A061910 -id:A061910 - cp's OEIS frontend

A076503 Prime numbers whose squares have square digit-sums.

2, 3, 11, 13, 23, 31, 41, 59, 67, 101, 103, 113, 131, 139, 157, 193, 211, 229, 239, 257, 283, 311, 337, 347, 373, 401, 409, 419, 463, 491, 499, 509, 571, 599, 643, 653, 661, 743, 751, 761, 769, 797, 1013, 1021, 1031, 1039, 1103, 1129, 1193, 1201, 1229, 1237

Offset: 1

Christopher Schloetz (cschloetz(AT)hotmail.com), Nov 09 2002

			13 is a member because 13 is prime and the digit-sum of its square is 1+6+9=16, which is also square.

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

Cf. A000040, A061910.

Select[Prime[Range[250]],IntegerQ[Sqrt[Total[IntegerDigits[#^2]]]]&] (* Harvey P. Dale, Aug 30 2016 *)

isok(n) = if (! isprime(n), 0, d = digits(n^2); issquare(sum(i=1, #d, d[i]))) \\ Michel Marcus, Jun 20 2013

More terms from Michel ten Voorde Jun 13 2003

A153747 Numbers k such that there are 9 digits in k^2 and for each factor f of 9 (1,3) the sum of digit groupings of size f is a square.

Original entry on oeis.org

10000, 10001, 10002, 10003, 10004, 10005, 10010, 10011, 10012, 10013, 10020, 10021, 10022, 10030, 10031, 10200, 10284, 10287, 10300, 10353, 10356, 10359, 10433, 10578, 10588, 10617, 10623, 10642, 10679, 10683, 10686, 10692, 10734

Offset: 1

Doug Bell, Dec 31 2008

This sequence is a subsequence of both A153745 and A061910.

Last term is a(474) = 31493. - Giovanni Resta, Jun 06 2015

			10433^2 = 108847489; 1+0+8+8+4+7+4+8+9 = 49 = 7^2; and 108+847+489 = 1444 = 38^2.

Giovanni Resta, Table of n, a(n) for n = 1..474 (full sequence)

Cf. A061910, A153745.

dgfsQ[n_]:=Module[{idn2=IntegerDigits[n^2]},AllTrue[{Sqrt[ Total[ idn2]], Sqrt[ Total[ FromDigits/@ Partition[idn2,3]]]},IntegerQ]]; Select[ Range[ 10000,31622],dgfsQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 19 2018 *)

A153748 Numbers k such that there are 10 digits in k^2 and for each factor f of 10 (1, 2, 5) the sum of digit groupings of size f is a square.

Original entry on oeis.org

33018, 33051, 33081, 33084, 33150, 33153, 33477, 33573, 33579, 33582, 33603, 33606, 33642, 33645, 33648, 36312, 39192, 41703, 44928, 47439, 53052, 53971, 55785, 56277, 60725, 63490, 66342, 66345, 66375, 66381, 66444, 66903, 66942, 67008, 69645, 76710, 77530

Offset: 1

Doug Bell, Dec 31 2008

This sequence is a subsequence of both A153745 and A061910.

Last term is a(48) = 99677. - Giovanni Resta, Jun 06 2015

Giovanni Resta, Table of n, a(n) for n = 1..48 (full sequence)

Cf. A061910, A153745.

More terms from Giovanni Resta, Jun 06 2015

A153750 Numbers k such that there are 14 digits in k^2 and for each factor f of 14 (1,2,7) the sum of digit groupings of size f is a square.

Original entry on oeis.org

3196200, 3330249, 3330348, 3330480, 3330801, 3331071, 3331367, 3331695, 3331731, 3331758, 3331803, 3331830, 3331860, 3331866, 3331929, 3331995, 3332025, 3332058, 3332061, 3332091, 3332124, 3332127, 3332160, 3332190

Offset: 1

Doug Bell, Dec 31 2008

This sequence is a subsequence of both A153745 and A061910.

Last term is a(266) = 9996830. - Giovanni Resta, Jun 06 2015

			3331367^2 = 11098006088689;
1+1+0+9+8+0+0+6+0+8+8+6+8+9 = 64 = 8^2;
11+09+80+06+08+86+89 = 289 = 17^2;
1109800+6088689 = 7198489 = 2683^2.

