Vincenzo Manto is the author of 32 sequences in the OEIS. His work has contributed significantly to the field of integer sequences, ranging from "Numbers m such that rad(m) * bigomega(m) = m." to "Primes prime(k) such that prime(k) - prime(k-1) is a factorial.".
His most prolific partners include James C. McMahon, Michael De Vlieger, James C. McMahon, James C. McMahon, Michael De Vlieger.
His contributions have been cited in 35 other sequences, showcasing the impact of his work on the OEIS community, making him a 92-percentile contributor
Numbers m such that rad(m) * bigomega(m) = m.
- Primes p are terms since rad(p) * bigomega(p) = p * 1 = p.
- From _Michael De Vlieger_, Jul 03 2026: (Start)
- Primes are the only squarefree numbers (in A005117) in the sequence. Squarefree k = rad(k) such that bigomega(k) = omega(k) > 2 do not appear in the sequence since k * omega(k) > k.
- The only powerful number k (in A001694) in this sequence is 4 = 2^2 = 2*2, since powerful numbers k imply rad(k)^2 | k, and further, bigomega(k) >= k/rad(k), where k/rad(k) >= rad(k). Attempting to increment the left hand side only increases the ratio RHS/LHS, since RHS increases by a prime factor.
- Consequences:
- 1. A175787 is a proper subset of this sequence.
- 2. There is no intersection of this sequence and A120944 (squarefree and composite).
- 3. {a(n)} \ A175787 is a proper subset of A332785.
- 4. 4 is the only perfect power (in A001597) and the only prime power (in A246547) in this sequence. (End)
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Numbers m such that rad(m) * omega(m) = m.
- Primes p are terms since rad(p) * omega(p) = p * 1 = p.
- Prime powers p^k (k > 1) are never terms since p * 1 != p^k.
- Composite terms classified by r = omega(m):
- - r = 2: m = 4*p for odd primes p (e.g., 12, 20, 28, 44).
- - r = 3: m = 9*p*q for distinct primes p, q != 3 (e.g., 90).
- - r = 4: m = 8*p*q*s for distinct odd primes p, q, s (e.g., 840).
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Primes prime(k) such that the sum of the k primes beginning with prime(k) is prime.
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Numbers k such that the sum of the k primes beginning with prime(k) is prime.
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Primes equal to the sum of the k primes beginning with prime(k) for some k.
- Subsequence of A161463.
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Numbers k that are neither squarefree nor powerful such that rad(k)*omega(k) = k, where rad = A007947 and omega = A001221.
- Composite terms in A397221.
- Intersection of A332785 and A397221.
- The smallest term in A361098 is a(3189) = 50050 = 2 * 5^2 * 7 * 11 * 13. This is to say that k = 50050 is such that k/rad(k) > A053669(k), k/rad(k) >= A119288(k), and k = rad(k) * omega(k) are all true.
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Numbers k that are neither squarefree nor powerful such that rad(k)*bigomega(k) = k, where rad = A007947 and bigomega = A001222.
- A396594 is the union of this sequence and A175787 (the primes and 4).
- Intersection of A332785 and A396594.
- A397658 \ {2, 4} is a proper subset.
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Numbers whose number of nonzero digits is identical in base 2 and 3.
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Smallest number k > n such that the binary Hamming distance between n and k is equal to the number of runs in the binary representation of n, and k has more runs than n.
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a(n) is the integer obtained by inserting the sum (modulo 10) of adjacent digits of n between them.
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Numbers k*P(j) such that bigomega(k) = k - j, where P = A002110 and bigomega = A001222.
- Intersection of A060735 and A396594.
- Numbers k in A060735 such that rad(k)*bigomega(k) = k, where rad = A007947 and bigomega = A001222.
- Apart from 2 and 4, this sequence is a subset of A332785 (neither squarefree nor powerful).
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a(n) = nearest prime to prime(n)^(3/2).
- There is no solution for p^3 = q^2 when p and q are distinct prime numbers.
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a(n) = prime^2 closest to prime(n)^3.
- a(n) = 1 (mod 24) for any n >= 2 because the nearest prime q is always >= 5 and the square of any prime q >= 5 is of the form 24k + 1.
- Empirically, the closest prime square to prime(n)^3 is always unique. It's conjectured that no tie occurs.
- The distance a(n) - prime(n)^3 is asymptotically bounded by O(p^1.5 * log(p)).
- A tie cannot occur: Suppose that p^2 + q^2 = 2*r^3 for p, q, r >= 5 primes, then r == 1 (mod 24), and so r can be written as a+b*i with a > b > 0. It is clear that gcd(p+q*i, p-q*i) = 1+i in the ring of Gaussian integers, so one of p+q*i and p-q*i must be associated to (a+b*i)^3*(1+i) and the other to (a-b*i)^3*(1+i). By comparing the real and imaginary parts, one of p and q is equal to |(a+b)*(a^2-4*a*b+b^2)| and the other to |(a-b)*(a^2+4*a*b+b^2)|. This implies that we have both a^2 - 4*a*b + b^2 = +-1 and a - b = 1, which is impossible. - _Jianing Song_, Jul 13 2026
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Numbers k such that k is coprime to each of its digits and to the sum of its digits.
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Primes prime(k) such that prime(k+1) - prime(k) is a perfect power.
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Maximum number of runs in the binary expansion of integers obtained by changing a single 0-bit to 1 in the binary representation of n considering bits up to and including the first leading 0.
