A395520 - OEIS

A395520

First differences of A036349.

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

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OFFSET

1,2

COMMENTS

Empirical observation: when only the prime-valued terms a(k) are examined, they appear to generate the primes in ascending order with at most finitely many exceptions. Computational search to N = 10^9 (covering 499932835 gaps) finds exactly two exceptions: gap value 13 precedes gap value 11 (prime 11 is delayed 1192 gap steps), and gap value 19 precedes gap value 17 (prime 17 is delayed 47205 gap steps). After these two early events, no further out-of-order prime gap values are observed across the remaining ~499.9 million gaps.

Empirical observation: the maximum delay of 47205 steps is set near n = 48000 and never exceeded. This suggests the existence of an absolute constant C bounding all such delays (the Bounded Prime Debt Conjecture). See MathOverflow question 510477.

Empirical observation: gap value 1 has density approximately 50% and gap value 2 has density approximately 25%, consistent with Theorem 3.1 of Alladi and Erdos (1977), which proves sopfr is equidistributed modulo 2.

REFERENCES

K. Alladi and P. Erdős, On an additive arithmetic function, Pacific Journal of Mathematics, Vol. 71, No. 2 (1977), pp. 275-294.

G. Tenenbaum, A note on the normal largest gap between prime factors, preprint, January 5, 2019.

A. Weingartner, A sieve problem and its application, Mathematika, Vol. 63, No. 1 (2017), pp. 213-229.

LINKS

Table of n, a(n) for n=1..96.

MathOverflow question 510477, Ordering property of prime-valued gaps in the even-sopfr sequence - is this known?

Wikipedia, Arithmetic function.

FORMULA

a(n) = A036349(n+1) - A036349(n).

EXAMPLE

The first terms of A036349 are 1, 2, 4, 8, 9, 15, 16, 18, 21, 25, 30, so the first gaps are: 2-1=1, 4-2=2, 8-4=4, 9-8=1, 15-9=6, 16-15=1, 18-16=2, 21-18=3, 25-21=4, 30-25=5, giving a(1)=1, a(2)=2, a(3)=4, a(4)=1, a(5)=6, a(6)=1, a(7)=2, a(8)=3, a(9)=4, a(10)=5.

The first prime-valued terms in order of appearance are 2, 3, 5, 7, 13, 11, ..., showing that gap value 13 precedes gap value 11 -- the first exception to the otherwise ascending prime ordering.

MAPLE

sopfr := proc(n) local f, p, e; f := ifactors(n)[2]; add(p[1]*p[2], p = f); end proc: E := select(n -> sopfr(n) mod 2 = 0, [$1..300]): seq(E[n+1]-E[n], n=1..nops(E)-1);

MATHEMATICA

sopfr[1]=0; sopfr[n_]:=Total[Flatten[Table[#[[1]], {#[[2]]}]&/@FactorInteger[n]]]; e=Select[Range[300], EvenQ[sopfr[#]]&]; Differences[e]

PROG

(Python)

from sympy import factorint

def sopfr(n): # from A001414

return sum(p*e for p, e in factorint(n).items()) # Chai Wah Wu, Jan 08 2016

E = [n for n in range(1, 300) if sopfr(n) % 2 == 0]

print([E[i+1] - E[i] for i in range(len(E) - 1)])

CROSSREFS

First differences of A036349.

Cf. A001414 (sopfr).