A370352 - OEIS

%I #27 Feb 17 2024 12:36:40

%S 1,1,6,22,46,526,838,838,5667,5667,20158,32127,56697,82617,174718,

%T 174718,314492,314492,415789,498957,1142398,1713598,1713598,2280067,

%U 2280067,4324316,4324316,5847653,6918908,6918908,6918908,9979197,15855829,24023995,28274398,28274398,28274398,28274398

%N The smallest number such that n or more numbers k exist with k - a(n) = sopfr(k) + sopfr(a(n)), where sopfr(m) is the sum of the primes dividing m with repetition.

%H Michael S. Branicky, <a href="/A370352/b370352.txt">Table of n, a(n) for n = 1..51</a>

%e a(1) = a(2) = 1 as 1 is the smallest number to have two numbers (k = 1, 6) such that 1 - 1 = 0 = sopfr(1) + sopfr(1) = 0, and 6 - 1 = 5 = sopfr(6) + sopfr(1) = 5 + 0 = 5.

%e a(3) = 6 as 6 is the smallest number to have three numbers (k = 20, 21, 26) such that 20 - 6 = 14 = sopfr(20) + sopfr(6) = 9 + 5 = 14, 21 - 6 = 15 = sopfr(21) + sopfr(6) = 10 + 5 = 15, and 26 - 6 = 20 = sopfr(26) + sopfr(6) = 15 + 5 = 20.

%o (Python)

%o from sympy import factorint

%o from itertools import count, islice

%o from collections import Counter

%o kcount, kmax = Counter(), 0

%o def sopfr(n): return sum(p*e for p, e in factorint(n).items())

%o def f(n):

%o     global kcount, kmax

%o     target = n + sopfr(n)

%o     for k in range(kmax+1, 2*target+5):

%o         kcount[k-sopfr(k)] += 1

%o         kmax += 1

%o     return kcount[target]

%o def agen(): # generator of terms

%o     adict, n = dict(), 1

%o     for m in count(1):

%o         v = f(m)

%o         if v not in adict: adict[v] = m

%o         for i in range(n, v+1): yield m; n += 1

%o print(list(islice(agen(), 16))) # _Michael S. Branicky_, Feb 17 2024

%Y Cf. A001414, A369349, A369351, A370351.

%K nonn

%O 1,3

%A _Scott R. Shannon_, Feb 16 2024

%E a(21) and beyond from _Michael S. Branicky_, Feb 16 2024