A340064 Every odd term k of the sequence is the cumulative sum of the prime digits used so far (the digits of k are included in the sum).

3, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 9, 22, 24, 26, 28, 19, 21, 30, 32, 34, 36, 38, 40, 42, 43, 44, 46, 48, 50, 52, 54, 63, 73, 56, 58, 60, 62, 64, 66, 68, 70, 72, 101, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 131, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 141, 126, 128, 153

Offset: 1

Author

Eric Angelini and Carole Dubois, Dec 28 2020

Keywords

• base
• nonn

Comments

This is the lexicographically earliest sequence of distinct positive terms with this property. The prime digits are 2, 3, 5 and 7.

The sequence is first extended with the smallest odd term not leading to a contradiction; if no such term exists, the sequence is extended with the smallest even term not yet present.

Examples

			Not a(1) = 1 as this 1, being odd, should be the sum of the prime digits so far -- which is wrong (there are none);
not a(1) = 2 as a(1) = 3 is odd and possible here;
a(12) = 9 as 9 is odd and the sum of the prime digits 3 + 2 + 2 + 2;
a(13) = 22 as 22 is the smallest even term available;
a(17) = 19 as 19 = 3 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2;
a(18) = 21 as 21 is the sum of 19 + 2 (the first digit of 21 itself); etc.
		

Crossrefs

CF. A338924

Programs

Python

def pds(k): return sum(int(d) for d in str(k) if d in "2357")
def aupto(nn):
  aset, alst, primesum, nexteven = set(), [], 0, 2
  for n in range(1, nn):
    k = 1
    found = False
    while not found:
      while k in aset: k += 2
      if k == primesum + pds(k): found = True; break
      if k > primesum + 7 * len(str(k)): break
      k += 2
    if found: ak = k
    else: ak = nexteven; nexteven += 2
    aset.add(ak); alst.append(ak); primesum += pds(ak)
  return alst
print(aupto(76)) # Michael S. Branicky, Dec 29 2020