A061910 -id:A061910 - cp's OEIS frontend

A061904 Numbers n such that the iterative cycle: n -> sum of digits of n^2 has only one distinct element.

1, 3, 6, 9, 10, 12, 15, 18, 21, 30, 39, 45, 48, 51, 60, 90, 100, 102, 105, 111, 120, 150, 180, 201, 210, 249, 300, 318, 321, 348, 351, 390, 450, 480, 501, 510, 549, 600, 900, 1000, 1002, 1005, 1011, 1020, 1050, 1101, 1110, 1149, 1200, 1500, 1761, 1800, 2001

Offset: 1

Asher Auel, May 17 2001

Since the only numbers invariant under this iteration are 1 and 9, n is contained in this sequence iff the sum of digits of n^2 is 1 or 9.

			6 -> 3+6 = 9 -> 8+1 = 9 thus 9 is the only element of the iterative cycle of 6. 12 -> 1+4+4 = 9 -> 8+1 = 9 ...

Cf. A007953, A004159, A061903 - A061910.

A061907 The iterative cycle: n -> sum of digits of n^2 has only four distinct elements.

Original entry on oeis.org

2, 11, 20, 101, 110, 134, 136, 163, 172, 197, 200, 217, 233, 242, 244, 262, 278, 287, 296, 298, 307, 313, 314, 316, 343, 359, 386, 397, 406, 413, 422, 424, 431, 433, 442, 458, 467, 469, 476, 478, 487, 514, 523, 541, 577, 583, 586, 593, 604, 613, 614, 622

Offset: 1

Asher Auel, May 17 2001

			a(2) = 4 since 2 -> 4 -> 1+6 = 7 -> 4+9 = 13 -> 1+6+9 = 16 -> 2+5+6 = 13, thus {4,7,13,16} are the distinct elements of the iterative cycle of 2.

Cf. A007953, A004159, A061903 - A061910.

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.

A259313 Numbers m for which there exists a k>=2 such that m equals the average of digitsum(m^p) for p from 1 to k.

Original entry on oeis.org

1, 9, 12, 13, 16, 19, 21, 49, 61, 67, 84, 106, 160, 191, 207, 250, 268, 373, 436, 783, 2321, 3133, 3786, 3805, 4842, 5128, 8167, 13599, 29431, 35308

Offset: 1

Pieter Post, Jun 24 2015

Digitsum = (A007953).

The 'k's are 2, 2, 4, 3, 4, 5, 7, 12, 15, 16, 19, 21, 57, 37, 38, 79, 48, 63, 72, 119, 306, 397, 469, 472, 582, 613, 927, 1461, 2926, 3449, ..., . - Robert G. Wilson v, Jul 30 2015

			Digitsum(9) is 9, digitsum(9^2) is 9. (9+9)/2 = 9. So 9 is in this sequence.
12^1 = 12, 12^2 = 144, 12^3 = 1728 and 12^4 = 20736. Digitsum(12) = 3, digitsum(144) = 9, digitsum(1728) = 18, digitsum(20736) = 18, (3+9+18+18)/4 = 12. So 12 is in this sequence.

Cf. A007953, A061910, A061209, A061210.

fQ[n_] := If[ IntegerQ@ Log10@ n, False, Block[{pwr = 2, s = Plus @@ IntegerDigits@ n}, While[s = s + Plus @@ IntegerDigits[n^pwr]; s < n*pwr, pwr++]; If[s == n*pwr, True, False]]]; k = 1; lst = {1}; While[k < 100001, If[fQ@ k, AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Jul 30 2015 *)

def sod(n):
    kk = 0
    while n > 0:
        kk= kk+(n%10)
        n =int(n//10)
    return kk
for c in range (2, 10**3):
    bb=0
    for a in range(1,200):
        bb=bb+sod(c**a)
        if bb==c*a:
            print (c,a)

a(21)-a(28) from Giovanni Resta, Jun 24 2015
a(1)-a(28) checked by Robert G. Wilson v, Jul 30 2015
a(29)-a(30) from Robert G. Wilson v, Jul 30 2015

A061806 Numbers n such that the iterative cycle: n -> sum of digits of n^2 has only three distinct elements.