Giovanni Resta, Table of n, a(n) for n = 1..266 (full sequence)

Cf. A061910, A153745.

sdgQ[n_]:=Module[{idn=IntegerDigits[n^2],t2,t7},t2=Total[FromDigits/@ Partition[ idn,2]];t7=Total[FromDigits/@Partition[idn,7]]; AllTrue[ {Sqrt[Total[idn]],Sqrt[t2],Sqrt[t7]},IntegerQ]]; Select[Range[ Round[ 3.16*10^6],Round[3.34*10^6]],sdgQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 07 2016 *)

A153751 Numbers k such that there are 15 digits in k^2 and for each factor f of 15 (1,3,5) the sum of digit groupings of size f is a square.

Original entry on oeis.org

10000000, 10000001, 10000002, 10000003, 10000004, 10000005, 10000010, 10000011, 10000012, 10000013, 10000020, 10000021, 10000022, 10000030, 10000031, 10000200, 10000300, 10011003, 10022000, 10035990, 10042440

Offset: 1

Doug Bell, Dec 31 2008

This sequence is a subsequence of both A153745 and A061910.

The last term is a(2782) = 31616301. - Giovanni Resta, Jun 06 2015

			10000011^2 = 100000220000121;
1+0+0+0+0+0+2+2+0+0+0+0+1+2+1 = 9 = 3^2;
100+000+220+000+121 = 441 = 21^2;
10000+02200+00121 = 12321 = 111^2.

Giovanni Resta, Table of n, a(n) for n = 1..2782 (all terms)

Cf. A004159, A061910, A153745.

Select[Range[10^7,31622776],AllTrue[{Sqrt[Total[IntegerDigits[#^2]]],Sqrt[Total[ FromDigits/@ Partition[IntegerDigits[#^2],3]]],Sqrt[Total[FromDigits/@Partition[IntegerDigits[#^2],5]]]},IntegerQ]&] (* Harvey P. Dale, Apr 11 2023 *)

A153752 Numbers k such that there are 16 digits in k^2 and for each factor f of 16 (1,2,4,8) the sum of digit groupings of size f is a square.

Original entry on oeis.org

31883334, 31886667, 31956690, 31970049, 32469999, 33338100, 33341067, 33870000, 34140000, 34149999, 34713042, 34763334, 34856667, 35780000, 36356249, 36356480, 36359065, 37523635, 37737452, 37949451, 38362409

Offset: 1

Doug Bell, Dec 31 2008

This sequence is a subsequence of both A153745 and A061910.

This sequence contains 124 terms, with a(124) = 9998956. - Giovanni Resta, Jun 06 2015

			31883334^2 = 1016546986955556;
1+0+1+6+5+4+6+9+8+6+9+5+5+5+5+6 = 81 = 9^2;
10+16+54+69+86+95+55+56 = 441 = 21^2;
1016+5469+8695+5556 = 20736 = 144^2;
10165469+86955556 = 97121025 = 9855^2.

Giovanni Resta, Table of n, a(n) for n = 1..124 (all terms)

Cf. A004159, A061910, A153745.

okQ[n_]:=Module[{n2=IntegerDigits[n^2]},And@@(IntegerQ[Sqrt[ #]]&/@ (Total/@(Table[ FromDigits/@Partition[n2,2^i],{i,0,3}])))]; Select[ Range[31622777,38400000],okQ] (* Harvey P. Dale, Aug 12 2012 *)

A223035 Prime numbers whose digits squared sum to a square.

Original entry on oeis.org

2, 3, 5, 7, 43, 263, 269, 1153, 1531, 1933, 2063, 2069, 2287, 2609, 3319, 3391, 3511, 3931, 4003, 4441, 4801, 4889, 5113, 5399, 5939, 6029, 6067, 6203, 6469, 6607, 8849, 9133, 9539, 10111, 10177, 10513, 10531, 10771, 11149, 11213, 11273, 11321, 11491, 11503

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, Apr 04 2013

			269 is a prime number, and 2^2+6^2+9^2 = 121 = 11^2.

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

Prime numbers from the sequence A175396.

Cf. A000040, A061910, A107288.

Select[Prime[Range[2000]], IntegerQ[Sqrt[Total[IntegerDigits[#]^2]]] &] (* T. D. Noe, Apr 05 2013 *)

ssod<-function(i) sum(as.numeric(strsplit(as.character(i),"")[[1]])^2)
issquare<-function(x) as.integer(sqrt(x))==sqrt(x)
x=as.bigz(c()); i=2
while(length(x)<10000) {if(issquare(ssod(i))) x=c(x,i); i=nextprime(i)}

A262712 Numbers k such that sum of digits of k^2 is 9.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 30, 39, 45, 48, 51, 60, 90, 102, 105, 111, 120, 150, 180, 201, 210, 249, 300, 318, 321, 348, 351, 390, 450, 480, 501, 510, 549, 600, 900, 1002, 1005, 1011, 1020, 1050, 1101, 1110, 1149, 1200, 1500, 1761, 1800, 2001, 2010, 2100, 2490

Offset: 1

Vincenzo Librandi, Sep 28 2015

Subsequence of A008585.

