A082662 -id:A082662 - cp's OEIS frontend

A353154 Sum of the even-indexed terms in the n-th row of the triangle A196020.

0, 0, 1, 0, 3, 0, 5, 0, 7, 1, 9, 0, 11, 3, 13, 0, 15, 5, 17, 0, 20, 7, 21, 0, 23, 9, 28, 0, 27, 11, 29, 0, 36, 13, 33, 1, 35, 15, 44, 0, 39, 17, 41, 3, 52, 19, 45, 0, 47, 21, 60, 5, 51, 23, 54, 0, 68, 25, 57, 7, 59, 27, 76, 0, 66, 29, 65, 9, 84, 31, 69, 0, 71, 33, 97, 11, 75, 36, 77, 0

Offset: 1

Omar E. Pol, Apr 27 2022

nonn

Conjecture: indices of zeros give A082662.
a(n) is the total number of steps in all even-indexed double-staircases of the diagram of A196020 with n levels.
a(n) is also the total number of steps in all even-indexed double-staircases of the diagram described in A335616 with n levels that have at least one step in the bottom level of the diagram.
The graph of the sum-of-divisors function A000203 is intermediate between the graph of A353149 and the graph of this sequence (see the Links section). - Omar E. Pol, May 13 2022

			For n = 15 the 15th row of the triangle A196020 is [29, 13, 7, 0, 1]. The sum of the even-indexed terms is 13 + 0 = 13, so a(15) = 13.

OEIS Plot 2, A353154 vs A000203

Cf. A000203, A057641, A082662, A196020, A209246, A211343, A235791, A236104, A237591, A237593, A285901, A335616, A347186, A353149.

a(n) = A353149(n) - A000203(n).

a(n) = A209246(n) - A353149(n).

A378470 a(n) is the smallest number k for which the width pattern of the symmetric representation of sigma(k), SRS(k), consists of two unimodal parts of maximum width n.

Original entry on oeis.org

3, 78, 10728, 28920, 53752896, 4157280, 278628512256, 90323520, 1658908800, 21499810560, 7487812494923563008, 13005699840, 155267279705546496147456, 111451576596480, 8599694054400, 468208581120, 4172630516011611848266349543424, 5202323481600, 21630916595004029113587563614961664, 67421367982080

Offset: 1

Hartmut F. W. Hoft, Nov 27 2024

nonn

Let the nonincreasing multiset cL = { c_1, ... , c_s } be a factorization of n, let dL = { d_1, ... , d_s } be any set of s distinct odd primes, let q = dL^(cL - 1) = d_1^(c_1 - 1) * ... * d_s^(c_s - 1), and let k satisfy 2^k < q < 2^(k+1). Then SRS(2^k * q) is unimodal of maximum height n, 2^k * q has 2n odd divisors and its width pattern has 2n-1 entries. The smallest possible choice for 2^k * q is with the increasing sequence of odd primes d_i = p_(i+1), 1 <= i <= s. The overall smallest 2^k * q is the minimum among all factorizations of n. The smallest number m for which SRS(m) has two unimodal parts of maximum width n requires the additional prime factor r > 2^(k+1) * q which yields m = 2^k * q * r.

This sequence is column 2 in the array of A367377 and a(2) = A370206(1).

			a(2) = 78  is in the sequence since SRS(78) consists of two parts with width pattern 1 2 1 0 1 2 1 and 78 is the smallest number with those properties.
a(3) = 10728 = 2^3 * 3^2 * 149 is in the sequence since SRS(10728) consists of two parts with width pattern 1 2 3 2 1 0 1 2 3 2 1 and 10728 is the smallest number with those properties.
a(6) = 4157280 = 2^5 * 3^2 * 5 * 2887 is in the sequence. The two factorizations of 6 are {6} and {3, 2} so that with 3^5 = 243 and 3^2 * 5^1 = 45 the inequality 2^5 < 45 < 2^6 determines the single unimodular SRS(32 * 45) of maximum width 6, A250071(6) = 1440. Since 2887 is the smallest prime exceeding 2^6 * 3^2 * 5, 4157280 is the smallest number with SRS(4157280) consisting of two unimodular parts of maximum width 6.

Cf. A003056, A082662, A162247, A235791, A237270, A237591, A237593, A239929, A250071, A341969, A370206.

(* function f[ ] by T. D. Noe in A162247 *)
sF[n_] := Min[Map[Apply[Times, Prime[Range[2, Length[#]+1]]^#]&, Map[Reverse[#-1]&, f[n]]]]
f2U[n_] := Module[{s=sF[n], k, p}, k=Floor[Log[2, s]]; p=NextPrime[2^(k+1) s]; 2^k s p]
a378470[n_] := Map[f2U, Range[n]]
a378470[20]

a(p) = 2^k * 3^(p-1) * r, for odd primes p, with 2^k < 3^(p-1) < 2^(k+1) and r > 2^(k+1) * 3^(p-1) least prime, i.e., k = floor( (p-1)*(log_2 (3)) ) and r = prime( primepi(2^(k+1) * 3^(p-1)) + 1 ).

cp's OEIS Frontend

A353154 Sum of the even-indexed terms in the n-th row of the triangle A196020.

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