A033936 -id:A033936 - cp's OEIS frontend

A096293 Number of iterations of n -> n + (sum of squares of digits of n) needed for the trajectory of n to join the trajectory of A033936.

0, 0, 8, 76, 72, 0, 330, 8, 73, 77, 76, 7, 75, 73, 72, 66, 6, 62, 25, 75, 67, 72, 74, 74, 8, 66, 38, 70, 74, 71, 72, 62, 72, 70, 61, 73, 7, 75, 70, 330, 71, 0, 329, 73, 61, 62, 73, 71, 72, 74, 71, 73, 65, 7, 74, 329, 73, 70, 69, 70, 62, 0, 3, 39, 60, 65, 5, 328, 60, 72, 75, 7, 73

Offset: 1

Jason Earls, Jun 24 2004

Conjecture: For any positive integer starting value n, iterations of n -> n + (sum of squares of digits of n) will eventually join A033936 (verified for all n up to 20000).

			a(3)=8 because the trajectory for 1 (sequence A033936) starts
1->2->6->42->62->102->107->157->232->249->350->384->473...
and the trajectory for 3 starts
3->12->17->67->152->182->251->281->350->384->473...
so the sequence beginning with 3 joins A033936 after 8 steps.

A258881 a(n) = n + the sum of the squared digits of n.

Original entry on oeis.org

0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 11, 13, 17, 23, 31, 41, 53, 67, 83, 101, 24, 26, 30, 36, 44, 54, 66, 80, 96, 114, 39, 41, 45, 51, 59, 69, 81, 95, 111, 129, 56, 58, 62, 68, 76, 86, 98, 112, 128, 146, 75, 77, 81, 87, 95, 105, 117, 131, 147, 165, 96, 98, 102

Offset: 0

M. F. Hasler, Jul 19 2015

Carlos Rivera, Puzzle 776. Ten consecutive integers such that..., The Prime Puzzles and Problems Connection, March 2015.

Cf. A003132, A062028, A259391, A259567, A033936, A076161 (indices of primes), A329179 (indices of squares).

Total[Flatten@ {#, IntegerDigits[#]^2}] & /@ Range[0, 61] (* Michael De Vlieger, Jul 20 2015 *)
Table[n+Total[IntegerDigits[n]^2],{n,0,100}] (* Harvey P. Dale, Nov 27 2022 *)

PARI
```
A258881(n)=n+norml2(digits(n))
```

def ssd(n): return sum(int(d)**2 for d in str(n))
def a(n): return n + ssd(n)
print([a(n) for n in range(63)]) # Michael S. Branicky, Jan 30 2021

A259567 Number of subsequent numbers, starting with n, for which A258881(x) = x + (sum of squares of digits of x) is prime.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 0

M. F. Hasler, Jul 19 2015

This sequence is motivated by sequence A259391 and the "Prime puzzle 776".

			For n = 0, A258881(0) = 0 is not prime.
For n = 1, A258881(1) = 1+1 = 2 is prime, but A258881(2) = 2+4 is not prime, therefore a(1)=1.
For n = 10, A258881(10) = 10 + 1^2 + 0^2 = 11, A258881(11) = 11 + 1^2 + 1^2 = 13, A258881(12) = 12 + 1^2 + 2^2 = 17, ..., A258881(19) = 19 + 1^2 + 9^2 = 101 are all prime, but A258881(20) = 20 + 2^2 + 0^2 is not prime, therefore a(10) = 10.
The next value of 10 occurs at index n = 1761702690, see A259391.

C. Rivera, Puzzle 776. Ten consecutive integers such that..., PrimePuzzles.net, March 2015.

Cf. A259391, A258881, A003132, A076161, A033936.

a(n)=for(m=n,n+9e9,isprime(A258881(m))||return(m-n))

If a(n) > 0, then a(n+1) = a(n)-1.

a(n) > 0 iff n is in A076161.

cp's OEIS Frontend

A096293 Number of iterations of n -> n + (sum of squares of digits of n) needed for the trajectory of n to join the trajectory of A033936.

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A258881 a(n) = n + the sum of the squared digits of n.

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A259567 Number of subsequent numbers, starting with n, for which A258881(x) = x + (sum of squares of digits of x) is prime.

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