A392974 Numbers k such that sopfr(k-1) = sopfr(k+1), where sopfr = A001414.

11, 17, 31, 155, 251, 1429, 1897, 2661, 3041, 3725, 4983, 6497, 8371, 8383, 9316, 10811, 11121, 11919, 13735, 16561, 16927, 21295, 24703, 24824, 29175, 33865, 34161, 36981, 37522, 40901, 49402, 49456, 49791, 50428, 53285, 53299, 56897, 57619, 66039, 75657, 75763, 82523, 85249

Offset

1,1

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Wikipedia, Ruth-Aaron pair.

Formula

{k : A001414(k-1) = A001414(k+1)}.

Example

11 is a term because sopfr(10) = sopfr(2*5) = 2+5 = 7 and sopfr(12) = sopfr(2^2*3) = 2+2+3 = 7.
17 is a term because sopfr(16) = sopfr(2^4) = 2+2+2+2 = 8 and sopfr(18) = sopfr(2*3^2) = 2+3+3 = 8.
31 is a term because sopfr(30) = sopfr(2*3*5) = 2+3+5 = 10 and sopfr(32) = sopfr(2^5) = 2+2+2+2+2 = 10.
155 is a term because sopfr(154) = sopfr(2*7*11) = 2+7+11 = 20 and sopfr(156) = sopfr(2^2*3*13) = 2+2+3+13 = 20.
251 is a term because sopfr(250) = sopfr(2*5^3) = 2+5+5+5 = 17 and sopfr(252) = sopfr(2^2*3^2*7) = 2+2+3+3+7 = 17.

Mathematica

sopfr[n_] := If[n == 1, 0, Total[Flatten[Table[#[[1]], #[[2]]] & /@ FactorInteger[n]]]]; Select[Range[2, 100000], sopfr[# - 1] == sopfr[# + 1] &]

Prog

(PARI) sopfr(n) = (n=factor(n))[, 1]~*n[, 2]; \\ A001414
isok(k) = sopfr(k-1) == sopfr(k+1); \\ Michel Marcus, Feb 16 2026
(Python)
from sympy import factorint
from itertools import count, islice
def sopfr(k): return sum(p*e for p, e in factorint(k).items())
def agen(STARTK=1): # generator of terms
    sk1, sk2, sk3 = sopfr(STARTK), sopfr(STARTK+1), sopfr(STARTK+2)
    for k in count(STARTK+1):
        if sk1 == sk3: yield k
        sk1, sk2, sk3 = sk2, sk3, sopfr(k+2)
print(list(islice(agen(), 43))) # Michael S. Branicky, Jul 17 2026

Crossrefs

Cf. A001414, A006145, A039752, A076155.

Sequence in context: A127703 A165667 A339180 * A086711 A039514 A098797

Adjacent sequences: A392971 A392972 A392973 * A392975 A392976 A392977

Keyword

nonn,changed

Author

Nishant R. Gautam, Jan 29 2026

Status

approved