ Range[2, 10^4], And[PalindromeQ@ #, PrimeQ@ Total[IntegerDigits[#]^2]] &] (* Michael De Vlieger, Oct 20 2021 *)

isok(p) = my(d=digits(p)); isprime(p) && (d==Vecrev(d)) && isprime(sum(k=1, #d, d[k]^2)); \\ Michel Marcus, Oct 17 2021

from sympy import isprime
def ok(n):
    s = str(n)
    return s==s[::-1] and isprime(n) and isprime(sum(int(d)**2 for d in s))
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Nov 23 2021

# second version for going to large terms
from sympy import isprime
from itertools import product
def ok(pal):
    return isprime(pal) and isprime(sum(int(d)**2 for d in str(pal)))
def agentod(maxdigs):
    yield 11
    for d in range(3, maxdigs+1, 2):
        pal = 10**(d-1) + 1
        if ok(pal): yield pal
        for first in "1379":
            for left in product("0123456789", repeat=(d-3)//2):
                left = "".join(left)
                for mid in "13579":
                    pal = int(first + left + mid + left[::-1] + first)
                    if ok(pal): yield pal
print([an for an in agentod(5)]) # Michael S. Branicky, Nov 23 2021

A091362 Primes p such that the sum of the digits of p is not prime, but the sum of the squares of the digits of p is prime.

Original entry on oeis.org

997, 1699, 2887, 5569, 5659, 5839, 5857, 6199, 6883, 6991, 7477, 8287, 8539, 8863, 8999, 9619, 9907, 11779, 11887, 13399, 13669, 14479, 14767, 14947, 15559, 16369, 16477, 16693, 16747, 16963, 17377, 17449, 17467, 17737, 17791, 17827, 17881

Offset: 1

Chuck Seggelin, Jan 03 2004

Apparently if the squares of the digits of a prime sum to a prime, it is more likely that the digits themselves also sum to a prime. In the first 10,000 primes there are 1558 primes p such that the squares of the digits of p sum to a prime. Of these, only 360 are such that the sums of the digits are not prime. Interestingly, all of these primes have a digit sum of 25 or 35. Essentially this sequence is the terms of A052034 (primes whose digits squared sum to a prime) that do not also appear in A046704 (primes whose digits sum to a prime).

			a(1)=997 because 9+9+7 = 25 which is not prime, but 9^2+9^2+7^2 = 211 which is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A046704 (primes whose digits sum to a prime), A052034 (primes whose digits squared sum to a prime).

ssdQ[n_]:=Module[{idn=IntegerDigits[n]},!PrimeQ[Total[idn]]&&PrimeQ[ Total[ idn^2]]]; Select[Prime[Range[2100]],ssdQ] (* Harvey P. Dale, Jun 28 2011 *)

A225535 Numbers whose cubed digits sum to a cube, and have more than one nonzero digit.

Original entry on oeis.org

168, 186, 345, 354, 435, 453, 534, 543, 618, 681, 816, 861, 1068, 1086, 1156, 1165, 1516, 1561, 1608, 1615, 1651, 1680, 1806, 1860, 3045, 3054, 3405, 3450, 3504, 3540, 4035, 4053, 4305, 4350, 4503, 4530, 5034, 5043, 5116, 5161, 5304, 5340, 5403, 5430, 5611

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 10 2013

			5^3 + 6^3 + 1^3 + 1^3 = 343, which is 7^3.

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

Cf. A225534 (cubed digits sum to a prime), A197039 (square), A046459. A055012.
Cf. A165330 (cube cycle), A046197 (cubic fixed points), A000578 (cubes).
Cf. A052034 (squared digits sum to a prime), A028839, A117685.
Cf. A164882 (n such that sum of the cubes of the digits of n^3 is perfect cube). - Zak Seidov, May 21 2013

fQ[n_] := Module[{d = IntegerDigits[n]}, Count[d, 0] + 1 < Length[d] && IntegerQ[Total[d^3]^(1/3)]]; Select[Range[5611], fQ] (* T. D. Noe, May 19 2013 *)

y=rep(0,10000); len=0; x=0; library(gmp);
digcubesum<-function(x) sum(as.numeric(unlist(strsplit(as.character(as.bigz(x)),split="")))^3);
iscube<-function(x) ifelse(as.bigz(x)<2,T,all(table(as.numeric(factorize(x)))%%3==0));
nonzerodig<-function(x) sum(strsplit(as.character(x),split="")[[1]]!="0");
which(sapply(1:6000,function(x) nonzerodig(x)>1 & iscube(digcubesum(x))))

A176179 Primes such that the sum of digits, the sum of the squares of digits and the sum of 3rd powers of their digits is also a prime.

Original entry on oeis.org

11, 101, 113, 131, 199, 223, 311, 337, 353, 373, 449, 461, 463, 641, 643, 661, 733, 829, 883, 919, 991, 1013, 1031, 1103, 1301, 1439, 1451, 1471, 1493, 1499, 1697, 1741, 1949, 2089, 2111, 2203, 2333, 2441, 2557, 3011, 3037, 3307, 3323, 3347, 3491, 3583, 3637, 3659, 3673, 3853, 4049, 4111, 4139, 4241, 4337, 4373, 4391, 4409

Offset: 1

Michel Lagneau, Apr 10 2010

See A091365 for the exceptions for the case where the sum of the digits of p is not prime, but the sum of the cubes of the digits of p is prime.

			For the prime number n =5693 we obtain :
5 + 6 + 9 + 3 = 23 ;
5^2 + 6^2 + 9^2 + 3^2 = 151 ;
5^3 + 6^3 + 9^3 + 3^3 = 1097.

Charles W. Trigg, Journal of Recreational Mathematics, Vol. 20(2), 1988.

Michael S. Branicky, 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. A109181, A046704, A052034, A091366.

with(numtheory):for n from 2 to 10000 do:l:=evalf(floor(ilog10(n))+1):n0:=n:s1:=0:s2:=0:s3:=0:for m from 1 to l do:q:=n0:u:=irem(q,10):v:=iquo(q,10):n0:=v :s1:=s1+u:s2:=s2+u^2:s3:=s3+u^3:od:if type(n,prime)=true and type(s1,prime)=true and type(s2,prime)=true and type(s3,prime)=true then print(n):else fi:od:

okQ[n_]:=Module[{idn=IntegerDigits[n]}, And@@PrimeQ[Total/@{idn,idn^2,idn^3}]]; Select[Prime[Range[600]],okQ]  (* Harvey P. Dale, Jan 18 2011 *)

from sympy import isprime, primerange
def ok(p):
    return all(isprime(sum(int(d)**k for d in str(p))) for k in [1, 2, 3])
def aupto(limit): return [p for p in primerange(1, limit+1) if ok(p)]
print(aupto(4409)) # Michael S. Branicky, Nov 23 2021

Corrected and extended by Harvey P. Dale, Jan 18 2011

cp's OEIS Frontend

A109181 a(n) = A003132(A052034(n)).

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A083444 Erroneous duplicate of A052034.

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A003132 Sum of squares of digits of n.

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

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Mathematica

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A091366 Primes p such that the sum of the cubes of the digits of p is prime.

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Mathematica

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A091367 Primes p such that the sum of the digits raised to the 4th power is prime.

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A052035 Palindromic primes whose sum of squared digits is also prime.

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Python