			6 is in sequence because 6^2 = 36 and 3+6 = 9.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. similar sequences listed in A262711.

Cf. A004159, A008585, A061910.

[n: n in [1..2*10^4] | &+Intseq(n^2) eq 9 ];

filter:= proc(n) convert(convert(n^2,base,10),`+`) = 9 end proc:select(filter, [$1..10^5]); # Robert Israel, Jan 04 2024

Select[Range[10^5], Total[IntegerDigits[#^2]] == 9 &]

for(n=1, 1e8, if (sumdigits(n^2) == 9, print1(n", "))) \\ Altug Alkan, Sep 28 2015

A262713 Numbers k such that the sum of digits of k^2 is 10.

Original entry on oeis.org

8, 19, 35, 46, 55, 71, 80, 145, 152, 179, 190, 251, 332, 350, 361, 449, 451, 460, 548, 550, 649, 710, 800, 1450, 1520, 1790, 1900, 2510, 3320, 3500, 3610, 4490, 4499, 4510, 4600, 5480, 5500, 6490, 7100, 8000, 14500, 15200, 17900, 19000, 20249, 20251, 24499

Offset: 1

Vincenzo Librandi, Sep 28 2015

From Altug Alkan, Sep 29 2015: (Start)
Subsequence of A001651.
If a(n)+1 mod 9 != 0 then a(n)-1 mod 9 = 0;
if a(n)-1 mod 9 != 0 then a(n)+1 mod 9 = 0;
a(n)^2 - 1 mod 9 = 0. (End)
A135027(n)*10^k is a term for all n > 0, k >= 0. - Michael S. Branicky, Aug 19 2021

			19 is in sequence because 19^2 = 361 and 3+6+1 = 10.

Michael S. Branicky, Table of n, a(n) for n = 1..244
Michael S. Branicky, Python program

Cf. similar sequences listed in A262711.

Cf. A004159, A061910, A135027.

[n: n in [1..3*10^4] | &+Intseq(n^2) eq 10 ];

Select[Range[10^5], Total[IntegerDigits[#^2]] == 10 &]

for(n=1, 1e6, if (sumdigits(n^2) == 10, print1(n", "))) \\ Altug Alkan, Sep 28 2015

# See linked program to go to larger numbers
def ok(n): return sum(map(int, str(n*n))) == 10
print(list(filter(ok, range(25000)))) # Michael S. Branicky, Aug 19 2021

A371047 Numbers k such that the digital sum of k^4 is a fourth power.

Original entry on oeis.org

0, 1, 10, 11, 100, 101, 110, 1000, 1001, 1010, 1100, 1434, 1716, 1767, 1776, 1884, 2094, 2112, 2133, 2208, 2271, 2292, 2298, 2514, 2544, 2556, 2604, 2628, 2892, 2919, 2922, 2976, 3006, 3018, 3066, 3078, 3096, 3111, 3126, 3138, 3144, 3159, 3162, 3492, 3498, 3504

Offset: 1

Stefano Spezia, Mar 09 2024

Stefano Spezia, Table of n, a(n) for n = 1..10000

Cf. A000583, A007953, A011557 (subsequence), A038444 (subsequence), A061910, A237525, A371004.

Select[Range[0,4000]^4,IntegerQ[DigitSum[#]^(1/4)]&]^(1/4)

a(n) = A371004(n)^(1/4).

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A076503 Prime numbers whose squares have square digit-sums.

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A153747 Numbers k such that there are 9 digits in k^2 and for each factor f of 9 (1,3) the sum of digit groupings of size f is a square.

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A153748 Numbers k such that there are 10 digits in k^2 and for each factor f of 10 (1, 2, 5) the sum of digit groupings of size f is a square.

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A153750 Numbers k such that there are 14 digits in k^2 and for each factor f of 14 (1,2,7) the sum of digit groupings of size f is a square.

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A153751 Numbers k such that there are 15 digits in k^2 and for each factor f of 15 (1,3,5) the sum of digit groupings of size f is a square.

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A153752 Numbers k such that there are 16 digits in k^2 and for each factor f of 16 (1,2,4,8) the sum of digit groupings of size f is a square.

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A223035 Prime numbers whose digits squared sum to a square.

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A262712 Numbers k such that sum of digits of k^2 is 9.

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