OFFSET
1,1
COMMENTS
For other known terms, see the Examples. These are derived using the expression f(x) given below; it is not known if other terms exist which are not values of f(x). It is conjectured that only numbers that are values of f(x) for some x are terms of this sequence.
The expression for f(x) was derived by noting the terms a(2), a(3) and a(4) appeared as terms in A221075, separated by 12 terms in that sequence. Using the formulas given there one can derive f(x). However, note that not all values of f(x) are terms of this sequence: f(7) = 10114032809617941274224 is not a term, as f(7)+2 has a prime factor 13^2, and the exponent 2 appears in f(7)+3. Likewise f(10) = 19951...359024 is not a term as f(10) itself contains 2 and 4 as prime exponents, and 2 appears in f(10)+1. Likewise, up to f(39), it can also be shown f(18), f(19), f(20), f(32), f(33), f(35), f(37), f(38) are not terms for similar reasons. Note that f(39) = 658951...618224 is a 132 digit number that is a term, so the formula is good at producing very large terms for many values of x. Note that if f(x) and f(x+1) are terms of the sequence then, for large x, f(x+1)/f(x) is approximately (2 + sqrt(3))^6 = 2701.999629903... .
Of note are the properties of f(x)+2: from the values studied it appears that f(x)+2 always produces a number k such that k-1 is a perfect square, while k+1 is a perfect square times a prime p, where p^3 also divides the number.
Of the terms produced by f(x), no group of 5 consecutive numbers satisfying the prime exponent criteria was found, other than f(1) = 24. See A388968.
The number 99612037019888, which lies between f(4) and f(5), is a term of this sequence and is a counterexample to the conjecture above that only numbers that are values of f(n) for some n are terms of this sequence. - Peter Bala, Sep 30 2025
Two positive integers x and y have the "Distinct Prime Exponents" (or DPE) property if their prime factorizations x = Product p_i^e_i and y = Product q_i^f_i are such that the exponent sets {e_i} and {f_i} are disjoint. The present sequence contains the numbers k such that k, k+1, k+2, k+3, ... is a maximal string of at least four consecutive numbers such that each successive pair has the DPE property. See A387531 for numbers k such that A387527(k) >= 3 (this includes nonmaximal k). - N. J. A. Sloane, Oct 03 2025
If k is a term, both numbers in one of the pairs (k, k+2), (k+1, k+2), or (k+1, k+3) are in A001694. - Pontus von Brömssen, Oct 06 2025
FORMULA
The formula
f(x) = 1/2*((2 + sqrt(3))^(6*x-3) + (2 - sqrt(3))^(6*x-3)) - 2, for integer x >= 1,
produces terms for x = 1..6, 8, 9, 11..17, 21..31, 34, 36, 39, and likely for infinitely more values of x. For the terms studied all values produced by the formula end in 24.
EXAMPLE
7 is a term as 7 = 7^1, 8 = 2^3, 9 = 2^2, 10 = 2^1*5^1, and the prime exponents of 8, 9 and 10 do not equal any of those of 7, 8 and 9 respectively.
24 is a term as 24 = 2^3*3^1, 25 = 5^2, 26 = 2^1*13^1, 27 = 3^3, 28 = 2^2*7^1, and the prime exponents of 25, 26, 27 and 28 do not equal any of those of 24, 25, 26 and 27 respectively.
70224 is a term as 70224 = 2^4*3^1*7^1*11^1*19*1, 70225 = 5^2*53^2, 70226 = 2^1*13^1*37^1*73^1, 70227 = 3^5*17^2, and the prime exponents of 70225, 70226 and 70227 do not equal any of those of 70224, 70225 and 70226 respectively.
Other known terms obtained from f(x) are 512706121224, 1385331749802024, 3743165875258953024, 27328112908421802064005624, 73840550964522899559001927224, and 539095242162880600113350770354161024.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Scott R. Shannon, Sep 22 2025
EXTENSIONS
a(5)-a(8) confirmed by Pontus von Brömssen, Oct 06 2025
Grammatical improvements from N. J. A. Sloane, Oct 07 2025
STATUS
approved