A153745 -id:A153745 - cp's OEIS frontend

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.

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 *)

A258660 Numbers n such that the number of digits d in n is not prime and for each factor f of d the sum of the d/f digit groupings of size f is a square.

Original entry on oeis.org

1, 4, 9, 1521, 3600, 7396, 8100, 103041, 120409, 160801, 11471769, 11655396, 12802084, 15210000, 22724289, 36000000, 42889401, 42928704, 45481536, 45968400, 46009089, 54567769, 61811044, 62236321, 70006689, 73925604, 73960000, 76965529, 79174404, 81000000, 85008400, 97693456, 97713225, 100000000

Offset: 1

Doug Bell, Jun 06 2015

If a(n) has m = p^k digits, then a(n)*10^((p-1)*m) is also a member of the sequence. For instance, 1521*10^(2^k-4) is in the sequence for all integers k >=2. # Chai Wah Wu, Jun 08 2015

Chai Wah Wu, Table of n, a(n) for n = 1..3730

Cf. A153745.

from sympy import divisors
from gmpy2 import is_prime, isqrt, isqrt_rem, is_square
A258660_list = []
for l in range(1,17):
    if not is_prime(l):
        fs = divisors(l)
        a, b = isqrt_rem(10**(l-1))
        if b > 0:
            a += 1
        for n in range(a,isqrt(10**l-1)+1):
            n2 = n**2
            ns = str(n2)
            for g in fs:
                y = 0
                for h in range(0,l,g):
                    y += int(ns[h:h+g])
                if not is_square(y):
                    break
            else:
                A258660_list.append(n2) # Chai Wah Wu, Jun 08 2015

a(n) = A153745(n)^2.

Corrected a(13)-a(14) by Chai Wah Wu, Jun 08 2015

A153746 Numbers k such that there are 8 digits in k^2 and for each factor f of 8 (1,2,4) the sum of digit groupings of size f is a square.

Original entry on oeis.org

3387, 3414, 3578, 3900, 4767, 6000, 6549, 6552, 6744, 6780, 6783, 7387, 7862, 7889, 8367, 8598, 8600, 8773, 8898, 9000, 9220, 9884, 9885

Offset: 1

Doug Bell, Dec 31 2008

This is the complete sequence. This sequence is a subsequence of both A153745 and A061910.

Cf. A061910, A153745.

A153749 Numbers k such that there are 12 digits in k^2 and for each factor f of 12 (1,2,3,4,6) the sum of digit groupings of size f is a square.

Original entry on oeis.org

316713, 334401, 658635

Offset: 1

Doug Bell, Dec 31 2008

This is the complete sequence. This sequence is a subsequence of both A153745 and A061910.

			316713^2 = 100307124369;
1+0+0+3+0+7+1+2+4+3+6+9 = 36 = 6^2;
10+03+07+12+43+69 = 144 = 12^2;
100+307+124+369 = 900 = 30^2;
1003+0712+4369 = 6084 = 78^2;
100307+124369 = 224676 = 474^2.

Cf. A061910, A153745.

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.

cp's OEIS Frontend

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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A258660 Numbers n such that the number of digits d in n is not prime and for each factor f of d the sum of the d/f digit groupings of size f is a square.

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A153746 Numbers k such that there are 8 digits in k^2 and for each factor f of 8 (1,2,4) the sum of digit groupings of size f is a square.

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A153749 Numbers k such that there are 12 digits in k^2 and for each factor f of 12 (1,2,3,4,6) the sum of digit groupings of size f is a square.

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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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