A140731 and A140732 are the same sequence

N. J. A. Sloane njas at research.att.com
Sat Jul 19 04:30:46 CEST 2008

```Partial sum of squares of lucky numbers.

1, 10, 59, 140, 309, 534, 975, 1600, 2561, 3650, 5019, 6868, 9269,
11870, 15839, 20328, 25089, 30418, 36043, 42284, 49853, 58502, 68303,
79328, 91649, 104874, 121003, 137644, 155333, 173558, 193439, 216240,
241521, 268090, 296651

Partial sum of A032598 Squares of lucky numbers.

SUM[i=1..n] A000959(i)^2.

This is to A024450 as A000959 is to A000040.

The values which are themselves lucky numbers begin: a(1) = 1, a(7) =
975 = A000959(148), a(11) = 5019 = A000959(613), a(19) = 36043 =
A000959(3493).

a(n) is prime for n beginning: 3, 21, 29, 34.

Example:

a(3) =  1^2 + 3^2 + 7^2 = 59 is prime.

a(13) =
1^2 + 3^2 + 7^2 + 9^2 + 13^2 + 15^2 + 21^2 + 25^2 + 31^2 + 33^2 + 37^2
+  43^2 + 49^2 = 9269 = 13 * 23 * 31.

The sum of the squares of the first 13 lucky numbers is a 3-brilliant
number whose factors include both 13 and its reversal.

a(21) = 1^2 + 3^2 + 7^2 + 9^2 + 13^2 + 15^2 + 21^2 + 25^2 + 31^2 +
33^2 + 37^2 + 43^2 + 49^2 + 51^2 + 63^2 + 67^2 + 69^2 + 73^2 + 75^2 +
79^2 + 87^2 = 49853 is a prime.

a(25) =
1^2 + 3^2 + 7^2 + 9^2 + 13^2 + 15^2 + 21^2 + 25^2 + 31^2 + 33^2 + 37^2
+  43^2 + 49^2 + 51^2 +  63^2 +  67^2 +  69^2 +  73^2 +  75^2 +  79^2
+  87^2 +  93^2 +  99^2 +  105^2 +  111^2 = 91649 = 37 * 2477.

a(29) =
1^2 + 3^2 + 7^2 + 9^2 + 13^2 + 15^2 + 21^2 + 25^2 + 31^2 + 33^2 + 37^2
+ 43^2 + 49^2 + 51^2 + 63^2 + 67^2 + 69^2 + 73^2 + 75^2 + 79^2 + 87^2
+ 93^2 + 99^2 + 105^2 + 111^2 + 115^2 + 127^2 + 129^2 + 133^2 = 155333
is prime.

a(34) =
1^2 + 3^2 + 7^2 + 9^2 + 13^2 + 15^2 + 21^2 + 25^2 + 31^2 + 33^2 + 37^2
+ 43^2 + 49^2 + 51^2 + 63^2 + 67^2 + 69^2 + 73^2 + 75^2 + 79^2 + 87^2
+ 93^2 + 99^2 + 105^2 + 111^2 + 115^2 + 127^2 + 129^2 + 133^2 + 135^2
+ 141^2 + 151^2 + 159^2 + 163^2 + 169^2 = 296651
is  prime.

===============

By the way, A032639 is to A000959 as A006450  is to A000040.

A032639 Lucky numbers indexed by the lucky numbers (Lucky numbers with
lucky number subscripts).
1, 7, 21, 31, 49, 63, 87, 111, 141, 159, 189, 219, 259, 267,

Further, the A006450 Primes with prime subscripts.
3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, 179, 191, 211

are in many derived seqs known as Prime-Index-Primes or PIPs, but that
is never mentioned in A006450 itself.

Best,

Jonathan Vos Post

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