Original entry on oeis.org

4, 5, 17, 26, 28, 32, 37, 40, 43, 44, 49, 50, 53, 62, 63, 64, 67, 73, 74, 76, 77, 82, 83, 86, 87, 88, 89, 91, 92, 93, 94, 97, 98, 107, 109, 113, 114, 116, 117, 118, 122, 124, 125, 126, 127, 128, 133, 137, 141, 143, 149, 154, 157, 158, 161, 164, 166, 167, 169, 170

Offset: 1

Asher Auel, May 17 2001

			4 -> 1+6 = 7 -> 4+9 = 13 -> 1+6+9 = 16 -> 2+5+6 = 13, thus {7,13,16} are the only distinct elements of the iterative cycle of 4.

Cf. A007953, A004159, A061903 - A061910.

A061905 The iterative cycle: n -> sum of digits of n^2 has only two distinct elements.

Original entry on oeis.org

7, 8, 13, 14, 16, 19, 22, 23, 24, 25, 27, 29, 31, 33, 34, 35, 36, 38, 41, 42, 46, 47, 52, 54, 55, 56, 57, 58, 59, 61, 65, 66, 68, 69, 70, 71, 72, 75, 78, 79, 80, 81, 84, 85, 95, 96, 99, 103, 104, 106, 108, 112, 115, 119, 121, 123, 129, 130, 131, 132, 135, 138, 139, 140

Offset: 1

Asher Auel, May 17 2001

It seems that {10,1}, {13,16} and {9,18} are the only iterative cycles with 2 distinct elements.

			7 -> 4+9 = 13 -> 1+6+9 = 16 -> 2+5+6 = 13, thus only {13,16} are contained in the iterative cycle of 7. 24 -> 5+7+6 = 18 -> 3+2+4 = 9 -> 8+1 = 9, thus {18,9} are the only elements of the iterative cycle of 24.

Cf. A007953, A004159, A061903 - A061910.

A168231 a(n)^2 = digital sum of (A076503(n))^2.

Original entry on oeis.org

2, 3, 2, 4, 4, 4, 4, 4, 5, 2, 4, 5, 4, 4, 5, 5, 4, 4, 4, 5, 5, 5, 5, 4, 5, 4, 5, 5, 5, 4, 4, 5, 4, 5, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 5, 4, 4, 5, 5, 5, 5, 5, 4, 5, 5, 4, 4, 5, 5, 5, 5, 7, 5, 4, 5, 5, 5, 5, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 7, 5, 5, 7, 5, 5, 7, 5, 5, 5, 7, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Zak Seidov, Nov 21 2009

a(n)^2 = A007953((A076503(n))^2).

Cf. A076503 Prime numbers whose squares have square digital sums, A061910 Numbers n such that sum of digits of n^2 is a square, A007953 Digital sum (i.e. sum of digits) of n, A000040 The prime numbers.

A174274 Each pair of adjacent digits of n sums to a square.

Original entry on oeis.org

10, 13, 18, 22, 27, 31, 36, 40, 45, 54, 63, 72, 79, 81, 88, 90, 97, 100, 101, 104, 109, 131, 136, 181, 188, 222, 227, 272, 279, 310, 313, 318, 363, 400, 401, 404, 409, 454, 540, 545, 631, 636, 722, 727, 790, 797, 810, 813, 818, 881, 888, 900, 901, 904, 909, 972, 979, 1000, 1001, 1004, 1009, 1010, 1013, 1018, 1040, 1045, 1090, 1097, 1310, 1313, 1318, 1363, 1810, 1813, 1818, 1881, 1888

Offset: 1

Zak Seidov, Nov 27 2010

Cf. A028839 Sum of digits of n is a square,

Cf. A061910 (numbers n such that sum of digits of n^2 is a square).

s={};Do[id=IntegerDigits[n];If[Union[IntegerQ/@Sqrt[Rest[id]+Most[id]]]=={True},AppendTo[s,n]],{n,10,2000}];s

cp's OEIS Frontend

A061904 Numbers n such that the iterative cycle: n -> sum of digits of n^2 has only one distinct element.

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A061907 The iterative cycle: n -> sum of digits of n^2 has only four distinct elements.

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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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A259313 Numbers m for which there exists a k>=2 such that m equals the average of digitsum(m^p) for p from 1 to k.

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A061806 Numbers n such that the iterative cycle: n -> sum of digits of n^2 has only three distinct elements.

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A061905 The iterative cycle: n -> sum of digits of n^2 has only two distinct elements.

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A168231 a(n)^2 = digital sum of (A076503(n))^2.

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A174274 Each pair of adjacent digits of n sums to a square.

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Mathematica