A068809 -id:A068809 - cp's OEIS frontend

A068949 Digit sums of A068809.

1, 4, 9, 13, 16, 19, 22, 27, 31, 34, 36, 40, 43, 46, 49, 52, 54, 55, 58, 61, 63, 64, 67, 70, 73, 76, 79, 81, 82, 85, 88, 90, 91, 94, 97, 100, 103, 106, 108, 109, 112, 115, 117, 118, 121, 124, 127, 130, 133, 136, 139, 142, 144, 148, 153, 154, 157

Offset: 1

Francois Jooste (phukraut(AT)hotmail.com), Mar 15 2002

			a(4)=13 since A068809(4)=49, whose digit sum is 4+9 = 13.

Giovanni Resta, Table of n, a(n) for n = 1..80

Cf. A068809, A068950, A068952, A069324.

a(42)-a(57) from Kevin Buzzard (k.buzzard(AT)imperial.ac.uk), Jun 20 2008

A068947 Square roots of A068809.

Original entry on oeis.org

1, 2, 3, 7, 13, 17, 43, 63, 83, 167, 264, 313, 707, 836, 1667, 2236, 3114, 4472, 6833, 8167, 8937, 16667, 21886, 29614, 41833, 74833, 89437, 94863, 134164, 191833, 298327, 545793, 547613, 947617, 987917, 1643167, 3143167, 3162083, 5477133

Offset: 1

Francois Jooste (phukraut(AT)hotmail.com), Mar 15 2002

			13^2 = 169 = A068809(5), so a(5)=13.

Giovanni Resta, Table of n, a(n) for n = 1..80

Cf. A068809, A068948, A068949, A068950, A068952, A069324, A040047.

A007953 := proc(n) option remember: return add(d, d=convert(n, base, 10)): end: A068947 := proc(n) option remember: local k,p: if(n=1)then return 1: fi: k:=procname(n-1): p:=A007953(k^2): do k:=k+1: if(A007953(k^2)>p)then return k: fi: od: end: seq(A068947(n),n=1..20); # Nathaniel Johnston, May 04 2011

A068950 Digital roots of A068809.

Original entry on oeis.org

1, 4, 9, 4, 7, 1, 4, 9, 4, 7, 9, 4, 7, 1, 4, 7, 9, 1, 4, 7, 9, 1, 4, 7, 1, 4, 7, 9, 1, 4, 7, 9, 1, 4, 7, 1, 4, 7, 9, 1, 4, 7, 9, 1, 4, 7, 1, 4, 7, 1, 4, 7, 9, 4, 9, 1, 4, 7, 9, 1, 4, 9, 1, 4, 7, 9, 1, 4, 7, 9, 1, 4, 7, 9, 1, 4, 7, 9, 1, 4

Offset: 1

Francois Jooste (phukraut(AT)hotmail.com), Mar 15 2002

			a(4)=4 since A068809(4)=49 and so 4+9=13 and 1+3=4.

Cf. A068809, A068949.

a(43)-a(80) from Giovanni Resta, Jun 27 2018

A068952 Squares in A068949.

Original entry on oeis.org

1, 4, 9, 16, 36, 49, 64, 81, 100, 121, 144, 196

Offset: 1

Francois Jooste (phukraut(AT)hotmail.com), Mar 15 2002

This sequence, so far, looks like A000548, except with the addition of 100. Is there a relationship?

			a(4)=16 since the fourth square in A068949 is 16.

Cf. A068809, A068949, A000548.

a(12) from Giovanni Resta, Jun 27 2018

A069324 Primes in A068949.

Original entry on oeis.org

13, 19, 31, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 157, 163, 181, 193, 199, 211

Offset: 1

Francois Jooste (phukraut(AT)hotmail.com), Mar 15 2002

Differs from A040047 at the 13th term. - Kevin Buzzard (k.buzzard(AT)imperial.ac.uk), Jun 20 2008

			a(4)=43 since the fourth prime in A068949 is 43.

Cf. A068809, A068949, A068952, A040047.

disum(n)= { local(resul) ; resul=0 ; while(n>0, resul += n%10 ; n = (n-n%10)/10 ; ) ; return(resul) ; }
A069324(maxs)= { local(ssqu,su) ; su=1 ; for(s=1,maxs, ssqu=s^2 ; if (disum(ssqu) > su, su=disum(ssqu) ; if( isprime(su), print1(su,",") ; ) ; ) ; ) ; }
A069324(200000000) ; \\ R. J. Mathar, May 19 2006

a(12) from R. J. Mathar, May 19 2006
a(13)-a(14) from Kevin Buzzard (k.buzzard(AT)imperial.ac.uk), Jun 20 2008
a(15)-a(19) from Giovanni Resta, Jun 27 2018

A068948 Primes in A068947.

