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A090252 and A354169 are similar in many ways. This sequence and A355892 illustrate this.

This compressed format only make sense if all e_i are less than 10, that is, for n <= 24574.

It is believed that 6 does not appear in A090252, so 11 is missing from the present sequence.

Previous name: Numbers k such that there exists a solution to (p_m ^ a_m)*(p_m-1 ^ a_m-1)*...*(3^a_1)*(2^a_0) = (B^m)*a_m + (B^m-1)*a_m-1 + ... + (B^1)*a_1 + (B^0)*a_0 where k = (p_m ^ a_m)*(p_m-1 ^ a_m-1)*...*(3^a_1)*(2^a_0); a_m >= 1; a_(i= 0; p_0, p_1, ..., p_m are prime numbers; a_0, a_1, ..., a_m, B are integers.

B is the base in which we can express k as Sum_{i=0..m} B^i * a_i. B may also be seen as the variable in a polynomial, and k is then also an encoding of the polynomial (defined by the product of primes formula).

For k = (2^r)*3 we have B = (2^r)*3 - r.

A167221(n) is the smallest positive integer that yields a solution for k = a(n).

Negative B's can be obtained when the polynomial is an even function. This happens for instance when for k = 10, 100, 3125, ... - Michel Marcus, Aug 10 2022

From Peter Munn, Aug 13 2022: (Start)

Positive integers k such that k is a fixed point of a completely additive function f_B:N+ -> Z, B > 0, where f_B(prime(i+1)) = B^i for all i >= 0. Equivalently, since row B of A104244 is f_B, {a(n)} lists the columns of A104244 that contain their own column number.

If we require B to be negative instead, the sequence appears to start 10, 100, 3125, 1799875, 65610000, ... . Of these, 1799875 = 5^3 * 7 * 11^2 * 17 is the only k with only negative solutions (B = -11); the solutions for 65610000 are {4049, -4051}.

(End)

If p is the (k+1)-th prime and p is congruent to 1 modulo k, then p^p is a term with p^((p-1)/k) a solution for B. The list of such primes starts 3, 5, 7, 31, 97, 101, 331, ... . I suspect this list is infinite, meaning the greatest prime factor of the terms would be unbounded. - Peter Munn, Aug 15 2022

a(n) is even for n > 0.

We observe numbers of the decimal form (abcabc) = 360360, 720720 and numbers of the decimal form (abcabc0) = 1081080, 2162160, 5405400, 4324320, 6126120.

Observation and questions: many terms are products of powers of a contiguous set of the smallest primes. Many early terms of a(n) are in A002182; e.g., a(35) - A002182(44). The smallest exception outside of the empty product a(0) = 1 is a(22) = 491400 = 2^3 * 3^3 * 5^2 * 7 * 13. In other words, many terms have A006530(a(n)) < A053669(a(n)); a(22) is the smallest exception. Other exceptions include {471240, 1113840, 3341520, 2827440, 16576560, 28274400, ...}. A000720(A053669(a(22))) - A000720(A006530(a(22))) = 1, but the first instance of 2 for this function is a(35) = 16576560. This is evident by mapping A054841 across a(n). Are there a finite number of terms of a(n) that are also in A002182? Are there a finite number of terms of a(n) that have A006530(a(n)) < A053669(a(n)); are they becoming less frequent as n increases? - Michael De Vlieger, May 20 2017

In other words, a(n) is the least integer with exactly n divisors that are oblong (A002378). - Bernard Schott, Jul 30 2022

From Michael De Vlieger, Oct 06 2017: (Start)

Terms in A025487 that set records in this sequence: {1, 6, 30, 210, 420, 1260, 2310, 4620, 13860, 30030, 60060, 180180, 510510, 900900, 1021020, 3063060, 6126120, 9699690, ...}.

Conjecture: prime signatures corresponding to primorials A002110(i) with i > 1 set records in this sequence. (End)

B is the base in which we can express k as Sum_{i=0..m} B^i * a_i. There is an isomorphism between (Z[B],+) and the positive rationals as the polynomials with integer coefficients considered as a group under addition are isomorphic to the positive rationals considered as a group under multiplication.

For prime decomposition of A002182(n) = 2^a * 3^b * 5^c * ..., a(n) = "abc..." converted to a decimal number.

In other words, each "place" read from left to right represents the n-th prime, starting with 2 at left and increasing to the right. A number in the "place" represents the multiplicity of the corresponding prime in A002182(n).

This notation is corrupt when any multiplicity exceeds 9. The smallest instance of this is at n = 221.

Similar to A054841 but multiplicities are in reverse order.

Given that the exponents e(i) (a,b,c... in the above) of the prime factorization are (weakly) decreasing, their concatenation remains unambiguous way beyond n = 221 (first instance where e(1) >= 10) and even beyond n = 8869 (first instance where e(2) >= 10). Only when e2 >= 10 + e(3) for the first time, in principle the first digit of e(2) could be mistaken for the last digit of e(1); yet it is unlikely if not impossible that e(2) < 10 and e(1) > 100. So the first ambiguous decomposition would require concat(e(1),e(2),e(3)) = concat(a',b',c') with a' > b' >= e(3), thus e(2) significantly larger than 10 + e(3) and e(1) much larger than 100. - M. F. Hasler, Jan 03 2020

The conventional a(1) = 0 represents the empty concatenation.

Due to simple concatenation, this encoding of the positive integers becomes ambiguous from n = 613 = prime(112)^1 on, which has the same encoding a(n) = 1121 as 6 = prime(1)^1*prime(2)^1. To get a unique encoding, one could use, e.g., the digit 9 as delimiter to separate indices and exponents, written in base 9 as to use only digits 0..8, as soon as a term would be the duplicate of an earlier term (or for all n >= 613). Then one would have, e.g., a(613) = prime(134_9)^1 = 13491.

Sequence A067599 is based on the same idea, but uses the primes instead of their indices. In A037276 the prime factors are repeated, instead of giving the exponent. In A080670 exponents 1 are omitted. In A124010 only the prime signature is given. In A054841 the sum e_i*10^(i-1) is given, i.e., exponents are used as digits in base 10, while they are listed individually in the rows of A067255.