- The bit-flip is allowed on any 0-bit within the standard binary representation of n, or on the first implicit leading zero (at position floor(log_2(n)) + 1). This identifies the maximum disruption to the run-length encoding (A005811) caused by a single-bit increment at Hamming distance 1. Therefore, this sequence provides a measure of how a single-bit change at Hamming distance 1 can disrupt the compression efficiency (RLE) of the binary representation of n.
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Numbers whose number of nonzero digits is identical in base 2 and 5.
- Numbers k such that A000120(k) = A276134(k).
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a(n) is the integer obtained by inserting the product (modulo 10) of adjacent digits of n between them.
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Sum of the cumulative number of previous occurrences of the digits of n in the sequence 1..n excluding the current occurrence of each digit.
- This sequence is the exclusive counterpart to A343644. While A343644 counts the occurrences of digits in the range [1,n], a(n) counts only the occurrences strictly preceding each digit of n. Formally a(n) = A343644(n) - A055642(n). Thus, this sequence represents the exclusive scan of digit occurrences, whereas A343644 is the inclusive scan. a(n) = 0 for all single-digit n as it measures the cumulative repetition of digits at the exact moment n is formed.
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Strictly increasing minimal sequence where consecutive terms can be added digit-by-digit without carrying in base 10.
- Many integers (e.g., 6, 7, 8, 9, 15, 16...) are never present because the greedy behavior and the strictly increasing condition bypass them to avoid carries.
- A subset of nonnegative integers with no decimal digits > 5 (A007092).
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Composites k not of the form 8*p or 9*q, odd prime p, prime q != 3, such that rad(k)*bigomega(k) = k, with rad = A007947 and bigomega = A001222.
- Numbers of the form 8*p, p = prime(i), i > 1, are such that rad(8*p)*bigomega(8*p) = (2*p)*4 = 8*p.
- Numbers of the form 9*q, q = prime(j), j != 2, are such that rad(9*q)*bigomega(9*q) = (3*q)*3 = 9*q.
- Composite numbers k such that k/rad(k) = bigomega(k), that are also not of the above 2 forms, appear in this sequence.
- The number 4 is the only powerful number in this sequence. Without 4, this sequence is a proper subset of A332785.
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Primes prime(k) such that prime(k) - prime(k-1) is a Fibonacci number.
- Fibonacci numbers grow exponentially while the average gap between consecutive primes grows logarithmically, thus this sequence's density decreases as k increases.
- It is conjectured that this sequence is infinite as a consequence of the Polignac's conjecture.
- Infinite if the Twin Prime conjecture holds. - _Michael S. Branicky_, May 24 2026
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Primes p such that the sum of the distinct prime factors of p-1 is prime.
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Numbers m with exactly two distinct prime factors q and p such that m = q^k * p^j and p > q^k with k, j >= 1.
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Numbers m whose squarefree core equals the number of duplicated prime factors.
- Numbers m such that bigomega(m) - omega(m) = core(m).
- In other words, numbers m such that A001222(m) - A001221(m) = A046660(m) = A007913(m).
- Squarefree numbers are naturally excluded since for any squarefree number m >= 1, bigomega(m) - omega(m) = 0, while core(m) = m >= 1.
- The difference bigomega(m) - omega(m) is sometimes called the prime factor multiplicity index of m (A046660).
- The set of prime squares A001248 is a proper subset since A046660(p^2) = 1 and core(p^2) = 1.
- Of the 56250 terms m <= 2^40 in A286708 (powerful but not a prime power), 56220 are Achilles numbers and 30 are in A386762 (perfect powers of numbers neither squarefree nor powerful). Most of the terms in the intersection with A386762 are cubes except 380204032 = 2^10 * 13^5 and 21924480357 = 3^10 * 13^5.
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Numbers k such that the sums of the digits of k^2, k^3 and k^4 coincide, excluding multiples of 10.
- While the sequence of numbers k such that the sums of the digits of k^2, k^3 and k^4 coincide is infinite, since it is always possible to append zeros to obtain a new term, it is not known whether there are an infinite number of terms without trailing zeros (when excluding multiples of 10, A008592).
- For any term k, k^2 == k^3 == k^4 (mod 9), which implies that k mod 9 must belong to {0, 1, 3, 6}.
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Numbers m such that rad(m)^omega(m) = m.
- A term m > 1 is in the sequence if and only if the exponent of every prime factor in its prime factorization is exactly equal to omega(m). Thus, m must be of the form (p_1 * p_2 * ... * p_k)^k for distinct primes p_i.
- Terms are mostly primes: for any p, omega(p) = 1 and rad(p) = p, so rad(p)^omega(p) = p ^ 1.
- Except for 1 and the prime numbers (A000040), all terms are perfect powers.
- Composite terms are the k-th powers of squarefree numbers having exactly k prime factors. For k = 2, these are the squares of squarefree semiprimes.
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Numbers such that each digit d_i is equal to the number of digits to its right that are strictly less than d_i.
- For a number with L digits d_1, d_2, ..., d_L, the rule is d_i = #{j > i : d_j < d_i}.
- The last digit d_L is always 0 because there are no digits to its right.
- The first digit d_1 is always between 1 and L-1.
- Therefore, a number of length L is in the sequence if its digits d_i (for i=0..L-1) satisfy d_0 = L-1, d_{L-1} = 0, and d_i is either 0 or L-1-i for 0 < i < L-1.
- Thus, each digit describes the 'downward slope' of the remaining digits to its right.
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Primes p such that the Fermat quotient q = (2^(p-1) - 1)/p mod p satisfies 1 < q < p and q divides p - 1.
- All known Mersenne primes (A000668) > 3 are terms of this sequence.
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