Original entry on oeis.org

2, 3, 7, 13, 17, 43, 83, 167, 313, 1667, 6833, 8167, 191833, 298327, 3143167, 197222917, 994927133, 3160522105583

Offset: 1

Francois Jooste (phukraut(AT)hotmail.com), Mar 15 2002

a(19) > 10^13. - Giovanni Resta, Jun 27 2018

			a(4)=13 since 13 is the fourth prime in A068947.

Cf. A068809, A068947.

a(16)-a(17) from Sean A. Irvine, Jun 07 2011

a(18) from Giovanni Resta, Jun 27 2018

A257652 The semiprimes which set new records for the sum of their decimal digits.

Original entry on oeis.org

4, 6, 9, 38, 39, 49, 69, 169, 278, 289, 299, 489, 589, 689, 699, 799, 899, 2899, 3899, 4989, 5899, 5999, 6999, 7999, 9899, 19999, 29999, 48999, 58999, 68999, 69999, 88999, 99899, 299899, 398999, 589989, 589999, 689999, 798999, 889999, 899999, 2899999, 3899999

Offset: 1

K. D. Bajpai, Jul 25 2015

The semiprimes that set new records in A175013. New records of digit sums of 4, 6, 9, 11, 12, 13, 15, 16, 17,.. are set by the semiprimes 4, 6, 9, 38, 39, 49, 69,...

			a(4) = 38 = 2 * 19, which is a semiprime with sum of digits = 3 + 8 = 11.
a(5) = 39 = 3 * 13, which is a semiprime with sum of digits = 3 + 9 = 12. Since 12 > 11, 38 and 39 are in list.

K. D. Bajpai, Table of n, a(n) for n = 1..64

Cf. A175013, A068809, A246569.

Subsequence of A213653.

t = {}; s = 0; Do[If[(x = Total[IntegerDigits[n]]) > s && PrimeOmega[n] == 2, AppendTo[t, n]; s = x], {n, 1000000}];t
DeleteDuplicates[{#,Total[IntegerDigits[#]]}&/@Select[Range[4*10^6],PrimeOmega[#] == 2&],GreaterEqual[ #1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Apr 12 2024 *)

A362264 Numbers > 9 with increasingly large digit average of their square, in base 10.

Original entry on oeis.org

10, 11, 12, 13, 17, 63, 83, 313, 94863, 3162083, 994927133

Offset: 0

M. F. Hasler, Apr 13 2023

The single-digit number 3, whose square is 9, has the highest possible digit average, therefore this "trivial solution" is excluded. However, the sequence could be defined as "numbers > 3 ..." in which case it would start 4, 6, 7, 63, ... see examples.
It is conjectured but not known that there are only finitely many numbers whose square has a digit average above 8.3.
Can it be proved or disproved that all terms > 17 end in a digit 3?
Next terms might be 707106074079263583 (da = 8.25) and 94180040294109027313 (da = 8.275), but there might be other terms in between.

			The respective digit averages are:
   n  |    a(n)   |       a(n)^2     | #digits | sum(digits) | digit average
  ----+-----------+------------------+---------+-------------+------------------
   -  |      4    |          16      |    2    |       7     |    7/2 = 3.5
   -  |      6    |          36      |    2    |       9     |    9/2 = 4.5
   -  |      7    |          49      |    2    |      13     |   13/2 = 6.5
   0  |     10    |         100      |    3    |       1     |    1/3 = 0.333...
   1  |     11    |         121      |    3    |       4     |    4/3 = 1.333...
   2  |     12    |         144      |    3    |       9     |     3  = 3.0
   3  |     13    |         169      |    3    |      16     |   16/3 = 3.333...
   4  |     17    |         289      |    3    |      19     |   19/3 = 6.333...
   5  |     63    |        3969      |    4    |      27     |   27/4 = 6.75
   6  |     83    |        6889      |    4    |      31     |   31/4 = 7.75
   7  |    313    |       97969      |    5    |      40     |     8  = 8.0
   8  |   94863   |     8998988769   |   10    |      81     |  81/10 = 8.1
   9  |  3162083  |   9998768898889  |   13    |     106     | 106/13 = 8.15...
  10  | 994927133 |989879999979599689|   18    |     148     |   74/9 = 8.222...

Cf. A164841, A068947, A068809.

m=0; for(k=10,oo, vecsum(d=digits(k^2))>m*#d && !print1(k", ") && m=vecsum(d)/#d)

cp's OEIS Frontend

A068949 Digit sums of A068809.

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A068947 Square roots of A068809.

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A068950 Digital roots of A068809.

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A068952 Squares in A068949.

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A069324 Primes in A068949.

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A068948 Primes in A068947.

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A257652 The semiprimes which set new records for the sum of their decimal digits.

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Mathematica

A362264 Numbers > 9 with increasingly large digit average of their square, in base 10.

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PARI